Cubic critical, narrow range

Percentage Accurate: 55.1% → 91.1%

Time: 18.1s

Alternatives: 11

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}} \cdot -0.375\\ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -3 \cdot \left(t\_0 + a \cdot \mathsf{fma}\left(-0.75, c \cdot \frac{t\_0}{{b}^{2}}, \mathsf{fma}\left(-0.2222222222222222, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{c}^{2}}, 0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right), \frac{1.5}{b}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ c (pow b 3.0)) -0.375)))
   (/
    1.0
    (fma
     -2.0
     (/ b c)
     (*
      a
      (fma
       a
       (*
        -3.0
        (+
         t_0
         (*
          a
          (fma
           -0.75
           (* c (/ t_0 (pow b 2.0)))
           (fma
            -0.2222222222222222
            (* b (/ (* (/ (pow c 4.0) (pow b 6.0)) 6.328125) (pow c 2.0)))
            (* 0.5625 (/ (pow c 2.0) (pow b 5.0))))))))
       (/ 1.5 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / pow(b, 3.0)) * -0.375;
	return 1.0 / fma(-2.0, (b / c), (a * fma(a, (-3.0 * (t_0 + (a * fma(-0.75, (c * (t_0 / pow(b, 2.0))), fma(-0.2222222222222222, (b * (((pow(c, 4.0) / pow(b, 6.0)) * 6.328125) / pow(c, 2.0))), (0.5625 * (pow(c, 2.0) / pow(b, 5.0)))))))), (1.5 / b))));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / (b ^ 3.0)) * -0.375)
	return Float64(1.0 / fma(-2.0, Float64(b / c), Float64(a * fma(a, Float64(-3.0 * Float64(t_0 + Float64(a * fma(-0.75, Float64(c * Float64(t_0 / (b ^ 2.0))), fma(-0.2222222222222222, Float64(b * Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125) / (c ^ 2.0))), Float64(0.5625 * Float64((c ^ 2.0) / (b ^ 5.0)))))))), Float64(1.5 / b)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]}, N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(a * N[(a * N[(-3.0 * N[(t$95$0 + N[(a * N[(-0.75 * N[(c * N[(t$95$0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.2222222222222222 * N[(b * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5625 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.2%

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

   1. sqr-neg 57.2%

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

   2. sqr-neg 57.2%

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

   3. associate-*l* 57.2%

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

3. Simplified 57.2%

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

   1. pow1 57.2%

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

   2. *-commutative 57.2%

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

   3. *-commutative 57.2%

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

   4. associate-*l* 57.2%

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

   5. *-commutative 57.2%

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

6. Applied egg-rr 57.2%

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

   1. unpow1 57.2%

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

   2. associate-*r* 57.2%

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

8. Simplified 57.2%

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

   1. +-commutative 57.2%

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

   2. add-cube-cbrt 53.6%

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

   3. fma-define 53.8%

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

   4. cbrt-prod 55.1%

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

   5. add-sqr-sqrt 55.1%

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

   6. pow2 55.1%

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

   7. associate-*l* 55.1%

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

   8. pow2 55.1%

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

   9. associate-*l* 55.1%

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

10. Applied egg-rr 55.1%

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

    1. clear-num 55.1%

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

    2. inv-pow 55.1%

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

12. Applied egg-rr 57.3%

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

    1. unpow-1 57.3%

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

    2. associate-/l* 57.3%

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

    3. unsub-neg 57.3%

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

    4. unpow2 57.3%

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

    5. fma-neg 57.4%

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

    6. associate-*l* 57.4%

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

    7. *-commutative 57.4%

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

    8. distribute-rgt-neg-in 57.4%

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

    9. *-commutative 57.4%

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

    10. distribute-lft-neg-in 57.4%

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

    11. metadata-eval 57.4%

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

    12. *-commutative 57.4%

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

14. Simplified 57.4%

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

15. Taylor expanded in a around 0 92.3%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-3 \cdot \left(a \cdot \left(-0.75 \cdot \frac{c \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(-0.2222222222222222 \cdot \frac{b \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + 0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) + -3 \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{1}{b}\right)}} \]

