Cubic critical, narrow range

Percentage Accurate: 55.0% → 91.4%

Time: 20.6s

Alternatives: 13

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: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot {a}^{2}\\ \frac{1}{-3 \cdot \frac{{a}^{3} \cdot \left(-0.75 \cdot \left(c \cdot \left(-0.75 \cdot c + c \cdot 0.375\right)\right) + \left(-0.2222222222222222 \cdot \frac{1.265625 \cdot {c}^{4} + {c}^{4} \cdot 5.0625}{{c}^{2}} + {c}^{2} \cdot 0.5625\right)\right)}{{b}^{5}} + \left(-3 \cdot \frac{-0.75 \cdot t_0 + 0.375 \cdot t_0}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (pow a 2.0))))
   (/
    1.0
    (+
     (*
      -3.0
      (/
       (*
        (pow a 3.0)
        (+
         (* -0.75 (* c (+ (* -0.75 c) (* c 0.375))))
         (+
          (*
           -0.2222222222222222
           (/ (+ (* 1.265625 (pow c 4.0)) (* (pow c 4.0) 5.0625)) (pow c 2.0)))
          (* (pow c 2.0) 0.5625))))
       (pow b 5.0)))
     (+
      (* -3.0 (/ (+ (* -0.75 t_0) (* 0.375 t_0)) (pow b 3.0)))
      (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * pow(a, 2.0);
	return 1.0 / ((-3.0 * ((pow(a, 3.0) * ((-0.75 * (c * ((-0.75 * c) + (c * 0.375)))) + ((-0.2222222222222222 * (((1.265625 * pow(c, 4.0)) + (pow(c, 4.0) * 5.0625)) / pow(c, 2.0))) + (pow(c, 2.0) * 0.5625)))) / pow(b, 5.0))) + ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / pow(b, 3.0))) + ((-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
    real(8) :: t_0
    t_0 = c * (a ** 2.0d0)
    code = 1.0d0 / (((-3.0d0) * (((a ** 3.0d0) * (((-0.75d0) * (c * (((-0.75d0) * c) + (c * 0.375d0)))) + (((-0.2222222222222222d0) * (((1.265625d0 * (c ** 4.0d0)) + ((c ** 4.0d0) * 5.0625d0)) / (c ** 2.0d0))) + ((c ** 2.0d0) * 0.5625d0)))) / (b ** 5.0d0))) + (((-3.0d0) * ((((-0.75d0) * t_0) + (0.375d0 * t_0)) / (b ** 3.0d0))) + (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * Math.pow(a, 2.0);
	return 1.0 / ((-3.0 * ((Math.pow(a, 3.0) * ((-0.75 * (c * ((-0.75 * c) + (c * 0.375)))) + ((-0.2222222222222222 * (((1.265625 * Math.pow(c, 4.0)) + (Math.pow(c, 4.0) * 5.0625)) / Math.pow(c, 2.0))) + (Math.pow(c, 2.0) * 0.5625)))) / Math.pow(b, 5.0))) + ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / Math.pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))));
}

Python

def code(a, b, c):
	t_0 = c * math.pow(a, 2.0)
	return 1.0 / ((-3.0 * ((math.pow(a, 3.0) * ((-0.75 * (c * ((-0.75 * c) + (c * 0.375)))) + ((-0.2222222222222222 * (((1.265625 * math.pow(c, 4.0)) + (math.pow(c, 4.0) * 5.0625)) / math.pow(c, 2.0))) + (math.pow(c, 2.0) * 0.5625)))) / math.pow(b, 5.0))) + ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / math.pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))))

