Cubic critical, narrow range

Percentage Accurate: 55.6% → 92.3%

Time: 18.3s

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: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ \mathbf{if}\;b \leq 0.108:\\ \;\;\;\;\frac{\frac{{b}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))))
   (if (<= b 0.108)
     (/ (/ (- (pow b 2.0) t_0) (- (- b) (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double tmp;
	if (b <= 0.108) {
		tmp = ((pow(b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (b <= 0.108)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.108], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.107999999999999999

   1. Initial program 87.2%

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

   3. Taylor expanded in a around 0 87.1%

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

      1. flip-+ 87.6%

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

      2. pow2 87.6%

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

      3. add-sqr-sqrt 89.3%

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

      4. cancel-sign-sub-inv 89.3%

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

      5. fma-def 88.7%

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

      6. metadata-eval 88.7%

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

      7. cancel-sign-sub-inv 88.7%

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

      8. fma-def 88.7%

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

      9. metadata-eval 88.7%

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

   5. Applied egg-rr 88.7%

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

      1. unpow2 88.7%

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

      2. sqr-neg 88.7%

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

      3. unpow2 88.7%

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

      4. fma-udef 89.3%

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

      5. unpow2 89.3%

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

      6. +-commutative 89.3%

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

      7. *-commutative 89.3%

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

      8. associate-*r* 89.2%

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

      9. fma-def 89.2%

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

      10. fma-udef 89.2%

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

      11. unpow2 89.2%

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

      12. +-commutative 89.2%

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

      13. *-commutative 89.2%

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

      14. associate-*r* 89.2%

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

      15. fma-def 89.2%

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

   7. Simplified 89.2%

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

   if 0.107999999999999999 < b

   1. Initial program 53.0%

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

   3. Taylor expanded in b around inf 94.0%

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

   4. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-out 94.0%

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

      2. associate-*r* 94.0%

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

      3. *-commutative 94.0%

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

      4. times-frac 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.108:\\ \;\;\;\;\frac{\frac{{b}^{2} - \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.0)
     (/ (/ (- (pow b 2.0) t_0) (- (- b) (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -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 t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.0) {
		tmp = ((pow(b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.0)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + 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_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.0], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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 3 a) c)))) (*.f64 3 a)) < -4

   1. Initial program 87.3%

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

   3. Taylor expanded in a around 0 87.4%

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

      1. flip-+ 87.4%

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

      2. pow2 87.4%

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

      3. add-sqr-sqrt 89.5%

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

      4. cancel-sign-sub-inv 89.5%

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

      5. fma-def 88.7%

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

      6. metadata-eval 88.7%

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

      7. cancel-sign-sub-inv 88.7%

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

      8. fma-def 88.7%

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

      9. metadata-eval 88.7%

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

   5. Applied egg-rr 88.7%

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

      1. unpow2 88.7%

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

      2. sqr-neg 88.7%

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

      3. unpow2 88.7%

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

      4. fma-udef 89.6%

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

      5. unpow2 89.6%

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

      6. +-commutative 89.6%

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

      7. *-commutative 89.6%

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

      8. associate-*r* 89.5%

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

      9. fma-def 89.5%

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

      10. fma-udef 89.5%

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

      11. unpow2 89.5%

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

      12. +-commutative 89.5%

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

      13. *-commutative 89.5%

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

      14. associate-*r* 89.5%

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

      15. fma-def 89.5%

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

   7. Simplified 89.5%

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.7%

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

   3. Taylor expanded in b around inf 91.5%

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

4. Final simplification 91.3%

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

Alternative 3: 89.8% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* -3.0 (* a c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.0)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -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 t_0 = fma(b, b, (-3.0 * (a * c)));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.0) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(-3.0 * Float64(a * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + 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_] := Block[{t$95$0 = N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.0], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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 3 a) c)))) (*.f64 3 a)) < -4