17. Simplified 92.3%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -3 \cdot \left(a \cdot \mathsf{fma}\left(-0.75, c \cdot \frac{\frac{c}{{b}^{3}} \cdot -0.375}{{b}^{2}}, \mathsf{fma}\left(-0.2222222222222222, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{c}^{2}}, 0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right) + \frac{c}{{b}^{3}} \cdot -0.375\right), \frac{1.5}{b}\right)\right)}} \]

18. Final simplification 92.3%

    \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -3 \cdot \left(\frac{c}{{b}^{3}} \cdot -0.375 + a \cdot \mathsf{fma}\left(-0.75, c \cdot \frac{\frac{c}{{b}^{3}} \cdot -0.375}{{b}^{2}}, \mathsf{fma}\left(-0.2222222222222222, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{c}^{2}}, 0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right), \frac{1.5}{b}\right)\right)} \]
19. Add Preprocessing

Alternative 2: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
    (*
     a
     (+
      (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
      (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0)))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0)))))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0)))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + (-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 7.0)))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

   1. sqr-neg 57.2%

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

   2. sqr-neg 57.2%

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

   3. associate-*l* 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in a around 0 92.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{a \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

6. Taylor expanded in c around 0 92.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]

7. Final simplification 92.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
8. Add Preprocessing

Alternative 3: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(-0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 3.0)))))
   (if (<= b 1.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (/
      1.0
      (/
       (fma
        -2.0
        b
        (*
         c
         (fma
          -3.0
          (* c (* -0.375 (/ (pow a 2.0) (pow b 3.0))))
          (/ (* a 1.5) b))))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 3.0));
	double tmp;
	if (b <= 1.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (c * fma(-3.0, (c * (-0.375 * (pow(a, 2.0) / pow(b, 3.0)))), ((a * 1.5) / b)))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 3.0)))
	tmp = 0.0
	if (b <= 1.0)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(c * fma(-3.0, Float64(c * Float64(-0.375 * Float64((a ^ 2.0) / (b ^ 3.0)))), Float64(Float64(a * 1.5) / b)))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.0], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(c * N[(-3.0 * N[(c * N[(-0.375 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. sqr-neg 83.8%

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

      3. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

      1. pow1 83.9%

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

      2. *-commutative 83.9%

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

      3. *-commutative 83.9%

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

      4. associate-*l* 83.8%

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

      5. *-commutative 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. unpow1 83.8%

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

      2. associate-*r* 83.9%

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

   8. Simplified 83.9%

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

      1. flip-+ 83.2%

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

      2. pow2 83.2%

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

      3. add-sqr-sqrt 84.8%

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

      4. pow2 84.8%

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

      5. associate-*l* 84.9%

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

      6. pow2 84.9%

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

      7. associate-*l* 84.8%

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

   10. Applied egg-rr 84.8%

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

   if 1 < b

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 52.9%

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

   3. Simplified 52.9%

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

      1. pow1 52.9%

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

      2. *-commutative 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*l* 52.9%

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

      5. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. unpow1 52.9%

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

      2. associate-*r* 52.9%

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

   8. Simplified 52.9%

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

      1. +-commutative 52.9%

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

      2. add-cube-cbrt 49.5%

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

      3. fma-define 49.6%

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

      4. cbrt-prod 51.0%

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

      5. add-sqr-sqrt 51.0%

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

      6. pow2 51.0%

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

      7. associate-*l* 51.0%

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

      8. pow2 51.0%

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

      9. associate-*l* 51.0%

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

   10. Applied egg-rr 51.0%

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

       1. clear-num 51.0%

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

       2. inv-pow 51.0%

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

   12. Applied egg-rr 52.9%

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

       1. unpow-1 52.9%

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

       2. associate-/l* 52.9%

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

       3. unsub-neg 52.9%

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

       4. unpow2 52.9%

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

       5. fma-neg 53.0%

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

       6. associate-*l* 53.0%

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

       7. *-commutative 53.0%

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

       8. distribute-rgt-neg-in 53.0%

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

       9. *-commutative 53.0%

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

       10. distribute-lft-neg-in 53.0%

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

       11. metadata-eval 53.0%

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

       12. *-commutative 53.0%

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

   14. Simplified 53.0%

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

   15. Taylor expanded in c around 0 91.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + c \cdot \left(-3 \cdot \left(c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{a}{b}\right)}{c}}} \]
   16. Step-by-step derivation

