Cubic critical, narrow range

Percentage Accurate: 54.6% → 99.5%

Time: 27.5s

Alternatives: 12

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: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{3 \cdot a}{3 \cdot a} \cdot \frac{c}{\left(-b\right) - \sqrt{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ (* 3.0 a) (* 3.0 a))
  (/ c (- (- b) (sqrt (* a (fma c -3.0 (/ (pow b 2.0) a))))))))

C

double code(double a, double b, double c) {
	return ((3.0 * a) / (3.0 * a)) * (c / (-b - sqrt((a * fma(c, -3.0, (pow(b, 2.0) / a))))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(3.0 * a) / Float64(3.0 * a)) * Float64(c / Float64(Float64(-b) - sqrt(Float64(a * fma(c, -3.0, Float64((b ^ 2.0) / a)))))))
end

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in a around inf 54.7%

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

   1. flip-+ 54.7%

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

   2. pow2 54.7%

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

   3. add-sqr-sqrt 55.6%

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

   4. cancel-sign-sub-inv 55.6%

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

   5. metadata-eval 55.6%

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

   6. *-commutative 55.6%

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

   7. cancel-sign-sub-inv 55.6%

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

   8. metadata-eval 55.6%

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

   9. *-commutative 55.6%

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

5. Applied egg-rr 55.6%

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

6. Taylor expanded in b around 0 99.1%

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

   1. *-un-lft-identity 99.1%

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

   2. associate-/l/ 99.1%

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

   3. associate-*r* 99.3%

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

   4. sqrt-prod 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-undefine 99.3%

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

   7. sqrt-prod 99.3%

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

8. Applied egg-rr 99.3%

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

   1. *-lft-identity 99.3%

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

   2. times-frac 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

    \[\leadsto \frac{3 \cdot a}{3 \cdot a} \cdot \frac{c}{\left(-b\right) - \sqrt{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)}} \]
12. Add Preprocessing

Alternative 2: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/
    1.0
    (*
     a
     (/
      (+
       (* -2.0 (/ b c))
       (* a (+ (* 1.125 (/ (* a c) (pow b 3.0))) (* 1.5 (/ 1.0 b)))))
      a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (a * ((1.125 * ((a * c) / pow(b, 3.0))) + (1.5 * (1.0 / b))))) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(a * Float64(Float64(Float64(-2.0 * Float64(b / c)) + Float64(a * Float64(Float64(1.125 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(1.5 * Float64(1.0 / b))))) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(1.125 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $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)) < -0.5

   1. Initial program 86.2%

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

      1. /-rgt-identity 86.2%

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

      2. metadata-eval 86.2%

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

   3. Simplified 86.2%

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

   if -0.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   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 inf 49.2%