Julia

function code(a, b, c)
	t_0 = Float64(c * (a ^ 2.0))
	return Float64(1.0 / Float64(Float64(-3.0 * Float64(Float64((a ^ 3.0) * Float64(Float64(-0.75 * Float64(c * Float64(Float64(-0.75 * c) + Float64(c * 0.375)))) + Float64(Float64(-0.2222222222222222 * Float64(Float64(Float64(1.265625 * (c ^ 4.0)) + Float64((c ^ 4.0) * 5.0625)) / (c ^ 2.0))) + Float64((c ^ 2.0) * 0.5625)))) / (b ^ 5.0))) + Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * t_0) + Float64(0.375 * t_0)) / (b ^ 3.0))) + Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (a ^ 2.0);
	tmp = 1.0 / ((-3.0 * (((a ^ 3.0) * ((-0.75 * (c * ((-0.75 * c) + (c * 0.375)))) + ((-0.2222222222222222 * (((1.265625 * (c ^ 4.0)) + ((c ^ 4.0) * 5.0625)) / (c ^ 2.0))) + ((c ^ 2.0) * 0.5625)))) / (b ^ 5.0))) + ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / (b ^ 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(-3.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(-0.75 * N[(c * N[(N[(-0.75 * c), $MachinePrecision] + N[(c * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.2222222222222222 * N[(N[(N[(1.265625 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * 5.0625), $MachinePrecision]), $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 2.0], $MachinePrecision] * 0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 * N[(N[(N[(-0.75 * t$95$0), $MachinePrecision] + N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in a around 0 51.7%

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

   1. *-commutative 51.7%

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

5. Simplified 51.7%

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

   1. clear-num 51.7%

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

   2. inv-pow 51.7%

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

   3. *-commutative 51.7%

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

   4. neg-mul-1 51.7%

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

   5. fma-def 51.7%

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

   6. pow2 51.7%

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

   7. *-commutative 51.7%

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

   8. associate-*r* 51.7%

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

   9. *-commutative 51.7%

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

7. Applied egg-rr 51.7%

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

   1. unpow-1 51.7%

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

   2. *-commutative 51.7%

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

   3. *-lft-identity 51.7%

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

   4. times-frac 51.7%

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

   5. metadata-eval 51.7%

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

   6. *-commutative 51.7%

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

   7. associate-*r* 51.7%

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

   8. *-commutative 51.7%

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

   9. *-commutative 51.7%

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

   10. cancel-sign-sub-inv 51.7%

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

   11. metadata-eval 51.7%

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

   12. +-commutative 51.7%

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

   13. fma-def 51.7%

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

9. Simplified 51.7%

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

10. Taylor expanded in b around inf 93.1%

    \[\leadsto \frac{1}{\color{blue}{-3 \cdot \frac{-0.75 \cdot \left(a \cdot \left(c \cdot \left(-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.2222222222222222 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {c}^{2}} + 0.5625 \cdot \left({a}^{3} \cdot {c}^{2}\right)\right)}{{b}^{5}} + \left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)}} \]

11. Taylor expanded in a around 0 93.1%

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

12. Final simplification 93.1%

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

Alternative 2: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot -0.375 + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+
   (* -0.5 (/ c b))
   (+
    (* (/ (* a (pow c 2.0)) (pow b 3.0)) -0.375)
    (*
     -0.16666666666666666
     (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((((a * pow(c, 2.0)) / pow(b, 3.0)) * -0.375) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / 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 = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((((a * (c ** 2.0d0)) / (b ** 3.0d0)) * (-0.375d0)) + ((-0.16666666666666666d0) * ((((a * c) ** 4.0d0) / a) * (6.328125d0 / (b ** 7.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)) * -0.375) + (-0.16666666666666666 * ((Math.pow((a * c), 4.0) / a) * (6.328125 / Math.pow(b, 7.0))))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((((a * math.pow(c, 2.0)) / math.pow(b, 3.0)) * -0.375) + (-0.16666666666666666 * ((math.pow((a * c), 4.0) / a) * (6.328125 / math.pow(b, 7.0))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)) * -0.375) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((((a * (c ^ 2.0)) / (b ^ 3.0)) * -0.375) + (-0.16666666666666666 * ((((a * c) ^ 4.0) / a) * (6.328125 / (b ^ 7.0))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in b around inf 92.8%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

4. Taylor expanded in c around 0 92.8%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
5. Step-by-step derivation

   1. distribute-rgt-in 92.8%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left(1.265625 \cdot {a}^{4}\right) \cdot {c}^{4} + \left(5.0625 \cdot {a}^{4}\right) \cdot {c}^{4}}}{a \cdot {b}^{7}}\right)\right) \]

   2. associate-*r* 92.8%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right)} + \left(5.0625 \cdot {a}^{4}\right) \cdot {c}^{4}}{a \cdot {b}^{7}}\right)\right) \]

   3. associate-*r* 92.8%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}}{a \cdot {b}^{7}}\right)\right) \]