       1. fma-define 91.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-2, b, c \cdot \left(-3 \cdot \left(c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{a}{b}\right)\right)}}{c}} \]

       2. fma-define 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \color{blue}{\mathsf{fma}\left(-3, c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), 1.5 \cdot \frac{a}{b}\right)}\right)}{c}} \]

       3. distribute-rgt-out 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \color{blue}{\left(\frac{{a}^{2}}{{b}^{3}} \cdot \left(-0.75 + 0.375\right)\right)}, 1.5 \cdot \frac{a}{b}\right)\right)}{c}} \]

       4. metadata-eval 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot \color{blue}{-0.375}\right), 1.5 \cdot \frac{a}{b}\right)\right)}{c}} \]

       5. associate-*r/ 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \color{blue}{\frac{1.5 \cdot a}{b}}\right)\right)}{c}} \]

       6. *-commutative 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \frac{\color{blue}{a \cdot 1.5}}{b}\right)\right)}{c}} \]

   17. Simplified 91.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 3\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(-0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(-0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.02)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) a))
   (/
    1.0
    (/
     (fma
      -2.0
      b
      (*
       c
       (fma
        -3.0
        (* c (* -0.375 (/ (pow a 2.0) (pow b 3.0))))
        (/ (* a 1.5) b))))
     c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.02) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (c * (a * -3.0)))) - b) / a);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (c * fma(-3.0, (c * (-0.375 * (pow(a, 2.0) / pow(b, 3.0)))), ((a * 1.5) / b)))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.02)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / a));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(c * fma(-3.0, Float64(c * Float64(-0.375 * Float64((a ^ 2.0) / (b ^ 3.0)))), Float64(Float64(a * 1.5) / b)))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.02], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(c * N[(-3.0 * N[(c * N[(-0.375 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.02

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. sqr-neg 83.8%

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

      3. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

      1. pow1 83.9%

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

      2. *-commutative 83.9%

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

      3. *-commutative 83.9%

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

      4. associate-*l* 83.8%

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

      5. *-commutative 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. unpow1 83.8%

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

      2. associate-*r* 83.9%

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

   8. Simplified 83.9%

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

      1. +-commutative 83.9%

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

      2. add-cube-cbrt 79.0%

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

      3. fma-define 79.1%

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

      4. cbrt-prod 80.0%

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

      5. add-sqr-sqrt 80.2%

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

      6. pow2 80.2%

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

      7. associate-*l* 80.2%

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

      8. pow2 80.2%

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

      9. associate-*l* 80.2%

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

   10. Applied egg-rr 80.2%

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

       1. div-inv 80.2%

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

       2. fma-undefine 80.5%

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

       3. cbrt-prod 81.5%

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

       4. add-sqr-sqrt 81.9%

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

       5. add-cbrt-cube 83.8%

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

       6. associate-*r* 83.9%

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

       7. *-commutative 83.9%

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

   12. Applied egg-rr 83.9%

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

       1. associate-*r/ 83.9%

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

       2. *-commutative 83.9%

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

       3. *-commutative 83.9%

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

       4. times-frac 84.0%

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

       5. metadata-eval 84.0%

          \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\sqrt{{b}^{2} - \left(c \cdot 3\right) \cdot a} + \left(-b\right)}{a} \]

