Cubic critical, narrow range

Percentage Accurate: 55.5% → 91.6%

Time: 20.7s

Alternatives: 14

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: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} \cdot \mathsf{fma}\left(a, c \cdot \frac{-3}{{b}^{2}}, 1\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (pow b 2.0) (fma a (* c (/ -3.0 (pow b 2.0))) 1.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -18.0)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (/
      (+
       (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
       (+
        (* c -0.5)
        (+
         (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))
         (*
          -0.16666666666666666
          (* (pow (* a c) 4.0) (/ 6.328125 (* a (pow b 6.0))))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) * fma(a, (c * (-3.0 / pow(b, 2.0))), 1.0);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -18.0) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))) + (-0.16666666666666666 * (pow((a * c), 4.0) * (6.328125 / (a * pow(b, 6.0)))))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) * fma(a, Float64(c * Float64(-3.0 / (b ^ 2.0))), 1.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -18.0)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) + Float64(-0.16666666666666666 * Float64((Float64(a * c) ^ 4.0) * Float64(6.328125 / Float64(a * (b ^ 6.0)))))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] * N[(a * N[(c * N[(-3.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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], -18.0], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -18

   1. Initial program 86.4%

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

   3. Taylor expanded in b around inf 86.4%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

      3. associate-*r* 86.2%

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

   5. Simplified 86.2%

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

      1. flip-+ 86.1%

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

      2. pow2 86.1%

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

      3. add-sqr-sqrt 87.9%

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

      4. +-commutative 87.9%

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

      5. associate-/l* 88.1%

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

      6. fma-define 88.1%

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

      7. +-commutative 88.1%

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

      8. associate-/l* 88.2%

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

   7. Applied egg-rr 88.2%

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

      1. unpow2 88.2%

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

      2. sqr-neg 88.2%

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

      3. unpow2 88.2%

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

      4. fma-undefine 88.1%

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

      5. fma-undefine 88.2%

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

      6. associate-/l* 88.2%

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

      7. fma-undefine 88.2%

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

      8. fma-undefine 88.2%

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

      9. associate-/l* 88.2%

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

   9. Simplified 88.2%

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

   if -18 < (/.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 51.1%

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

   3. Taylor expanded in b around inf 93.7%

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

      1. div-inv 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\left(\left(1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right) \cdot \frac{1}{a \cdot {b}^{6}}\right)}\right)\right)}{b} \]

      2. distribute-rgt-out 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left(\color{blue}{\left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)\right)} \cdot \frac{1}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

      3. pow-prod-down 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left(\left(\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(1.265625 + 5.0625\right)\right) \cdot \frac{1}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

      4. metadata-eval 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left(\left({\left(a \cdot c\right)}^{4} \cdot \color{blue}{6.328125}\right) \cdot \frac{1}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

   5. Applied egg-rr 93.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\left(\left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right) \cdot \frac{1}{a \cdot {b}^{6}}\right)}\right)\right)}{b} \]
   6. Step-by-step derivation

      1. associate-*r/ 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\frac{\left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right) \cdot 1}{a \cdot {b}^{6}}}\right)\right)}{b} \]

      2. *-rgt-identity 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4} \cdot 6.328125}}{a \cdot {b}^{6}}\right)\right)}{b} \]

      3. associate-/l* 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)}\right)\right)}{b} \]

   7. Simplified 93.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18:\\ \;\;\;\;\frac{\frac{{b}^{2} \cdot \mathsf{fma}\left(a, c \cdot \frac{-3}{{b}^{2}}, 1\right) - {b}^{2}}{b + \sqrt{{b}^{2} \cdot \mathsf{fma}\left(a, c \cdot \frac{-3}{{b}^{2}}, 1\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{b}^{2}}, \sqrt{\mathsf{fma}\left(a, \frac{c \cdot -3}{{b}^{2}}, 1\right)}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -18.0)
   (/
    (fma
     (sqrt (pow b 2.0))
     (sqrt (fma a (/ (* c -3.0) (pow b 2.0)) 1.0))
     (- b))
    (* 3.0 a))
   (/
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
     (+
      (* c -0.5)
      (+
       (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))
       (*
        -0.16666666666666666
        (* (pow (* a c) 4.0) (/ 6.328125 (* a (pow b 6.0))))))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -18.0) {
		tmp = fma(sqrt(pow(b, 2.0)), sqrt(fma(a, ((c * -3.0) / pow(b, 2.0)), 1.0)), -b) / (3.0 * a);
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))) + (-0.16666666666666666 * (pow((a * c), 4.0) * (6.328125 / (a * pow(b, 6.0)))))))) / b;
	}
	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)) <= -18.0)
		tmp = Float64(fma(sqrt((b ^ 2.0)), sqrt(fma(a, Float64(Float64(c * -3.0) / (b ^ 2.0)), 1.0)), Float64(-b)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) + Float64(-0.16666666666666666 * Float64((Float64(a * c) ^ 4.0) * Float64(6.328125 / Float64(a * (b ^ 6.0)))))))) / b);
	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], -18.0], N[(N[(N[Sqrt[N[Power[b, 2.0], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(a * N[(N[(c * -3.0), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -18