   4. distribute-rgt-out 92.8%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

   5. times-frac 92.8%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{1.265625 + 5.0625}{{b}^{7}}\right)}\right)\right) \]

6. Simplified 92.8%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right) \]

7. Final simplification 92.8%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot -0.375 + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot {a}^{2}\\ t_1 := c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;b \leq 0.0023:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_1 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_1}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{-0.75 \cdot t_0 + 0.375 \cdot t_0}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (pow a 2.0))) (t_1 (* c (* a 3.0))))
   (if (<= b 0.0023)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_1 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_1))))
      (* a 3.0))
     (/
      1.0
      (+
       (* -3.0 (/ (+ (* -0.75 t_0) (* 0.375 t_0)) (pow b 3.0)))
       (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * pow(a, 2.0);
	double t_1 = c * (a * 3.0);
	double tmp;
	if (b <= 0.0023) {
		tmp = ((pow(-b, 2.0) + (t_1 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_1)))) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / pow(b, 3.0))) + ((-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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (a ** 2.0d0)
    t_1 = c * (a * 3.0d0)
    if (b <= 0.0023d0) then
        tmp = (((-b ** 2.0d0) + (t_1 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_1)))) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (((-3.0d0) * ((((-0.75d0) * t_0) + (0.375d0 * t_0)) / (b ** 3.0d0))) + (((-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 t_0 = c * Math.pow(a, 2.0);
	double t_1 = c * (a * 3.0);
	double tmp;
	if (b <= 0.0023) {
		tmp = ((Math.pow(-b, 2.0) + (t_1 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_1)))) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / Math.pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * math.pow(a, 2.0)
	t_1 = c * (a * 3.0)
	tmp = 0
	if b <= 0.0023:
		tmp = ((math.pow(-b, 2.0) + (t_1 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_1)))) / (a * 3.0)
	else:
		tmp = 1.0 / ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / math.pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * (a ^ 2.0))
	t_1 = Float64(c * Float64(a * 3.0))
	tmp = 0.0
	if (b <= 0.0023)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_1 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_1)))) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * t_0) + Float64(0.375 * t_0)) / (b ^ 3.0))) + 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)
	t_0 = c * (a ^ 2.0);
	t_1 = c * (a * 3.0);
	tmp = 0.0;
	if (b <= 0.0023)
		tmp = (((-b ^ 2.0) + (t_1 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_1)))) / (a * 3.0);
	else
		tmp = 1.0 / ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / (b ^ 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0023], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$1 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-3.0 * N[(N[(N[(-0.75 * t$95$0), $MachinePrecision] + N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0023