   14. Simplified 84.1%

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

   if 1.02 < b

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 52.9%

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

   3. Simplified 52.9%

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

      1. pow1 52.9%

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

      2. *-commutative 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*l* 52.9%

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

      5. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. unpow1 52.9%

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

      2. associate-*r* 52.9%

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

   8. Simplified 52.9%

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

      1. +-commutative 52.9%

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

      2. add-cube-cbrt 49.5%

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

      3. fma-define 49.6%

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

      4. cbrt-prod 51.0%

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

      5. add-sqr-sqrt 51.0%

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

      6. pow2 51.0%

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

      7. associate-*l* 51.0%

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

      8. pow2 51.0%

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

      9. associate-*l* 51.0%

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

   10. Applied egg-rr 51.0%

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

       1. clear-num 51.0%

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

       2. inv-pow 51.0%

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

   12. Applied egg-rr 52.9%

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

       1. unpow-1 52.9%

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

       2. associate-/l* 52.9%

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

       3. unsub-neg 52.9%

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

       4. unpow2 52.9%

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

       5. fma-neg 53.0%

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

       6. associate-*l* 53.0%

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

       7. *-commutative 53.0%

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

       8. distribute-rgt-neg-in 53.0%

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

       9. *-commutative 53.0%

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

       10. distribute-lft-neg-in 53.0%

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

       11. metadata-eval 53.0%

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

       12. *-commutative 53.0%

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

   14. Simplified 53.0%

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

   15. Taylor expanded in c around 0 91.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + c \cdot \left(-3 \cdot \left(c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{a}{b}\right)}{c}}} \]
   16. Step-by-step derivation

       1. fma-define 91.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-2, b, c \cdot \left(-3 \cdot \left(c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{a}{b}\right)\right)}}{c}} \]

       2. fma-define 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \color{blue}{\mathsf{fma}\left(-3, c \cdot \left(-0.75 \cdot \frac{{a}^{2}}{{b}^{3}} + 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), 1.5 \cdot \frac{a}{b}\right)}\right)}{c}} \]

       3. distribute-rgt-out 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \color{blue}{\left(\frac{{a}^{2}}{{b}^{3}} \cdot \left(-0.75 + 0.375\right)\right)}, 1.5 \cdot \frac{a}{b}\right)\right)}{c}} \]

       4. metadata-eval 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot \color{blue}{-0.375}\right), 1.5 \cdot \frac{a}{b}\right)\right)}{c}} \]

       5. associate-*r/ 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \color{blue}{\frac{1.5 \cdot a}{b}}\right)\right)}{c}} \]

       6. *-commutative 91.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \frac{\color{blue}{a \cdot 1.5}}{b}\right)\right)}{c}} \]

   17. Simplified 91.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(\frac{{a}^{2}}{{b}^{3}} \cdot -0.375\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(-3, c \cdot \left(-0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right), \frac{a \cdot 1.5}{b}\right)\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ \mathbf{if}\;b \leq 1:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1.5 \cdot \frac{-1}{b} - -3 \cdot \left(a \cdot \left(-0.75 \cdot t\_0 + t\_0 \cdot 0.375\right)\right)\right) - -2 \cdot \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))))
   (if (<= b 1.0)
     (* 0.3333333333333333 (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) a))
     (/
      -1.0
      (-
       (*
        a
        (- (* 1.5 (/ -1.0 b)) (* -3.0 (* a (+ (* -0.75 t_0) (* t_0 0.375))))))
       (* -2.0 (/ b c)))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	double tmp;
	if (b <= 1.0) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (c * (a * -3.0)))) - b) / a);
	} else {
		tmp = -1.0 / ((a * ((1.5 * (-1.0 / b)) - (-3.0 * (a * ((-0.75 * t_0) + (t_0 * 0.375)))))) - (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	tmp = 0.0
	if (b <= 1.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / a));
	else
		tmp = Float64(-1.0 / Float64(Float64(a * Float64(Float64(1.5 * Float64(-1.0 / b)) - Float64(-3.0 * Float64(a * Float64(Float64(-0.75 * t_0) + Float64(t_0 * 0.375)))))) - Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.0], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(a * N[(N[(1.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] - N[(-3.0 * N[(a * N[(N[(-0.75 * t$95$0), $MachinePrecision] + N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. sqr-neg 83.8%

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

      3. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

      1. pow1 83.9%

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

      2. *-commutative 83.9%

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

      3. *-commutative 83.9%

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

      4. associate-*l* 83.8%

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

      5. *-commutative 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. unpow1 83.8%

