Cubic critical, narrow range

Percentage Accurate: 55.4% → 92.1%

Time: 15.3s

Alternatives: 9

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.7:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (fma c -3.0 (/ (pow b 2.0) a)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -2.7)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (+
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
        (*
         a
         (+
          (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
          (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * fma(c, -3.0, (pow(b, 2.0) / a));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.7) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * fma(c, -3.0, Float64((b ^ 2.0) / a)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -2.7)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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], -2.7], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.7000000000000002

   1. Initial program 86.7%

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

   3. Taylor expanded in a around inf 86.5%

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

      1. flip-+ 86.4%

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

      2. pow2 86.4%

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

      3. add-sqr-sqrt 87.2%

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

      4. cancel-sign-sub-inv 87.2%

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

      5. metadata-eval 87.2%

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

      6. cancel-sign-sub-inv 87.2%

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

      7. metadata-eval 87.2%

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

   5. Applied egg-rr 87.2%

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

      1. unpow2 87.2%

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

      2. sqr-neg 87.2%

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

      3. unpow2 87.2%

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

      4. +-commutative 87.2%

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

      5. *-commutative 87.2%

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

      6. fma-define 87.2%

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

      7. +-commutative 87.2%

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

      8. *-commutative 87.2%

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

      9. fma-define 87.3%

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

   7. Simplified 87.3%

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

   if -2.7000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.3%

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

   3. Taylor expanded in a around 0 95.8%

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

   4. Taylor expanded in c around 0 95.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.7:\\ \;\;\;\;\frac{\frac{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right) - {b}^{2}}{b + \sqrt{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.7:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (fma c -3.0 (/ (pow b 2.0) a)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -2.7)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (+
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
        (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * fma(c, -3.0, (pow(b, 2.0) / a));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.7) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * fma(c, -3.0, Float64((b ^ 2.0) / a)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -2.7)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0 + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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], -2.7], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.7000000000000002

   1. Initial program 86.7%

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

   3. Taylor expanded in a around inf 86.5%

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

      1. flip-+ 86.4%

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

      2. pow2 86.4%

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

      3. add-sqr-sqrt 87.2%

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

      4. cancel-sign-sub-inv 87.2%

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

      5. metadata-eval 87.2%

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

      6. cancel-sign-sub-inv 87.2%

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

      7. metadata-eval 87.2%

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

   5. Applied egg-rr 87.2%

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

      1. unpow2 87.2%

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

      2. sqr-neg 87.2%

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

      3. unpow2 87.2%

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

      4. +-commutative 87.2%

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

      5. *-commutative 87.2%

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

      6. fma-define 87.2%

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

      7. +-commutative 87.2%

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

      8. *-commutative 87.2%

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

      9. fma-define 87.3%

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

   7. Simplified 87.3%

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

   if -2.7000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.3%

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

   3. Taylor expanded in a around 0 93.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.7:\\ \;\;\;\;\frac{\frac{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right) - {b}^{2}}{b + \sqrt{a \cdot \mathsf{fma}\left(c, -3, \frac{{b}^{2}}{a}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.7:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (+ (/ (pow b 2.0) a) (* c -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -2.7)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (+
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
        (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * ((pow(b, 2.0) / a) + (c * -3.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.7) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
	}
	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 = a * (((b ** 2.0d0) / a) + (c * (-3.0d0)))
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-2.7d0)) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (3.0d0 * a)
    else
        tmp = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * ((Math.pow(b, 2.0) / a) + (c * -3.0));
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.7) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = a * ((math.pow(b, 2.0) / a) + (c * -3.0))
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.7:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (3.0 * a)
	else:
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0)))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(Float64((b ^ 2.0) / a) + Float64(c * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -2.7)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = a * (((b ^ 2.0) / a) + (c * -3.0));
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.7)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	else
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (-0.5625 * ((a * (c ^ 3.0)) / (b ^ 5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision] + N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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], -2.7], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.7000000000000002

   1. Initial program 86.7%

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

   3. Taylor expanded in a around inf 86.5%

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

      1. flip-+ 86.4%

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

      2. pow2 86.4%

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

      3. add-sqr-sqrt 87.2%

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

      4. cancel-sign-sub-inv 87.2%

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

      5. metadata-eval 87.2%

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

      6. cancel-sign-sub-inv 87.2%

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

      7. metadata-eval 87.2%

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

   5. Applied egg-rr 87.2%

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

   if -2.7000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.3%

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

   3. Taylor expanded in a around 0 93.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.7:\\ \;\;\;\;\frac{\frac{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right) - {\left(-b\right)}^{2}}{b + \sqrt{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -2.7:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -2.7)
     t_0
     (+
      (* -0.5 (/ c b))
      (*
       a
       (+
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
        (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))))))))

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 <= -2.7) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
	}
	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 <= (-2.7d0)) then
        tmp = t_0
    else
        tmp = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0)))))
    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 <= -2.7) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0)))));
	}
	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 <= -2.7:
		tmp = t_0
	else:
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0)))))
	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 <= -2.7)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))));
	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 <= -2.7)
		tmp = t_0;
	else
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (-0.5625 * ((a * (c ^ 3.0)) / (b ^ 5.0)))));
	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, -2.7], t$95$0, N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.7000000000000002