   1. Initial program 89.1%

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

   3. Taylor expanded in a around 0 89.2%

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

      1. *-commutative 89.2%

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

   5. Simplified 89.2%

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

      1. flip-+ 88.3%

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

      2. pow2 88.3%

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

      3. add-sqr-sqrt 89.7%

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

      4. pow2 89.7%

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

      5. *-commutative 89.7%

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

      6. associate-*r* 89.7%

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

      7. *-commutative 89.7%

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

      8. pow2 89.7%

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

      9. *-commutative 89.7%

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

      10. associate-*r* 89.6%

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

      11. *-commutative 89.6%

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

   7. Applied egg-rr 89.6%

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

   if 0.0023 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0 49.3%

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

      1. *-commutative 49.3%

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

   5. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. inv-pow 49.3%

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

      3. *-commutative 49.3%

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

      4. neg-mul-1 49.3%

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

      5. fma-def 49.3%

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

      6. pow2 49.3%

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

      7. *-commutative 49.3%

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

      8. associate-*r* 49.3%

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

      9. *-commutative 49.3%

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

   7. Applied egg-rr 49.3%

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

      1. unpow-1 49.3%

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

      2. *-commutative 49.3%

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

      3. *-lft-identity 49.3%

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

      4. times-frac 49.3%

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

      5. metadata-eval 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-*r* 49.4%

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

      8. *-commutative 49.4%

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

      9. *-commutative 49.4%

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

      10. cancel-sign-sub-inv 49.4%

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

      11. metadata-eval 49.4%

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

      12. +-commutative 49.4%

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

      13. fma-def 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in b around inf 92.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0023:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(a \cdot 3\right) - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{-0.75 \cdot \left(c \cdot {a}^{2}\right) + 0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot {a}^{2}\\ \mathbf{if}\;b \leq 0.0023:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(a \cdot 3\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{-0.75 \cdot t_0 + 0.375 \cdot t_0}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (pow a 2.0))))
   (if (<= b 0.0023)
     (/ 1.0 (/ (* a 3.0) (- (sqrt (fma b b (- (* c (* a 3.0))))) b)))
     (/
      1.0
      (+
       (* -3.0 (/ (+ (* -0.75 t_0) (* 0.375 t_0)) (pow b 3.0)))
       (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * pow(a, 2.0);
	double tmp;
	if (b <= 0.0023) {
		tmp = 1.0 / ((a * 3.0) / (sqrt(fma(b, b, -(c * (a * 3.0)))) - b));
	} else {
		tmp = 1.0 / ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * (a ^ 2.0))
	tmp = 0.0
	if (b <= 0.0023)
		tmp = Float64(1.0 / Float64(Float64(a * 3.0) / Float64(sqrt(fma(b, b, Float64(-Float64(c * Float64(a * 3.0))))) - b)));
	else
		tmp = Float64(1.0 / Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * t_0) + Float64(0.375 * t_0)) / (b ^ 3.0))) + Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0023], N[(1.0 / N[(N[(a * 3.0), $MachinePrecision] / N[(N[Sqrt[N[(b * b + (-N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-3.0 * N[(N[(N[(-0.75 * t$95$0), $MachinePrecision] + N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0023

   1. Initial program 89.1%

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

   3. Taylor expanded in a around 0 89.2%

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

      1. *-commutative 89.2%

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

   5. Simplified 89.2%

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

      1. add-cube-cbrt 88.9%

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

      2. pow3 88.9%

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

      3. neg-mul-1 88.9%

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

      4. fma-def 88.9%

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

      5. pow2 88.9%

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

      6. *-commutative 88.9%

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

      7. associate-*r* 88.7%

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

      8. *-commutative 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. rem-cube-cbrt 89.1%

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

      2. clear-num 89.2%

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

      3. inv-pow 89.2%

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

      4. *-commutative 89.2%

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

      5. associate-*l* 89.2%

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

   9. Applied egg-rr 89.2%

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

       1. unpow-1 89.2%

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

       2. fma-udef 89.2%

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

       3. neg-mul-1 89.2%

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

       4. +-commutative 89.2%

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

       5. unsub-neg 89.2%

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

       6. associate-*r* 89.2%

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

   11. Simplified 89.2%

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

       1. cancel-sign-sub-inv 89.2%

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

       2. unpow2 89.2%

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

       3. fma-def 89.5%

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

   13. Applied egg-rr 89.5%

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

   if 0.0023 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0 49.3%

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

      1. *-commutative 49.3%

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

   5. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. inv-pow 49.3%

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

      3. *-commutative 49.3%

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

      4. neg-mul-1 49.3%

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

      5. fma-def 49.3%

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

      6. pow2 49.3%

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

      7. *-commutative 49.3%

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

      8. associate-*r* 49.3%

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

      9. *-commutative 49.3%

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

   7. Applied egg-rr 49.3%

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

      1. unpow-1 49.3%

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

      2. *-commutative 49.3%

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

      3. *-lft-identity 49.3%

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

      4. times-frac 49.3%

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

      5. metadata-eval 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-*r* 49.4%

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

      8. *-commutative 49.4%

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

      9. *-commutative 49.4%

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

      10. cancel-sign-sub-inv 49.4%

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

      11. metadata-eval 49.4%

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

      12. +-commutative 49.4%

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

      13. fma-def 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in b around inf 92.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0023:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(a \cdot 3\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{-0.75 \cdot \left(c \cdot {a}^{2}\right) + 0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.00705:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot \left(-c\right)\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.00705)
   (/ (- (sqrt (fma b b (* 3.0 (* a (- c))))) 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 (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.00705) {
		tmp = (sqrt(fma(b, b, (3.0 * (a * -c)))) - 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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.00705)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(3.0 * Float64(a * Float64(-c))))) - 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

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.00705], N[(N[(N[Sqrt[N[(b * b + N[(3.0 * N[(a * (-c)), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.00704999999999999984

