Cubic critical, wide range

Percentage Accurate: 18.0% → 99.3%

Time: 16.1s

Alternatives: 6

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 18.0% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{a \cdot c} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot 3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   (/ a (* a c))
   (-
    (- b)
    (sqrt
     (* (fma (sqrt a) (sqrt (* c 3.0)) b) (- b (sqrt (* a (* c 3.0))))))))))

C

double code(double a, double b, double c) {
	return 1.0 / ((a / (a * c)) * (-b - sqrt((fma(sqrt(a), sqrt((c * 3.0)), b) * (b - sqrt((a * (c * 3.0))))))));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / Float64(a * c)) * Float64(Float64(-b) - sqrt(Float64(fma(sqrt(a), sqrt(Float64(c * 3.0)), b) * Float64(b - sqrt(Float64(a * Float64(c * 3.0)))))))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(a / N[(a * c), $MachinePrecision]), $MachinePrecision] * N[((-b) - N[Sqrt[N[(N[(N[Sqrt[a], $MachinePrecision] * N[Sqrt[N[(c * 3.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

   1. add-sqr-sqrt 19.1%

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

   2. difference-of-squares 19.2%

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

   3. associate-*l* 19.2%

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

   4. associate-*l* 19.2%

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

4. Applied egg-rr 19.2%

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

   1. associate-*r* 19.2%

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

   2. *-commutative 19.2%

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

   3. associate-*r* 19.2%

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

   4. *-commutative 19.2%

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

6. Simplified 19.2%

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

   1. flip-+ 19.1%

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

8. Applied egg-rr 19.4%

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

   1. unpow2 19.4%

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

   2. sqr-neg 19.4%

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

   3. unpow2 19.4%

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

   4. associate-*r* 19.4%

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

   5. *-commutative 19.4%

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

   6. associate-*r* 19.4%

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

   7. associate-*r* 19.4%

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

   8. *-commutative 19.4%

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

   9. associate-*r* 19.4%

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

10. Simplified 19.4%

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

11. Taylor expanded in b around 0 98.6%

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

    1. +-commutative 98.6%

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

    2. fma-define 98.6%

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

    3. unpow2 98.6%

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

    4. rem-square-sqrt 99.3%

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

    5. *-commutative 99.3%

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

    6. distribute-lft1-in 99.3%

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

    7. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \left(\left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right) \cdot b\right)\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    8. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \left(\color{blue}{0} \cdot b\right)\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    9. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \color{blue}{0}\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    10. metadata-eval 99.3%

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

13. Simplified 99.3%

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

    1. clear-num 99.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{\mathsf{fma}\left(a, c \cdot 3, 0\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}}} \]

    2. inv-pow 99.1%

       \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\frac{\mathsf{fma}\left(a, c \cdot 3, 0\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}\right)}^{-1}} \]

    3. *-commutative 99.1%

       \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\frac{\mathsf{fma}\left(a, c \cdot 3, 0\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}\right)}^{-1} \]

    4. +-commutative 99.1%

       \[\leadsto {\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(a, c \cdot 3, 0\right)}{\left(-b\right) - \sqrt{\color{blue}{\left(\sqrt{a \cdot \left(c \cdot 3\right)} + b\right)} \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}\right)}^{-1} \]

    5. sqrt-prod 99.1%

       \[\leadsto {\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(a, c \cdot 3, 0\right)}{\left(-b\right) - \sqrt{\left(\color{blue}{\sqrt{a} \cdot \sqrt{c \cdot 3}} + b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}\right)}^{-1} \]

    6. fma-define 99.1%

       \[\leadsto {\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(a, c \cdot 3, 0\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot 3}, b\right)} \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}\right)}^{-1} \]

15. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\frac{\mathsf{fma}\left(a, c \cdot 3, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot 3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}\right)}^{-1}} \]
16. Step-by-step derivation

    1. unpow-1 99.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\frac{\mathsf{fma}\left(a, c \cdot 3, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot 3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}}} \]

    2. associate-/r/ 99.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 3}{\mathsf{fma}\left(a, c \cdot 3, 0\right)} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot 3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}} \]

    3. *-commutative 99.1%

       \[\leadsto \frac{1}{\frac{\color{blue}{3 \cdot a}}{\mathsf{fma}\left(a, c \cdot 3, 0\right)} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot 3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)} \]

    4. fma-define 99.1%

       \[\leadsto \frac{1}{\frac{3 \cdot a}{\color{blue}{a \cdot \left(c \cdot 3\right) + 0}} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot 3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)} \]

    5. +-rgt-identity 99.1%

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

    6. associate-*r* 99.1%

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

    7. *-commutative 99.1%

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

    8. times-frac 99.3%

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

    9. metadata-eval 99.3%

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

    10. *-commutative 99.3%

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

17. Simplified 99.3%

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

18. Final simplification 99.3%

    \[\leadsto \frac{1}{\frac{a}{a \cdot c} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot 3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)} \]
19. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.1%