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

      1. clear-num 49.2%

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

      2. inv-pow 49.2%

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

      3. *-commutative 49.2%

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

      4. neg-mul-1 49.2%

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

      5. fma-define 49.2%

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

      6. cancel-sign-sub-inv 49.2%

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

      7. metadata-eval 49.2%

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

      8. *-commutative 49.2%

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

   5. Applied egg-rr 49.2%

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

      1. unpow-1 49.2%

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

      2. associate-/l* 49.2%

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

      3. rem-log-exp 45.2%

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

      4. fma-undefine 45.2%

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

      5. neg-mul-1 45.2%

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

      6. prod-exp 20.2%

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

      7. *-commutative 20.2%

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

      8. prod-exp 45.2%

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

      9. rem-log-exp 49.2%

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

      10. unsub-neg 49.2%

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

   7. Simplified 49.2%

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

      1. flip-- 49.3%

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

      2. add-sqr-sqrt 50.3%

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

      3. fma-undefine 50.3%

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

      4. *-commutative 50.3%

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

      5. fma-define 50.3%

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

      6. unpow2 50.3%

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

      7. fma-undefine 50.3%

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

      8. *-commutative 50.3%

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

      9. fma-define 50.3%

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

   9. Applied egg-rr 50.3%

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

   10. Taylor expanded in a around 0 91.8%

       \[\leadsto \frac{1}{a \cdot \color{blue}{\frac{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-2 \cdot \frac{b}{a} + c \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -0.5)
     t_0
     (/
      1.0
      (*
       a
       (/
        (+
         (* -2.0 (/ b a))
         (* c (+ (* 1.125 (/ (* a c) (pow b 3.0))) (* 1.5 (/ 1.0 b)))))
        c))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (a * (((-2.0 * (b / a)) + (c * ((1.125 * ((a * c) / pow(b, 3.0))) + (1.5 * (1.0 / b))))) / c));
	}
	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) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-0.5d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (a * ((((-2.0d0) * (b / a)) + (c * ((1.125d0 * ((a * c) / (b ** 3.0d0))) + (1.5d0 * (1.0d0 / b))))) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (a * (((-2.0 * (b / a)) + (c * ((1.125 * ((a * c) / Math.pow(b, 3.0))) + (1.5 * (1.0 / b))))) / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -0.5:
		tmp = t_0
	else:
		tmp = 1.0 / (a * (((-2.0 * (b / a)) + (c * ((1.125 * ((a * c) / math.pow(b, 3.0))) + (1.5 * (1.0 / b))))) / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(a * Float64(Float64(Float64(-2.0 * Float64(b / a)) + Float64(c * Float64(Float64(1.125 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(1.5 * Float64(1.0 / b))))) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = t_0;
	else
		tmp = 1.0 / (a * (((-2.0 * (b / a)) + (c * ((1.125 * ((a * c) / (b ^ 3.0))) + (1.5 * (1.0 / b))))) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], t$95$0, N[(1.0 / N[(a * N[(N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(1.125 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $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)) < -0.5

   1. Initial program 86.2%

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

   if -0.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   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 inf 49.2%

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

      1. clear-num 49.2%

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

      2. inv-pow 49.2%

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

      3. *-commutative 49.2%

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

      4. neg-mul-1 49.2%

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

      5. fma-define 49.2%

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

      6. cancel-sign-sub-inv 49.2%

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

      7. metadata-eval 49.2%

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

      8. *-commutative 49.2%

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

   5. Applied egg-rr 49.2%

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

      1. unpow-1 49.2%

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

      2. associate-/l* 49.2%

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

      3. rem-log-exp 45.2%

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

      4. fma-undefine 45.2%

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

      5. neg-mul-1 45.2%

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

      6. prod-exp 20.2%

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

      7. *-commutative 20.2%

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

      8. prod-exp 45.2%

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

      9. rem-log-exp 49.2%

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

      10. unsub-neg 49.2%

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

   7. Simplified 49.2%

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

      1. flip-- 49.3%

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

      2. add-sqr-sqrt 50.3%

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

      3. fma-undefine 50.3%

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

      4. *-commutative 50.3%

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

      5. fma-define 50.3%

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

      6. unpow2 50.3%

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

      7. fma-undefine 50.3%

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

      8. *-commutative 50.3%

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

      9. fma-define 50.3%

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

   9. Applied egg-rr 50.3%

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

   10. Taylor expanded in c around 0 91.7%

       \[\leadsto \frac{1}{a \cdot \color{blue}{\frac{-2 \cdot \frac{b}{a} + c \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-2 \cdot \frac{b}{a} + c \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -0.5)
     t_0
     (/
      1.0
      (*
       a
       (/
        (+
         (* -2.0 (/ b c))
         (* a (+ (* 1.125 (/ (* a c) (pow b 3.0))) (* 1.5 (/ 1.0 b)))))
        a))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (a * ((1.125 * ((a * c) / pow(b, 3.0))) + (1.5 * (1.0 / b))))) / a));
	}
	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) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-0.5d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (a * ((((-2.0d0) * (b / c)) + (a * ((1.125d0 * ((a * c) / (b ** 3.0d0))) + (1.5d0 * (1.0d0 / b))))) / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (a * ((1.125 * ((a * c) / Math.pow(b, 3.0))) + (1.5 * (1.0 / b))))) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -0.5:
		tmp = t_0
	else:
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (a * ((1.125 * ((a * c) / math.pow(b, 3.0))) + (1.5 * (1.0 / b))))) / a))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(a * Float64(Float64(Float64(-2.0 * Float64(b / c)) + Float64(a * Float64(Float64(1.125 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(1.5 * Float64(1.0 / b))))) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = t_0;
	else
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (a * ((1.125 * ((a * c) / (b ^ 3.0))) + (1.5 * (1.0 / b))))) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], t$95$0, N[(1.0 / N[(a * N[(N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(1.125 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $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)) < -0.5