   1. Initial program 86.7%

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

   if -2.7000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.3%

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

   3. Taylor expanded in a around 0 93.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.7:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -2.7:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -2.7)
     t_0
     (*
      c
      (+
       (*
        c
        (+
         (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
         (* -0.375 (/ a (pow b 3.0)))))
       (* 0.5 (/ -1.0 b)))))))

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 <= -2.7) {
		tmp = t_0;
	} else {
		tmp = c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / 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) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-2.7d0)) then
        tmp = t_0
    else
        tmp = c * ((c * (((-0.5625d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) + ((-0.375d0) * (a / (b ** 3.0d0))))) + (0.5d0 * ((-1.0d0) / 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) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -2.7) {
		tmp = t_0;
	} else {
		tmp = c * ((c * ((-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + (-0.375 * (a / Math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
	}
	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 <= -2.7:
		tmp = t_0
	else:
		tmp = c * ((c * ((-0.5625 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) + (-0.375 * (a / math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)))
	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 <= -2.7)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))));
	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 <= -2.7)
		tmp = t_0;
	else
		tmp = c * ((c * ((-0.5625 * ((c * (a ^ 2.0)) / (b ^ 5.0))) + (-0.375 * (a / (b ^ 3.0))))) + (0.5 * (-1.0 / 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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.7], t$95$0, N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / 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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.7000000000000002

   1. Initial program 86.7%

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

   if -2.7000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.3%

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

   3. Taylor expanded in c around 0 93.6%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.7:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.22:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \left(a \cdot -0.375\right) \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -0.22)
     t_0
     (/ (fma c -0.5 (* (* a -0.375) (pow (/ c b) 2.0))) b))))

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 <= -0.22) {
		tmp = t_0;
	} else {
		tmp = fma(c, -0.5, ((a * -0.375) * pow((c / b), 2.0))) / b;
	}
	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 <= -0.22)
		tmp = t_0;
	else
		tmp = Float64(fma(c, -0.5, Float64(Float64(a * -0.375) * (Float64(c / b) ^ 2.0))) / b);
	end
	return 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, -0.22], t$95$0, N[(N[(c * -0.5 + N[(N[(a * -0.375), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.220000000000000001

   1. Initial program 83.2%

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

   if -0.220000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.5%

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

   3. Taylor expanded in c around 0 89.6%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]

   4. Taylor expanded in a around inf 89.7%

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

   5. Taylor expanded in b around inf 90.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 90.0%

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

      2. fma-define 90.0%

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

      3. associate-*r/ 90.0%

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

      4. associate-*r* 90.0%

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

      5. associate-/l* 90.0%

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

      6. unpow2 90.0%

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

      7. unpow2 90.0%

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

      8. times-frac 90.0%

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

      9. unpow2 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.22:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \left(a \cdot -0.375\right) \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.22:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -0.22)
     t_0
     (* c (- (* -0.375 (/ (* a c) (pow b 3.0))) (/ 0.5 b))))))

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 <= -0.22) {
		tmp = t_0;
	} else {
		tmp = c * ((-0.375 * ((a * c) / pow(b, 3.0))) - (0.5 / 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) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-0.22d0)) then
        tmp = t_0
    else
        tmp = c * (((-0.375d0) * ((a * c) / (b ** 3.0d0))) - (0.5d0 / 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) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.22) {
		tmp = t_0;
	} else {
		tmp = c * ((-0.375 * ((a * c) / Math.pow(b, 3.0))) - (0.5 / b));
	}
	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 <= -0.22:
		tmp = t_0
	else:
		tmp = c * ((-0.375 * ((a * c) / math.pow(b, 3.0))) - (0.5 / b))
	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 <= -0.22)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 3.0))) - Float64(0.5 / b)));
	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 <= -0.22)
		tmp = t_0;
	else
		tmp = c * ((-0.375 * ((a * c) / (b ^ 3.0))) - (0.5 / 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[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.22], t$95$0, N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.220000000000000001

   1. Initial program 83.2%

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

   if -0.220000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.5%

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

   3. Taylor expanded in c around 0 89.6%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]

   4. Taylor expanded in a around inf 89.7%

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

   5. Taylor expanded in c around 0 89.9%

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

      1. associate-*r/ 89.9%

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

      2. metadata-eval 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.22:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in c around 0 84.4%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]

4. Taylor expanded in a around inf 84.5%

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

5. Taylor expanded in c around 0 84.6%

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

   1. associate-*r/ 84.6%

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

   2. metadata-eval 84.6%

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

7. Simplified 84.6%

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

8. Final simplification 84.6%

   \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right) \]
9. Add Preprocessing

Alternative 9: 64.4% 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 51.0%

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

3. Taylor expanded in b around inf 68.2%

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

   1. associate-*r/ 68.2%

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

   2. *-commutative 68.2%

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

5. Simplified 68.2%

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

6. Final simplification 68.2%

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

Reproduce

herbie shell --seed 2024081 
(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)))