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

   1. add-sqr-sqrt 19.1%

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

   2. difference-of-squares 19.2%

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

   3. associate-*l* 19.2%

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

   4. associate-*l* 19.2%

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

4. Applied egg-rr 19.2%

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

   1. associate-*r* 19.2%

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

   2. *-commutative 19.2%

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

   3. associate-*r* 19.2%

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

   4. *-commutative 19.2%

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

6. Simplified 19.2%

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

   1. flip-+ 19.1%

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

8. Applied egg-rr 19.4%

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

   1. unpow2 19.4%

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

   2. sqr-neg 19.4%

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

   3. unpow2 19.4%

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

   4. associate-*r* 19.4%

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

   5. *-commutative 19.4%

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

   6. associate-*r* 19.4%

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

   7. associate-*r* 19.4%

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

   8. *-commutative 19.4%

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

   9. associate-*r* 19.4%

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

10. Simplified 19.4%

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

11. Taylor expanded in b around 0 98.6%

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

    1. +-commutative 98.6%

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

    2. fma-define 98.6%

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

    3. unpow2 98.6%

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

    4. rem-square-sqrt 99.3%

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

    5. *-commutative 99.3%

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

    6. distribute-lft1-in 99.3%

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

    7. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \left(\left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right) \cdot b\right)\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    8. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \left(\color{blue}{0} \cdot b\right)\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    9. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \color{blue}{0}\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    10. metadata-eval 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in a around 0 99.2%

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

15. Final simplification 99.2%

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

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ t_1 := \sqrt{t\_0}\\ \frac{\frac{t\_0}{\left(-b\right) - \sqrt{\left(b - t\_1\right) \cdot \left(b + t\_1\right)}}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))) (t_1 (sqrt t_0)))
   (/ (/ t_0 (- (- b) (sqrt (* (- b t_1) (+ b t_1))))) (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double t_1 = sqrt(t_0);
	return (t_0 / (-b - sqrt(((b - t_1) * (b + t_1))))) / (a * 3.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    t_0 = a * (c * 3.0d0)
    t_1 = sqrt(t_0)
    code = (t_0 / (-b - sqrt(((b - t_1) * (b + t_1))))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double t_1 = Math.sqrt(t_0);
	return (t_0 / (-b - Math.sqrt(((b - t_1) * (b + t_1))))) / (a * 3.0);
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	t_1 = math.sqrt(t_0)
	return (t_0 / (-b - math.sqrt(((b - t_1) * (b + t_1))))) / (a * 3.0)

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	t_1 = sqrt(t_0)
	return Float64(Float64(t_0 / Float64(Float64(-b) - sqrt(Float64(Float64(b - t_1) * Float64(b + t_1))))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * 3.0);
	t_1 = sqrt(t_0);
	tmp = (t_0 / (-b - sqrt(((b - t_1) * (b + t_1))))) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(t$95$0 / N[((-b) - N[Sqrt[N[(N[(b - t$95$1), $MachinePrecision] * N[(b + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 19.1%

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

   1. add-sqr-sqrt 19.1%

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

   2. difference-of-squares 19.2%

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

   3. associate-*l* 19.2%

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

   4. associate-*l* 19.2%

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

4. Applied egg-rr 19.2%

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

   1. associate-*r* 19.2%

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

   2. *-commutative 19.2%

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

   3. associate-*r* 19.2%

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

   4. *-commutative 19.2%

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

6. Simplified 19.2%

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

   1. flip-+ 19.1%

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

8. Applied egg-rr 19.4%

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

   1. unpow2 19.4%

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

   2. sqr-neg 19.4%

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

   3. unpow2 19.4%

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

   4. associate-*r* 19.4%

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

   5. *-commutative 19.4%

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

   6. associate-*r* 19.4%

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

   7. associate-*r* 19.4%

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

   8. *-commutative 19.4%

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

   9. associate-*r* 19.4%

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

10. Simplified 19.4%

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

11. Taylor expanded in b around 0 98.6%

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

    1. +-commutative 98.6%

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

    2. fma-define 98.6%

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

    3. unpow2 98.6%

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

    4. rem-square-sqrt 99.3%

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

    5. *-commutative 99.3%

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

    6. distribute-lft1-in 99.3%

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

    7. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \left(\left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right) \cdot b\right)\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    8. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \left(\color{blue}{0} \cdot b\right)\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    9. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(a, c \cdot 3, -1 \cdot \color{blue}{0}\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    10. metadata-eval 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in a around 0 99.2%

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

    1. *-commutative 99.2%

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

    2. associate-*r* 99.3%

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

16. Simplified 99.3%

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

17. Final simplification 99.3%

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

Alternative 4: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 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 = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in b around inf 95.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Final simplification 95.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
5. Add Preprocessing

Alternative 5: 90.3% 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 19.1%

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

3. Taylor expanded in b around inf 89.7%

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

   1. *-commutative 89.7%

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

   2. associate-*l/ 89.7%

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

5. Simplified 89.7%

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

6. Final simplification 89.7%

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

Alternative 6: 3.3% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.1%

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

   1. add-sqr-sqrt 19.1%

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

   2. difference-of-squares 19.2%

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

   3. associate-*l* 19.2%

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

   4. associate-*l* 19.2%

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

4. Applied egg-rr 19.2%

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

   1. associate-*r* 19.2%

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

   2. *-commutative 19.2%

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

   3. associate-*r* 19.2%

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

   4. *-commutative 19.2%

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

6. Simplified 19.2%

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

7. Taylor expanded in b around inf 3.3%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.3%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.3%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.3%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.3%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.3%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Simplified 3.3%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

10. Final simplification 3.3%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))