   1. Initial program 86.4%

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

   3. Taylor expanded in b around inf 86.4%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

      3. associate-*r* 86.2%

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

   5. Simplified 86.2%

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

      1. +-commutative 86.2%

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

      2. sqrt-prod 85.9%

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

      3. fma-define 87.6%

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

      4. +-commutative 87.6%

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

      5. associate-/l* 87.7%

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

      6. fma-define 87.7%

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

   7. Applied egg-rr 87.7%

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

   if -18 < (/.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 51.1%

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

   3. Taylor expanded in b around inf 93.7%

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

      1. div-inv 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\left(\left(1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right) \cdot \frac{1}{a \cdot {b}^{6}}\right)}\right)\right)}{b} \]

      2. distribute-rgt-out 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left(\color{blue}{\left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)\right)} \cdot \frac{1}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

      3. pow-prod-down 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left(\left(\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(1.265625 + 5.0625\right)\right) \cdot \frac{1}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

      4. metadata-eval 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left(\left({\left(a \cdot c\right)}^{4} \cdot \color{blue}{6.328125}\right) \cdot \frac{1}{a \cdot {b}^{6}}\right)\right)\right)}{b} \]

   5. Applied egg-rr 93.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\left(\left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right) \cdot \frac{1}{a \cdot {b}^{6}}\right)}\right)\right)}{b} \]
   6. Step-by-step derivation

      1. associate-*r/ 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\frac{\left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right) \cdot 1}{a \cdot {b}^{6}}}\right)\right)}{b} \]

      2. *-rgt-identity 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4} \cdot 6.328125}}{a \cdot {b}^{6}}\right)\right)}{b} \]

      3. associate-/l* 93.7%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)}\right)\right)}{b} \]

   7. Simplified 93.7%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{b}^{2}}, \sqrt{\mathsf{fma}\left(a, \frac{c \cdot -3}{{b}^{2}}, 1\right)}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{b}^{2}}, \sqrt{\mathsf{fma}\left(a, \frac{c \cdot -3}{{b}^{2}}, 1\right)}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -18.0)
   (/
    (fma
     (sqrt (pow b 2.0))
     (sqrt (fma a (/ (* c -3.0) (pow b 2.0)) 1.0))
     (- b))
    (* 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 tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -18.0) {
		tmp = fma(sqrt(pow(b, 2.0)), sqrt(fma(a, ((c * -3.0) / pow(b, 2.0)), 1.0)), -b) / (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)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -18.0)
		tmp = Float64(fma(sqrt((b ^ 2.0)), sqrt(fma(a, Float64(Float64(c * -3.0) / (b ^ 2.0)), 1.0)), Float64(-b)) / 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_] := 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], -18.0], N[(N[(N[Sqrt[N[Power[b, 2.0], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(a * N[(N[(c * -3.0), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $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)) < -18

   1. Initial program 86.4%

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

   3. Taylor expanded in b around inf 86.4%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

      3. associate-*r* 86.2%

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

   5. Simplified 86.2%

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

      1. +-commutative 86.2%

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

      2. sqrt-prod 85.9%

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

      3. fma-define 87.6%

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

      4. +-commutative 87.6%

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

      5. associate-/l* 87.7%

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

      6. fma-define 87.7%

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

   7. Applied egg-rr 87.7%

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

   if -18 < (/.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 51.1%

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

   3. Taylor expanded in a around 0 93.7%

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

   4. Taylor expanded in c around 0 93.7%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -18:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{b}^{2}}, \sqrt{\mathsf{fma}\left(a, \frac{c \cdot -3}{{b}^{2}}, 1\right)}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.45)
   (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 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 tmp;
	if (b <= 1.45) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (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) :: tmp
    if (b <= 1.45d0) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (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 tmp;
	if (b <= 1.45) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (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):
	tmp = 0
	if b <= 1.45:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (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)
	tmp = 0.0
	if (b <= 1.45)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / 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)
	tmp = 0.0;
	if (b <= 1.45)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (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_] := If[LessEqual[b, 1.45], 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], 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 b < 1.44999999999999996

   1. Initial program 84.2%

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

   if 1.44999999999999996 < b

   1. Initial program 49.7%

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

   3. Taylor expanded in a around 0 92.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.36:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.36)
   (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
   (*
    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 tmp;
	if (b <= 1.36) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} 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) :: tmp
    if (b <= 1.36d0) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    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 tmp;
	if (b <= 1.36) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} 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):
	tmp = 0
	if b <= 1.36:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	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)
	tmp = 0.0
	if (b <= 1.36)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	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)
	tmp = 0.0;
	if (b <= 1.36)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	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_] := If[LessEqual[b, 1.36], 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], 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 b < 1.3600000000000001