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

      2. associate-*r* 83.9%

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

   8. Simplified 83.9%

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

      1. +-commutative 83.9%

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

      2. add-cube-cbrt 79.0%

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

      3. fma-define 79.1%

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

      4. cbrt-prod 80.0%

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

      5. add-sqr-sqrt 80.2%

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

      6. pow2 80.2%

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

      7. associate-*l* 80.2%

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

      8. pow2 80.2%

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

      9. associate-*l* 80.2%

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

   10. Applied egg-rr 80.2%

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

       1. div-inv 80.2%

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

       2. fma-undefine 80.5%

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

       3. cbrt-prod 81.5%

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

       4. add-sqr-sqrt 81.9%

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

       5. add-cbrt-cube 83.8%

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

       6. associate-*r* 83.9%

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

       7. *-commutative 83.9%

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

   12. Applied egg-rr 83.9%

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

       1. associate-*r/ 83.9%

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

       2. *-commutative 83.9%

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

       3. *-commutative 83.9%

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

       4. times-frac 84.0%

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

       5. metadata-eval 84.0%

          \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\sqrt{{b}^{2} - \left(c \cdot 3\right) \cdot a} + \left(-b\right)}{a} \]

   14. Simplified 84.1%

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

   if 1 < b

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 52.9%

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

   3. Simplified 52.9%

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

      1. pow1 52.9%

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

      2. *-commutative 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*l* 52.9%

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

      5. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. unpow1 52.9%

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

      2. associate-*r* 52.9%

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

   8. Simplified 52.9%

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

      1. +-commutative 52.9%

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

      2. add-cube-cbrt 49.5%

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

      3. fma-define 49.6%

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

      4. cbrt-prod 51.0%

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

      5. add-sqr-sqrt 51.0%

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

      6. pow2 51.0%

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

      7. associate-*l* 51.0%

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

      8. pow2 51.0%

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

      9. associate-*l* 51.0%

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

   10. Applied egg-rr 51.0%

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

       1. clear-num 51.0%

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

       2. inv-pow 51.0%

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

   12. Applied egg-rr 52.9%

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

       1. unpow-1 52.9%

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

       2. associate-/l* 52.9%

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

       3. unsub-neg 52.9%

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

       4. unpow2 52.9%

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

       5. fma-neg 53.0%

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

       6. associate-*l* 53.0%

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

       7. *-commutative 53.0%

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

       8. distribute-rgt-neg-in 53.0%

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

       9. *-commutative 53.0%

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

       10. distribute-lft-neg-in 53.0%

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

       11. metadata-eval 53.0%

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

       12. *-commutative 53.0%

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

   14. Simplified 53.0%

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

   15. Taylor expanded in a around 0 91.9%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(-3 \cdot \left(a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + 0.375 \cdot \frac{c}{{b}^{3}}\right)\right) + 1.5 \cdot \frac{1}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1.5 \cdot \frac{-1}{b} - -3 \cdot \left(a \cdot \left(-0.75 \cdot \frac{c}{{b}^{3}} + \frac{c}{{b}^{3}} \cdot 0.375\right)\right)\right) - -2 \cdot \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.02)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) a))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.02) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (c * (a * -3.0)))) - b) / a);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.02)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / a));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.02], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.02

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. sqr-neg 83.8%

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

      3. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

      1. pow1 83.9%

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

      2. *-commutative 83.9%

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

      3. *-commutative 83.9%

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

      4. associate-*l* 83.8%

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

      5. *-commutative 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. unpow1 83.8%

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

      2. associate-*r* 83.9%

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

   8. Simplified 83.9%

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

      1. +-commutative 83.9%

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

      2. add-cube-cbrt 79.0%

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

      3. fma-define 79.1%

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

      4. cbrt-prod 80.0%

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

      5. add-sqr-sqrt 80.2%

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

      6. pow2 80.2%

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

      7. associate-*l* 80.2%

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

      8. pow2 80.2%

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

      9. associate-*l* 80.2%

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

   10. Applied egg-rr 80.2%

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

       1. div-inv 80.2%

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

       2. fma-undefine 80.5%

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

       3. cbrt-prod 81.5%

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

       4. add-sqr-sqrt 81.9%

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

       5. add-cbrt-cube 83.8%

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

       6. associate-*r* 83.9%

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

       7. *-commutative 83.9%

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

   12. Applied egg-rr 83.9%

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

       1. associate-*r/ 83.9%

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

       2. *-commutative 83.9%

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

       3. *-commutative 83.9%

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

       4. times-frac 84.0%

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

       5. metadata-eval 84.0%

          \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\sqrt{{b}^{2} - \left(c \cdot 3\right) \cdot a} + \left(-b\right)}{a} \]

