Cubic critical, narrow range

Percentage Accurate: 55.4% → 99.5%

Time: 17.9s

Alternatives: 10

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: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / ((0.0d0 - b) - sqrt((c * ((a * (-3.0d0)) + ((b * b) / c)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

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

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

6. Applied egg-rr 57.4%

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

7. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

   3. --lowering--.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(0 - c\right)} \cdot \frac{1}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}} \]
10. Step-by-step derivation

    1. un-div-inv N/A

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

    2. sub0-neg N/A

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

    3. distribute-frac-neg N/A

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

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

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

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

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

    6. associate-*r* N/A

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

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

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

    8. associate-*r* N/A

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

    9. rem-square-sqrt N/A

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

    10. associate-*r* N/A

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

    11. associate-*r* N/A

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in c around inf

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

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

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

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

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

    3. *-commutative N/A

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

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

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

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

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

    6. unpow2 N/A

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

    7. *-lowering-*.f64 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

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

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

6. Applied egg-rr 57.4%

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

7. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

   3. --lowering--.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(0 - c\right)} \cdot \frac{1}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}} \]
10. Step-by-step derivation

    1. un-div-inv N/A

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

    2. sub0-neg N/A

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

    3. distribute-frac-neg N/A

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

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

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

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

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

    6. associate-*r* N/A

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

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

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

    8. associate-*r* N/A

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

    9. rem-square-sqrt N/A

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

    10. associate-*r* N/A

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

    11. associate-*r* N/A

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in a around inf

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

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

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

    2. +-commutative N/A

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

    3. rem-square-sqrt N/A

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

    4. unpow2 N/A

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

    5. *-commutative N/A

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

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

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

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

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

    8. unpow2 N/A

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

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

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

    10. unpow2 N/A

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

    11. rem-square-sqrt N/A

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

    12. *-lowering-*.f64 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / ((0.0d0 - b) - sqrt(((b * b) + ((-3.0d0) * (c * a)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

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

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

6. Applied egg-rr 57.4%

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

7. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

   3. --lowering--.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(0 - c\right)} \cdot \frac{1}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}} \]
10. Step-by-step derivation

    1. un-div-inv N/A

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

    2. sub0-neg N/A

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

    3. distribute-frac-neg N/A

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

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

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

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

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

    6. associate-*r* N/A

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

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

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

    8. associate-*r* N/A

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

    9. rem-square-sqrt N/A

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

    10. associate-*r* N/A

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

    11. associate-*r* N/A

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

11. Applied egg-rr 99.6%

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

    1. associate-*r* N/A

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

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

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

    3. *-lowering-*.f64 99.6%

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

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

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

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

6. Applied egg-rr 57.4%

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

7. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

   3. --lowering--.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(0 - c\right)} \cdot \frac{1}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}} \]
10. Step-by-step derivation

    1. un-div-inv N/A

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

    2. sub0-neg N/A

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

    3. distribute-frac-neg N/A

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

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

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

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

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

    6. associate-*r* N/A

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

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

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

    8. associate-*r* N/A

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

    9. rem-square-sqrt N/A

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

    10. associate-*r* N/A

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

    11. associate-*r* N/A

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 5: 88.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{c}{\left(a \cdot \left(\frac{c}{b} \cdot \left(0 - -1.5\right) - \frac{-1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right) - b\right) - b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  c
  (-
   (-
    (*
     a
     (- (* (/ c b) (- 0.0 -1.5)) (/ (* -1.125 (* a (* c c))) (* b (* b b)))))
    b)
   b)))

C

double code(double a, double b, double c) {
	return c / (((a * (((c / b) * (0.0 - -1.5)) - ((-1.125 * (a * (c * c))) / (b * (b * 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 * (((c / b) * (0.0d0 - (-1.5d0))) - (((-1.125d0) * (a * (c * c))) / (b * (b * b))))) - b) - b)
end function

Java

public static double code(double a, double b, double c) {
	return c / (((a * (((c / b) * (0.0 - -1.5)) - ((-1.125 * (a * (c * c))) / (b * (b * b))))) - b) - b);
}

Python

def code(a, b, c):
	return c / (((a * (((c / b) * (0.0 - -1.5)) - ((-1.125 * (a * (c * c))) / (b * (b * b))))) - b) - b)

Julia

function code(a, b, c)
	return Float64(c / Float64(Float64(Float64(a * Float64(Float64(Float64(c / b) * Float64(0.0 - -1.5)) - Float64(Float64(-1.125 * Float64(a * Float64(c * c))) / Float64(b * Float64(b * b))))) - b) - b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / (((a * (((c / b) * (0.0 - -1.5)) - ((-1.125 * (a * (c * c))) / (b * (b * b))))) - b) - b);
end

Wolfram

code[a_, b_, c_] := N[(c / N[(N[(N[(a * N[(N[(N[(c / b), $MachinePrecision] * N[(0.0 - -1.5), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.125 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