   1. Initial program 86.2%

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

   if -0.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   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 inf 49.2%

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

      1. clear-num 49.2%

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

      2. inv-pow 49.2%

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

      3. *-commutative 49.2%

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

      4. neg-mul-1 49.2%

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

      5. fma-define 49.2%

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

      6. cancel-sign-sub-inv 49.2%

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

      7. metadata-eval 49.2%

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

      8. *-commutative 49.2%

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

   5. Applied egg-rr 49.2%

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

      1. unpow-1 49.2%

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

      2. associate-/l* 49.2%

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

      3. rem-log-exp 45.2%

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

      4. fma-undefine 45.2%

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

      5. neg-mul-1 45.2%

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

      6. prod-exp 20.2%

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

      7. *-commutative 20.2%

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

      8. prod-exp 45.2%

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

      9. rem-log-exp 49.2%

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

      10. unsub-neg 49.2%

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

   7. Simplified 49.2%

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

      1. flip-- 49.3%

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

      2. add-sqr-sqrt 50.3%

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

      3. fma-undefine 50.3%

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

      4. *-commutative 50.3%

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

      5. fma-define 50.3%

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

      6. unpow2 50.3%

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

      7. fma-undefine 50.3%

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

      8. *-commutative 50.3%

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

      9. fma-define 50.3%

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

   9. Applied egg-rr 50.3%

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

   10. Taylor expanded in a around 0 91.8%

       \[\leadsto \frac{1}{a \cdot \color{blue}{\frac{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.0028:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -0.0028)
     t_0
     (/ 1.0 (* a (/ (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) a))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.0028) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (1.5 * (a / b))) / a));
	}
	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) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-0.0028d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (a * ((((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.0028) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (1.5 * (a / b))) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -0.0028:
		tmp = t_0
	else:
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (1.5 * (a / b))) / a))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.0028)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(a * Float64(Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.0028)
		tmp = t_0;
	else
		tmp = 1.0 / (a * (((-2.0 * (b / c)) + (1.5 * (a / b))) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0028], t$95$0, N[(1.0 / N[(a * N[(N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $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)) < -0.00279999999999999997

   1. Initial program 79.6%

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

   if -0.00279999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 44.1%

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

   3. Taylor expanded in a around inf 43.9%

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

      1. clear-num 43.9%

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

      2. inv-pow 43.9%

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

      3. *-commutative 43.9%

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

      4. neg-mul-1 43.9%

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

      5. fma-define 43.9%

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

      6. cancel-sign-sub-inv 43.9%

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

      7. metadata-eval 43.9%

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

      8. *-commutative 43.9%

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

   5. Applied egg-rr 43.9%

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

      1. unpow-1 43.9%

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

      2. associate-/l* 43.9%

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

      3. rem-log-exp 42.1%

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

      4. fma-undefine 42.1%

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

      5. neg-mul-1 42.1%

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

      6. prod-exp 19.2%

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

      7. *-commutative 19.2%

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

      8. prod-exp 42.1%

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

      9. rem-log-exp 43.9%

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

      10. unsub-neg 43.9%

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

   7. Simplified 43.9%

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

   8. Taylor expanded in a around 0 88.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)}}}{3 \cdot a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in a around inf 54.7%

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

   1. flip-+ 54.7%

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

   2. pow2 54.7%

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

   3. add-sqr-sqrt 55.6%

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

   4. cancel-sign-sub-inv 55.6%

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

   5. metadata-eval 55.6%

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

   6. *-commutative 55.6%

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

   7. cancel-sign-sub-inv 55.6%

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

   8. metadata-eval 55.6%

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

   9. *-commutative 55.6%

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

5. Applied egg-rr 55.6%

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

6. Taylor expanded in b around 0 99.1%

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

7. Final simplification 99.1%

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

Alternative 7: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -3 + \frac{{b}^{2}}{c}\right)}}}{3 \cdot a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in a around inf 54.7%