   1. Initial program 78.2%

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

      1. +-commutative 78.2%

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

      2. sqr-neg 78.2%

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

      3. unsub-neg 78.2%

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

      4. div-sub 77.2%

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

      5. --rgt-identity 77.2%

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

      6. div-sub 78.2%

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

   3. Simplified 78.3%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

      3. metadata-eval 78.4%

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

      4. distribute-lft-neg-in 78.4%

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

   6. Applied egg-rr 78.4%

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

   if -0.00704999999999999984 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.7%

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

   3. Taylor expanded in a around 0 42.7%

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

      1. *-commutative 42.7%

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

   5. Simplified 42.7%

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

      1. clear-num 42.7%

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

      2. inv-pow 42.7%

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

      3. *-commutative 42.7%

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

      4. neg-mul-1 42.7%

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

      5. fma-def 42.7%

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

      6. pow2 42.7%

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

      7. *-commutative 42.7%

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

      8. associate-*r* 42.7%

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

      9. *-commutative 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. unpow-1 42.7%

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

      2. *-commutative 42.7%

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

      3. *-lft-identity 42.7%

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

      4. times-frac 42.7%

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

      5. metadata-eval 42.7%

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

      6. *-commutative 42.7%

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

      7. associate-*r* 42.7%

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

      8. *-commutative 42.7%

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

      9. *-commutative 42.7%

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

      10. cancel-sign-sub-inv 42.7%

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

      11. metadata-eval 42.7%

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

      12. +-commutative 42.7%

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

      13. fma-def 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in b around inf 92.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.00705:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot \left(-c\right)\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 3} \leq -0.00705:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -t_0\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* a 3.0)) -0.00705)
     (/ (- (sqrt (fma b b (- t_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 t_0 = c * (a * 3.0);
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (a * 3.0)) <= -0.00705) {
		tmp = (sqrt(fma(b, b, -t_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)
	t_0 = Float64(c * Float64(a * 3.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 3.0)) <= -0.00705)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-t_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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.00705], N[(N[(N[Sqrt[N[(b * b + (-t$95$0)), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.00704999999999999984

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0 78.1%

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

      1. *-commutative 78.1%

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

   5. Simplified 78.1%

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

      1. fma-neg 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*r* 78.4%

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

      4. *-commutative 78.4%

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

   7. Applied egg-rr 78.4%

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

   if -0.00704999999999999984 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.7%

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

   3. Taylor expanded in a around 0 42.7%

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

      1. *-commutative 42.7%

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

   5. Simplified 42.7%

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

      1. clear-num 42.7%

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

      2. inv-pow 42.7%

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

      3. *-commutative 42.7%

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

      4. neg-mul-1 42.7%

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

      5. fma-def 42.7%

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

      6. pow2 42.7%

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

      7. *-commutative 42.7%

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

      8. associate-*r* 42.7%

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

      9. *-commutative 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. unpow-1 42.7%

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

      2. *-commutative 42.7%

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

      3. *-lft-identity 42.7%

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

      4. times-frac 42.7%

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

      5. metadata-eval 42.7%

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

      6. *-commutative 42.7%

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

      7. associate-*r* 42.7%

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

      8. *-commutative 42.7%

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

      9. *-commutative 42.7%

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

      10. cancel-sign-sub-inv 42.7%

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

      11. metadata-eval 42.7%

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

      12. +-commutative 42.7%

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

      13. fma-def 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in b around inf 92.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.00705:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(a \cdot 3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.00705:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.00705)
   (/ (- (sqrt (fma b b (* a (* -3.0 c)))) 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 (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.00705) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - 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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.00705)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - 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

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.00705], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.00704999999999999984

   1. Initial program 78.2%

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

      1. +-commutative 78.2%

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

      2. sqr-neg 78.2%

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

      3. unsub-neg 78.2%

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

      4. div-sub 77.2%

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

      5. --rgt-identity 77.2%

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

      6. div-sub 78.2%

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

   3. Simplified 78.3%

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

   if -0.00704999999999999984 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.7%

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

   3. Taylor expanded in a around 0 42.7%

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

      1. *-commutative 42.7%

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

   5. Simplified 42.7%

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

      1. clear-num 42.7%

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

      2. inv-pow 42.7%

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

      3. *-commutative 42.7%

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

      4. neg-mul-1 42.7%

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

      5. fma-def 42.7%

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

      6. pow2 42.7%

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

      7. *-commutative 42.7%

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

      8. associate-*r* 42.7%

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

      9. *-commutative 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. unpow-1 42.7%