   14. Simplified 84.1%

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

   if 1.02 < b

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 52.9%

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

   3. Simplified 52.9%

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

      1. pow1 52.9%

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

      2. *-commutative 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*l* 52.9%

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

      5. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. unpow1 52.9%

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

      2. associate-*r* 52.9%

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

   8. Simplified 52.9%

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

      1. +-commutative 52.9%

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

      2. add-cube-cbrt 49.5%

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

      3. fma-define 49.6%

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

      4. cbrt-prod 51.0%

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

      5. add-sqr-sqrt 51.0%

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

      6. pow2 51.0%

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

      7. associate-*l* 51.0%

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

      8. pow2 51.0%

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

      9. associate-*l* 51.0%

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

   10. Applied egg-rr 51.0%

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

       1. clear-num 51.0%

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

       2. inv-pow 51.0%

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

   12. Applied egg-rr 52.9%

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

       1. unpow-1 52.9%

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

       2. associate-/l* 52.9%

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

       3. unsub-neg 52.9%

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

       4. unpow2 52.9%

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

       5. fma-neg 53.0%

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

       6. associate-*l* 53.0%

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

       7. *-commutative 53.0%

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

       8. distribute-rgt-neg-in 53.0%

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

       9. *-commutative 53.0%

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

       10. distribute-lft-neg-in 53.0%

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

       11. metadata-eval 53.0%

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

       12. *-commutative 53.0%

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

   14. Simplified 53.0%

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

   15. Taylor expanded in a around 0 85.7%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.0) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.0d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.0)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. sqr-neg 83.8%

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

      3. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

   if 1 < b

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 52.9%

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

   3. Simplified 52.9%

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

      1. pow1 52.9%

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

      2. *-commutative 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*l* 52.9%

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

      5. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. unpow1 52.9%

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

      2. associate-*r* 52.9%

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

   8. Simplified 52.9%

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

      1. +-commutative 52.9%

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

      2. add-cube-cbrt 49.5%

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

      3. fma-define 49.6%

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

      4. cbrt-prod 51.0%

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

      5. add-sqr-sqrt 51.0%

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

      6. pow2 51.0%

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

      7. associate-*l* 51.0%

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

      8. pow2 51.0%

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

      9. associate-*l* 51.0%

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

   10. Applied egg-rr 51.0%

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

       1. clear-num 51.0%

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

       2. inv-pow 51.0%

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

   12. Applied egg-rr 52.9%

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

       1. unpow-1 52.9%

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

       2. associate-/l* 52.9%

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

       3. unsub-neg 52.9%

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

       4. unpow2 52.9%

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

       5. fma-neg 53.0%

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

       6. associate-*l* 53.0%

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

       7. *-commutative 53.0%

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

       8. distribute-rgt-neg-in 53.0%

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

       9. *-commutative 53.0%

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

       10. distribute-lft-neg-in 53.0%

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

       11. metadata-eval 53.0%

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

       12. *-commutative 53.0%

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

   14. Simplified 53.0%

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

   15. Taylor expanded in a around 0 85.7%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.02)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.02) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.02d0) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.02) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.02:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.02)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.02)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.02], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.02

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. sqr-neg 83.8%

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

      3. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

      1. pow1 83.9%

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

      2. *-commutative 83.9%

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

      3. *-commutative 83.9%

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

      4. associate-*l* 83.8%

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

      5. *-commutative 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. unpow1 83.8%

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

      2. associate-*r* 83.9%

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

   8. Simplified 83.9%

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

   if 1.02 < b

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 52.9%

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

   3. Simplified 52.9%

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

      1. pow1 52.9%

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

      2. *-commutative 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-*l* 52.9%

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

      5. *-commutative 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. unpow1 52.9%