   1. Initial program 84.2%

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

   if 1.3600000000000001 < b

   1. Initial program 49.7%

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

   3. Taylor expanded in c around 0 91.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.36:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 110:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 110.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 110.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 110.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 110.0], 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[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 110

   1. Initial program 78.8%

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

      1. /-rgt-identity 78.8%

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

      2. metadata-eval 78.8%

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

   3. Simplified 79.0%

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

   if 110 < b

   1. Initial program 45.2%

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

   3. Taylor expanded in a around 0 90.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 110:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 105.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 105.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 105.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 105.0], 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[(N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 105

   1. Initial program 78.8%

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

      1. /-rgt-identity 78.8%

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

      2. metadata-eval 78.8%

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

   3. Simplified 79.0%

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

   if 105 < b

   1. Initial program 45.2%

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

   3. Taylor expanded in b around inf 90.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 105.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 105.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 105.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 105.0], 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[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 105

   1. Initial program 78.8%

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

      1. /-rgt-identity 78.8%

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

      2. metadata-eval 78.8%

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

   3. Simplified 79.0%

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

   if 105 < b

   1. Initial program 45.2%

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

   3. Taylor expanded in a around 0 90.2%

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

   4. Taylor expanded in c around 0 90.0%

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

      1. associate-/l* 90.0%

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

      2. associate-*r/ 90.0%

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

      3. metadata-eval 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 105:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 105.0)
   (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) (* 3.0 a))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 105.0) {
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} 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) :: tmp
    if (b <= 105.0d0) then
        tmp = (sqrt(((b * b) - (a * (3.0d0 * c)))) - b) / (3.0d0 * a)
    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 tmp;
	if (b <= 105.0) {
		tmp = (Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 105.0:
		tmp = (math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a)
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 105.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 105.0)
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 105.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 105

   1. Initial program 78.8%

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

   3. Taylor expanded in a around 0 78.7%

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

      1. *-commutative 78.7%

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

      2. metadata-eval 78.7%

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

      3. distribute-rgt-neg-in 78.7%

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

      4. associate-*r* 78.8%

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

      5. distribute-rgt-neg-in 78.8%

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

      6. distribute-rgt-neg-in 78.8%

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

      7. metadata-eval 78.8%

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

   5. Simplified 78.8%

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

   if 105 < b

   1. Initial program 45.2%

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

   3. Taylor expanded in a around 0 90.2%

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

   4. Taylor expanded in c around 0 90.0%

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

      1. associate-/l* 90.0%

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

      2. associate-*r/ 90.0%

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

      3. metadata-eval 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 105:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 120:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 120.0)
   (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 120.0) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} 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) :: tmp
    if (b <= 120.0d0) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    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 tmp;
	if (b <= 120.0) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 120.0:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 120.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 120.0)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 120.0], 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], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 120

   1. Initial program 78.8%

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

   if 120 < b

   1. Initial program 45.2%

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

   3. Taylor expanded in a around 0 90.2%

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

   4. Taylor expanded in c around 0 90.0%

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

      1. associate-/l* 90.0%

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

      2. associate-*r/ 90.0%

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

      3. metadata-eval 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 120:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.2% accurate, 1.0× speedup

Math

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

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(a * Float64(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[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in a around 0 83.1%

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

4. Taylor expanded in c around 0 82.9%

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

   1. associate-/l* 82.9%

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

   2. associate-*r/ 82.9%

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

   3. metadata-eval 82.9%

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

6. Simplified 82.9%

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

7. Final simplification 82.9%

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

Alternative 12: 64.2% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b} \end{array} \]

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(c * Float64(-0.5 / b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in a around 0 83.1%

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

4. Taylor expanded in c around 0 65.7%

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

   1. associate-*r/ 65.7%

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

   2. *-commutative 65.7%

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

   3. associate-/l* 65.7%

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

6. Simplified 65.7%

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

7. Final simplification 65.7%

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

Alternative 13: 64.2% 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 54.1%

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

3. Taylor expanded in b around inf 65.7%

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

   1. associate-*r/ 65.7%

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

   2. *-commutative 65.7%

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

5. Simplified 65.7%

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

6. Final simplification 65.7%

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

Alternative 14: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in b around inf 54.1%

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

   1. associate-*r/ 54.1%

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

   2. *-commutative 54.1%

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

   3. associate-*r* 54.1%

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

5. Simplified 54.1%

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

   1. add-cube-cbrt 54.1%

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

   2. pow3 54.1%

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

7. Applied egg-rr 54.1%

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

8. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{0.3333333333333333}\right)}^{3} \cdot \left(b + -1 \cdot b\right)}{a}} \]
9. Step-by-step derivation

   1. rem-cube-cbrt 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

10. Simplified 3.2%

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

11. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024079 
(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)))