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

      2. *-commutative 42.7%

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

      3. *-lft-identity 42.7%

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

      4. times-frac 42.7%

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

      5. metadata-eval 42.7%

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

      6. *-commutative 42.7%

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

      7. associate-*r* 42.7%

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

      8. *-commutative 42.7%

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

      9. *-commutative 42.7%

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

      10. cancel-sign-sub-inv 42.7%

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

      11. metadata-eval 42.7%

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

      12. +-commutative 42.7%

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

      13. fma-def 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in b around inf 92.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.00705:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0023:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(a \cdot 3\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot \left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right) - \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{3}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0023)
   (/ 1.0 (/ (* a 3.0) (- (sqrt (fma b b (- (* c (* a 3.0))))) b)))
   (/
    1.0
    (*
     3.0
     (-
      (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5))
      (/ (* (* c (pow a 2.0)) -0.375) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0023) {
		tmp = 1.0 / ((a * 3.0) / (sqrt(fma(b, b, -(c * (a * 3.0)))) - b));
	} else {
		tmp = 1.0 / (3.0 * (fma(-0.6666666666666666, (b / c), ((a / b) * 0.5)) - (((c * pow(a, 2.0)) * -0.375) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0023)
		tmp = Float64(1.0 / Float64(Float64(a * 3.0) / Float64(sqrt(fma(b, b, Float64(-Float64(c * Float64(a * 3.0))))) - b)));
	else
		tmp = Float64(1.0 / Float64(3.0 * Float64(fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)) - Float64(Float64(Float64(c * (a ^ 2.0)) * -0.375) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0023], N[(1.0 / N[(N[(a * 3.0), $MachinePrecision] / N[(N[Sqrt[N[(b * b + (-N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(3.0 * N[(N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0023

   1. Initial program 89.1%

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

   3. Taylor expanded in a around 0 89.2%

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

      1. *-commutative 89.2%

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

   5. Simplified 89.2%

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

      1. add-cube-cbrt 88.9%

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

      2. pow3 88.9%

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

      3. neg-mul-1 88.9%

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

      4. fma-def 88.9%

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

      5. pow2 88.9%

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

      6. *-commutative 88.9%

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

      7. associate-*r* 88.7%

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

      8. *-commutative 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. rem-cube-cbrt 89.1%

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

      2. clear-num 89.2%

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

      3. inv-pow 89.2%

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

      4. *-commutative 89.2%

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

      5. associate-*l* 89.2%

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

   9. Applied egg-rr 89.2%

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

       1. unpow-1 89.2%

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

       2. fma-udef 89.2%

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

       3. neg-mul-1 89.2%

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

       4. +-commutative 89.2%

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

       5. unsub-neg 89.2%

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

       6. associate-*r* 89.2%

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

   11. Simplified 89.2%

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

       1. cancel-sign-sub-inv 89.2%

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

       2. unpow2 89.2%

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

       3. fma-def 89.5%

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

   13. Applied egg-rr 89.5%

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

   if 0.0023 < b

   1. Initial program 49.4%

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

   3. Taylor expanded in a around 0 49.3%

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

      1. *-commutative 49.3%

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

   5. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. inv-pow 49.3%

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

      3. *-commutative 49.3%

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

      4. neg-mul-1 49.3%

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

      5. fma-def 49.3%

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

      6. pow2 49.3%

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

      7. *-commutative 49.3%

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

      8. associate-*r* 49.3%

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

      9. *-commutative 49.3%

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

   7. Applied egg-rr 49.3%

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

      1. unpow-1 49.3%

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

      2. *-commutative 49.3%

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

      3. *-lft-identity 49.3%

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

      4. times-frac 49.3%

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

      5. metadata-eval 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-*r* 49.4%

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

      8. *-commutative 49.4%

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

      9. *-commutative 49.4%

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

      10. cancel-sign-sub-inv 49.4%

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

      11. metadata-eval 49.4%

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

      12. +-commutative 49.4%

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

      13. fma-def 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in b around inf 92.3%

       \[\leadsto \frac{1}{3 \cdot \color{blue}{\left(-1 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}} + \left(-0.6666666666666666 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 92.3%

          \[\leadsto \frac{1}{3 \cdot \color{blue}{\left(\left(-0.6666666666666666 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right) + -1 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}}\right)}} \]