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

   1. flip-+ 54.7%

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

   2. pow2 54.7%

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

   3. add-sqr-sqrt 55.6%

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

   4. cancel-sign-sub-inv 55.6%

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

   5. metadata-eval 55.6%

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

   6. *-commutative 55.6%

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

   7. cancel-sign-sub-inv 55.6%

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

   8. metadata-eval 55.6%

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

   9. *-commutative 55.6%

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

5. Applied egg-rr 55.6%

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

6. Taylor expanded in b around 0 99.1%

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

7. Taylor expanded in c around inf 99.1%

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

8. Final simplification 99.1%

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

Alternative 8: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)}}}{3 \cdot a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in a around inf 54.7%

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

   1. flip-+ 54.7%

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

   2. pow2 54.7%

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

   3. add-sqr-sqrt 55.6%

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

   4. cancel-sign-sub-inv 55.6%

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

   5. metadata-eval 55.6%

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

   6. *-commutative 55.6%

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

   7. cancel-sign-sub-inv 55.6%

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

   8. metadata-eval 55.6%

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

   9. *-commutative 55.6%

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

5. Applied egg-rr 55.6%

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

6. Taylor expanded in b around 0 99.1%

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

   1. pow1 99.1%

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

   2. associate-*r* 99.3%

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

8. Applied egg-rr 99.3%

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

   1. unpow1 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 9: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} + -3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in a around inf 54.7%

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

   1. flip-+ 54.7%

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

   2. pow2 54.7%

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

   3. add-sqr-sqrt 55.6%

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

   4. cancel-sign-sub-inv 55.6%

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

   5. metadata-eval 55.6%

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

   6. *-commutative 55.6%

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

   7. cancel-sign-sub-inv 55.6%

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

   8. metadata-eval 55.6%

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

   9. *-commutative 55.6%

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

5. Applied egg-rr 55.6%

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

6. Taylor expanded in b around 0 99.1%

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

7. Taylor expanded in a around 0 99.1%

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

8. Final simplification 99.1%

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

Alternative 10: 82.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in a around inf 54.7%

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

   1. clear-num 54.7%

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

   2. inv-pow 54.7%

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

   3. *-commutative 54.7%

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

   4. neg-mul-1 54.7%

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

   5. fma-define 54.7%

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

   6. cancel-sign-sub-inv 54.7%

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

   7. metadata-eval 54.7%

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

   8. *-commutative 54.7%

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

5. Applied egg-rr 54.7%

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

   1. unpow-1 54.7%

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

   2. associate-/l* 54.7%

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

   3. rem-log-exp 50.2%

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

   4. fma-undefine 50.2%

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

   5. neg-mul-1 50.2%

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

   6. prod-exp 24.2%

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

   7. *-commutative 24.2%

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

   8. prod-exp 50.2%

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

   9. rem-log-exp 54.7%

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

   10. unsub-neg 54.7%

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

7. Simplified 54.7%

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

8. Taylor expanded in c around 0 80.9%

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

9. Final simplification 80.9%

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

Alternative 11: 82.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / (a * (((-2.0 * (b / c)) + (1.5 * (a / b))) / 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 = 1.0d0 / (a * ((((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in a around inf 54.7%

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

   1. clear-num 54.7%

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

   2. inv-pow 54.7%

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

   3. *-commutative 54.7%

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

   4. neg-mul-1 54.7%

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

   5. fma-define 54.7%

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

   6. cancel-sign-sub-inv 54.7%

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

   7. metadata-eval 54.7%

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

   8. *-commutative 54.7%

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

5. Applied egg-rr 54.7%

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

   1. unpow-1 54.7%

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

   2. associate-/l* 54.7%

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

   3. rem-log-exp 50.2%

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

   4. fma-undefine 50.2%

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

   5. neg-mul-1 50.2%

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

   6. prod-exp 24.2%

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

   7. *-commutative 24.2%

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

   8. prod-exp 50.2%

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

   9. rem-log-exp 54.7%

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

   10. unsub-neg 54.7%

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

7. Simplified 54.7%

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

8. Taylor expanded in a around 0 81.0%

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

9. Final simplification 81.0%

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

Alternative 12: 65.0% 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 54.9%

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

3. Taylor expanded in b around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. *-commutative 64.4%

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

5. Simplified 64.4%

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

6. Final simplification 64.4%

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

Reproduce

herbie shell --seed 2024053 
(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)))