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

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

6. Applied egg-rr 57.4%

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

7. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

   3. --lowering--.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(0 - c\right)} \cdot \frac{1}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}} \]
10. Step-by-step derivation

    1. un-div-inv N/A

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

    2. sub0-neg N/A

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

    3. distribute-frac-neg N/A

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

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

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

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

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

    6. associate-*r* N/A

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

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

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

    8. associate-*r* N/A

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

    9. rem-square-sqrt N/A

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

    10. associate-*r* N/A

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

    11. associate-*r* N/A

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in a around 0

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

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

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

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

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

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

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

    4. *-commutative N/A

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

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

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

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

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

    7. associate-*r/ N/A

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

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

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

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

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

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

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

    11. unpow2 N/A

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

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

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

    13. cube-mult N/A

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

    14. unpow2 N/A

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

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

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

    16. unpow2 N/A

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

    17. *-lowering-*.f64 87.5%

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

14. Simplified 87.5%

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

15. Final simplification 87.5%

    \[\leadsto \frac{c}{\left(a \cdot \left(\frac{c}{b} \cdot \left(0 - -1.5\right) - \frac{-1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right) - b\right) - b} \]
16. Add Preprocessing

Alternative 6: 88.3% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

   1. clear-num N/A

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

   2. associate-/l* N/A

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

   3. associate-/r* N/A

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

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

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

   5. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 55.8%

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

6. Applied egg-rr 55.8%

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

7. Taylor expanded in a around 0

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

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

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

   2. associate-*r/ N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

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

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

   12. mul-1-neg N/A

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

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

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

   14. distribute-rgt-out N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

9. Simplified 87.2%

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

10. Final simplification 87.2%

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

Alternative 7: 82.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{\left(\frac{c \cdot a}{b} \cdot \left(0 - -1.5\right) - b\right) - b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ c (- (- (* (/ (* c a) b) (- 0.0 -1.5)) b) b)))

C

double code(double a, double b, double c) {
	return c / (((((c * a) / b) * (0.0 - -1.5)) - 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 / (((((c * a) / b) * (0.0d0 - (-1.5d0))) - b) - b)
end function

Java

public static double code(double a, double b, double c) {
	return c / (((((c * a) / b) * (0.0 - -1.5)) - b) - b);
}

Python

def code(a, b, c):
	return c / (((((c * a) / b) * (0.0 - -1.5)) - b) - b)

Julia

function code(a, b, c)
	return Float64(c / Float64(Float64(Float64(Float64(Float64(c * a) / b) * Float64(0.0 - -1.5)) - b) - b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / (((((c * a) / b) * (0.0 - -1.5)) - b) - b);
end

Wolfram

code[a_, b_, c_] := N[(c / N[(N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * N[(0.0 - -1.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

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

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

6. Applied egg-rr 57.4%

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

7. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(0 - c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

   3. --lowering--.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, c\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(0 - c\right)} \cdot \frac{1}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}} \]
10. Step-by-step derivation

    1. un-div-inv N/A

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

    2. sub0-neg N/A

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

    3. distribute-frac-neg N/A

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

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

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

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

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

    6. associate-*r* N/A

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

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

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

    8. associate-*r* N/A

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

    9. rem-square-sqrt N/A

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

    10. associate-*r* N/A

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

    11. associate-*r* N/A

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in c around 0

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

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

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

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

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

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

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

    4. *-commutative N/A

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

    5. *-lowering-*.f64 81.8%

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

14. Simplified 81.8%

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

15. Final simplification 81.8%

    \[\leadsto \frac{c}{\left(\frac{c \cdot a}{b} \cdot \left(0 - -1.5\right) - b\right) - b} \]
16. Add Preprocessing

Alternative 8: 82.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\frac{b \cdot -0.6666666666666666}{c} + \frac{a \cdot 0.5}{b}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

   1. clear-num N/A

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

   2. associate-/l* N/A

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

   3. associate-/r* N/A

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

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

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

   5. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 55.8%

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

6. Applied egg-rr 55.8%

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

7. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

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

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

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

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

   4. associate-*r/ N/A

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

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

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

   12. *-lowering-*.f64 81.6%

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

9. Simplified 81.6%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{b \cdot -0.6666666666666666}{c} + \frac{0.5 \cdot a}{b}}} \]

10. Final simplification 81.6%

    \[\leadsto \frac{0.3333333333333333}{\frac{b \cdot -0.6666666666666666}{c} + \frac{a \cdot 0.5}{b}} \]
11. Add Preprocessing

Alternative 9: 64.5% 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 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 64.3%

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

7. Simplified 64.3%

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

Alternative 10: 64.4% 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 55.8%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 64.3%

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

7. Simplified 64.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f64 64.3%

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

9. Applied egg-rr 64.3%

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

10. Final simplification 64.3%

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

Reproduce

herbie shell --seed 2024163 
(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)))