       2. mul-1-neg 92.3%

          \[\leadsto \frac{1}{3 \cdot \left(\left(-0.6666666666666666 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right) + \color{blue}{\left(-\frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}}\right)}\right)} \]

       3. unsub-neg 92.3%

          \[\leadsto \frac{1}{3 \cdot \color{blue}{\left(\left(-0.6666666666666666 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right) - \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}}\right)}} \]

       4. fma-def 92.4%

          \[\leadsto \frac{1}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right)} - \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}}\right)} \]

       5. distribute-rgt-out 92.4%

          \[\leadsto \frac{1}{3 \cdot \left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right) - \frac{\color{blue}{\left({a}^{2} \cdot c\right) \cdot \left(-0.75 + 0.375\right)}}{{b}^{3}}\right)} \]

       6. *-commutative 92.4%

          \[\leadsto \frac{1}{3 \cdot \left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right) - \frac{\color{blue}{\left(c \cdot {a}^{2}\right)} \cdot \left(-0.75 + 0.375\right)}{{b}^{3}}\right)} \]

       7. metadata-eval 92.4%

          \[\leadsto \frac{1}{3 \cdot \left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right) - \frac{\left(c \cdot {a}^{2}\right) \cdot \color{blue}{-0.375}}{{b}^{3}}\right)} \]

   12. Simplified 92.4%

       \[\leadsto \frac{1}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right) - \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{3}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0023:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(a \cdot 3\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot \left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right) - \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{3}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.00705:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.00705) t_0 (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.00705) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.00705d0)) then
        tmp = t_0
    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 t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.00705) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.00705:
		tmp = t_0
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.00705)
		tmp = t_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)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.00705)
		tmp = t_0;
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.00705], t$95$0, 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.00704999999999999984

   1. Initial program 78.2%

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

   if -0.00704999999999999984 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.7%

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

   3. Taylor expanded in a around 0 42.7%

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

      1. *-commutative 42.7%

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

   5. Simplified 42.7%

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

      1. clear-num 42.7%

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

      2. inv-pow 42.7%

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

      3. *-commutative 42.7%

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

      4. neg-mul-1 42.7%

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

      5. fma-def 42.7%

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

      6. pow2 42.7%

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

      7. *-commutative 42.7%

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

      8. associate-*r* 42.7%

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

      9. *-commutative 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. unpow-1 42.7%

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

      2. *-commutative 42.7%

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

      3. *-lft-identity 42.7%

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

      4. times-frac 42.7%

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

      5. metadata-eval 42.7%

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

      6. *-commutative 42.7%

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

      7. associate-*r* 42.7%

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

      8. *-commutative 42.7%

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

      9. *-commutative 42.7%

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

      10. cancel-sign-sub-inv 42.7%

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

      11. metadata-eval 42.7%

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

      12. +-commutative 42.7%

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

      13. fma-def 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in b around inf 92.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.00705:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.5% 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 51.7%

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

3. Taylor expanded in a around 0 51.7%

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

   1. *-commutative 51.7%

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

5. Simplified 51.7%

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

   1. clear-num 51.7%

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

   2. inv-pow 51.7%

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

   3. *-commutative 51.7%

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

   4. neg-mul-1 51.7%

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

   5. fma-def 51.7%

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

   6. pow2 51.7%

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

   7. *-commutative 51.7%

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

   8. associate-*r* 51.7%

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

   9. *-commutative 51.7%

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

7. Applied egg-rr 51.7%

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

   1. unpow-1 51.7%

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

   2. *-commutative 51.7%

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

   3. *-lft-identity 51.7%

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

   4. times-frac 51.7%

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

   5. metadata-eval 51.7%

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

   6. *-commutative 51.7%

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

   7. associate-*r* 51.7%

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

   8. *-commutative 51.7%

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

   9. *-commutative 51.7%

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

   10. cancel-sign-sub-inv 51.7%

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

   11. metadata-eval 51.7%

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

   12. +-commutative 51.7%

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

   13. fma-def 51.7%

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

9. Simplified 51.7%

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

10. Taylor expanded in b around inf 85.9%

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

11. Final simplification 85.9%

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

Alternative 11: 12.0% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \cdot -0.1111111111111111 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ b a) -0.1111111111111111))