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

      2. associate-*r* 52.9%

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

   8. Simplified 52.9%

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

      1. +-commutative 52.9%

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

      2. add-cube-cbrt 49.5%

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

      3. fma-define 49.6%

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

      4. cbrt-prod 51.0%

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

      5. add-sqr-sqrt 51.0%

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

      6. pow2 51.0%

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

      7. associate-*l* 51.0%

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

      8. pow2 51.0%

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

      9. associate-*l* 51.0%

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

   10. Applied egg-rr 51.0%

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

       1. clear-num 51.0%

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

       2. inv-pow 51.0%

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

   12. Applied egg-rr 52.9%

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

       1. unpow-1 52.9%

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

       2. associate-/l* 52.9%

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

       3. unsub-neg 52.9%

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

       4. unpow2 52.9%

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

       5. fma-neg 53.0%

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

       6. associate-*l* 53.0%

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

       7. *-commutative 53.0%

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

       8. distribute-rgt-neg-in 53.0%

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

       9. *-commutative 53.0%

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

       10. distribute-lft-neg-in 53.0%

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

       11. metadata-eval 53.0%

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

       12. *-commutative 53.0%

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

   14. Simplified 53.0%

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

   15. Taylor expanded in a around 0 85.7%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

   1. sqr-neg 57.2%

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

   2. sqr-neg 57.2%

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

   3. associate-*l* 57.2%

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

3. Simplified 57.2%

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

   1. pow1 57.2%

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

   2. *-commutative 57.2%

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

   3. *-commutative 57.2%

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

   4. associate-*l* 57.2%

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

   5. *-commutative 57.2%

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

6. Applied egg-rr 57.2%

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

   1. unpow1 57.2%

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

   2. associate-*r* 57.2%

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

8. Simplified 57.2%

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

   1. +-commutative 57.2%

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

   2. add-cube-cbrt 53.6%

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

   3. fma-define 53.8%

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

   4. cbrt-prod 55.1%

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

   5. add-sqr-sqrt 55.1%

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

   6. pow2 55.1%

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

   7. associate-*l* 55.1%

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

   8. pow2 55.1%

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

   9. associate-*l* 55.1%

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

10. Applied egg-rr 55.1%

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

    1. clear-num 55.1%

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

    2. inv-pow 55.1%

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

12. Applied egg-rr 57.3%

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

    1. unpow-1 57.3%

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

    2. associate-/l* 57.3%

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

    3. unsub-neg 57.3%

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

    4. unpow2 57.3%

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

    5. fma-neg 57.4%

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

    6. associate-*l* 57.4%

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

    7. *-commutative 57.4%

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

    8. distribute-rgt-neg-in 57.4%

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

    9. *-commutative 57.4%

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

    10. distribute-lft-neg-in 57.4%

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

    11. metadata-eval 57.4%

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

    12. *-commutative 57.4%

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

14. Simplified 57.4%

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

15. Taylor expanded in a around 0 82.0%

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

16. Final simplification 82.0%

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

Alternative 10: 64.7% 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 57.2%

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

   1. sqr-neg 57.2%

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

   2. sqr-neg 57.2%

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

   3. associate-*l* 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in b around inf 62.8%

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

   1. associate-*r/ 62.8%

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

   2. *-commutative 62.8%

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

7. Simplified 62.8%

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

8. Final simplification 62.8%

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

Alternative 11: 3.2% 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 57.2%

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

   1. sqr-neg 57.2%

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

   2. sqr-neg 57.2%

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

   3. associate-*l* 57.2%

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

3. Simplified 57.2%

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

   1. pow1 57.2%

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

   2. *-commutative 57.2%

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

   3. *-commutative 57.2%

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

   4. associate-*l* 57.2%

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

   5. *-commutative 57.2%

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

6. Applied egg-rr 57.2%

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

   1. unpow1 57.2%

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

   2. associate-*r* 57.2%

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

8. Simplified 57.2%

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

   1. div-inv 57.3%

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

   2. neg-mul-1 57.3%

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

   3. fma-define 57.3%

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

   4. pow2 57.3%

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

   5. associate-*l* 57.2%

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

10. Applied egg-rr 57.2%

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

    1. *-commutative 57.2%

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

    2. cancel-sign-sub-inv 57.2%

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

    3. distribute-lft-neg-in 57.2%

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

    4. +-commutative 57.2%

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

    5. associate-*r* 57.2%

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

    6. distribute-lft-neg-in 57.2%

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

    7. metadata-eval 57.2%

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

    8. fma-define 57.2%

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

    9. associate-/r* 57.2%

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

    10. metadata-eval 57.2%

        \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} \]

12. Simplified 57.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{0.3333333333333333}{a}} \]

13. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
14. Step-by-step derivation

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

15. Simplified 3.2%

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

16. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024067 
(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)))