C

double code(double a, double b, double c) {
	return (b / a) * -0.1111111111111111;
}

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 / a) * (-0.1111111111111111d0)
end function

Java

public static double code(double a, double b, double c) {
	return (b / a) * -0.1111111111111111;
}

Python

def code(a, b, c):
	return (b / a) * -0.1111111111111111

Julia

function code(a, b, c)
	return Float64(Float64(b / a) * -0.1111111111111111)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (b / a) * -0.1111111111111111;
end

Wolfram

code[a_, b_, c_] := N[(N[(b / a), $MachinePrecision] * -0.1111111111111111), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in a around 0 51.7%

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

   1. *-commutative 51.7%

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

5. Simplified 51.7%

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

   1. flip3-+ 51.6%

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

   2. pow1/2 51.6%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\color{blue}{\left({\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right)}^{0.5}\right)}}^{3}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a} \]

   3. pow-pow 52.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + \color{blue}{{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right)}^{\left(0.5 \cdot 3\right)}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a} \]

   4. pow2 52.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left(\color{blue}{{b}^{2}} - \left(a \cdot c\right) \cdot 3\right)}^{\left(0.5 \cdot 3\right)}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a} \]

   5. *-commutative 52.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left({b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}^{\left(0.5 \cdot 3\right)}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a} \]

   6. associate-*r* 52.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left({b}^{2} - \color{blue}{\left(3 \cdot a\right) \cdot c}\right)}^{\left(0.5 \cdot 3\right)}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a} \]

   7. *-commutative 52.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left({b}^{2} - \color{blue}{\left(a \cdot 3\right)} \cdot c\right)}^{\left(0.5 \cdot 3\right)}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a} \]

   8. metadata-eval 52.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left({b}^{2} - \left(a \cdot 3\right) \cdot c\right)}^{\color{blue}{1.5}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a} \]

   9. pow2 52.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left({b}^{2} - \left(a \cdot 3\right) \cdot c\right)}^{1.5}}{\color{blue}{{\left(-b\right)}^{2}} + \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}{3 \cdot a} \]

7. Applied egg-rr 52.7%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left({b}^{2} - \left(a \cdot 3\right) \cdot c\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left({b}^{2} - \left(a \cdot 3\right) \cdot c\right) - \left(-b\right) \cdot \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}}}{3 \cdot a} \]

8. Taylor expanded in b around inf 11.7%

   \[\leadsto \color{blue}{-0.1111111111111111 \cdot \frac{b}{a}} \]
9. Step-by-step derivation

   1. *-commutative 11.7%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.1111111111111111} \]

10. Simplified 11.7%

    \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.1111111111111111} \]

11. Final simplification 11.7%

    \[\leadsto \frac{b}{a} \cdot -0.1111111111111111 \]
12. Add Preprocessing

Alternative 12: 64.8% 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 51.7%

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

3. Taylor expanded in b around inf 67.9%

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

   1. *-commutative 67.9%

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

   2. associate-*l/ 67.9%

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

5. Simplified 67.9%

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

6. Final simplification 67.9%

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

Alternative 13: 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 51.7%

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

3. Taylor expanded in a around 0 51.7%

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

   1. *-commutative 51.7%

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

5. Simplified 51.7%

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

   1. clear-num 51.7%

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

   2. inv-pow 51.7%

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

   3. *-commutative 51.7%

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

   4. neg-mul-1 51.7%

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

   5. fma-def 51.7%

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

   6. pow2 51.7%

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

   7. *-commutative 51.7%

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

   8. associate-*r* 51.7%

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

   9. *-commutative 51.7%

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

7. Applied egg-rr 51.7%

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

   1. unpow-1 51.7%

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

   2. *-commutative 51.7%

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

   3. *-lft-identity 51.7%

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

   4. times-frac 51.7%

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

   5. metadata-eval 51.7%

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

   6. *-commutative 51.7%

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

   7. associate-*r* 51.7%

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

   8. *-commutative 51.7%

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

   9. *-commutative 51.7%

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

   10. cancel-sign-sub-inv 51.7%

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

   11. metadata-eval 51.7%

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

   12. +-commutative 51.7%

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

   13. fma-def 51.7%

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

9. Simplified 51.7%

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

10. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
11. 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%

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

12. Simplified 3.2%

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

13. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024010 
(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)))