   1. Initial program 87.3%

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

   3. Taylor expanded in a around 0 87.4%

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

      1. flip-+ 87.4%

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

      2. pow2 87.4%

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

      3. add-sqr-sqrt 89.5%

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

      4. cancel-sign-sub-inv 89.5%

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

      5. fma-def 88.7%

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

      6. metadata-eval 88.7%

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

      7. cancel-sign-sub-inv 88.7%

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

      8. fma-def 88.7%

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

      9. metadata-eval 88.7%

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

   5. Applied egg-rr 88.7%

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.7%

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

   3. Taylor expanded in b around inf 91.5%

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

4. Final simplification 91.2%

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

Alternative 4: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* -3.0 (* a c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4.0)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) (sqrt t_0))) (* a 3.0))
     (/
      (+
       (* -1.6875 (/ (pow (* a c) 3.0) (pow b 5.0)))
       (+
        (* -1.5 (/ (* a c) b))
        (* -1.125 (* (pow (* a c) 2.0) (pow b -3.0)))))
      (* a 3.0)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (-3.0 * (a * c)));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4.0) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = ((-1.6875 * (pow((a * c), 3.0) / pow(b, 5.0))) + ((-1.5 * ((a * c) / b)) + (-1.125 * (pow((a * c), 2.0) * pow(b, -3.0))))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(-3.0 * Float64(a * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-1.6875 * Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0))) + Float64(Float64(-1.5 * Float64(Float64(a * c) / b)) + Float64(-1.125 * Float64((Float64(a * c) ^ 2.0) * (b ^ -3.0))))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4.0], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.6875 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4

   1. Initial program 87.3%

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

   3. Taylor expanded in a around 0 87.4%

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

      1. flip-+ 87.4%

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

      2. pow2 87.4%

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

      3. add-sqr-sqrt 89.5%

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

      4. cancel-sign-sub-inv 89.5%

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

      5. fma-def 88.7%

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

      6. metadata-eval 88.7%

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

      7. cancel-sign-sub-inv 88.7%

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

      8. fma-def 88.7%

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

      9. metadata-eval 88.7%

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

   5. Applied egg-rr 88.7%

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.7%

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

   3. Taylor expanded in b around inf 91.1%

      \[\leadsto \frac{\color{blue}{-1.6875 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 91.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({a}^{3} \cdot {c}^{3}\right)\right)}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

      2. expm1-udef 89.6%

         \[\leadsto \frac{-1.6875 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({a}^{3} \cdot {c}^{3}\right)} - 1}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

      3. pow-prod-down 89.6%

         \[\leadsto \frac{-1.6875 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(a \cdot c\right)}^{3}}\right)} - 1}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

   5. Applied egg-rr 89.6%

      \[\leadsto \frac{-1.6875 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{3}\right)} - 1}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]
   6. Step-by-step derivation

      1. expm1-def 91.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot c\right)}^{3}\right)\right)}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

      2. expm1-log1p 91.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{3}}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

   7. Simplified 91.1%

      \[\leadsto \frac{-1.6875 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{3}}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 91.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}\right)}{3 \cdot a} \]

      2. expm1-udef 81.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)} - 1\right)}\right)}{3 \cdot a} \]

      3. div-inv 81.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}\right)} - 1\right)\right)}{3 \cdot a} \]

      4. pow-prod-down 81.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}\right)} - 1\right)\right)}{3 \cdot a} \]

      5. pow-flip 81.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}\right)} - 1\right)\right)}{3 \cdot a} \]

      6. metadata-eval 81.1%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}\right)} - 1\right)\right)}{3 \cdot a} \]

   9. Applied egg-rr 81.1%

      \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)} - 1\right)}\right)}{3 \cdot a} \]
   10. Step-by-step derivation

       1. expm1-def 91.1%

          \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)\right)}\right)}{3 \cdot a} \]

       2. expm1-log1p 91.1%

          \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)}\right)}{3 \cdot a} \]

   11. Simplified 91.1%

       \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)}\right)}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \left({\left(a \cdot c\right)}^{2} \cdot {b}^{-3}\right)\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;b \leq 32:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* -3.0 (* a c)))))
   (if (<= b 32.0)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) (sqrt t_0))) (* a 3.0))
     (+ (* -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 t_0 = fma(b, b, (-3.0 * (a * c)));
	double tmp;
	if (b <= 32.0) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 3.0);
	} 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)
	t_0 = fma(b, b, Float64(-3.0 * Float64(a * c)))
	tmp = 0.0
	if (b <= 32.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 3.0));
	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_] := Block[{t$95$0 = N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 32.0], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $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 < 32

   1. Initial program 77.0%

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

   3. Taylor expanded in a around 0 77.0%

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

      1. flip-+ 77.2%

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

      2. pow2 77.2%

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

      3. add-sqr-sqrt 78.9%

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

      4. cancel-sign-sub-inv 78.9%

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

      5. fma-def 78.5%

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

      6. metadata-eval 78.5%

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

      7. cancel-sign-sub-inv 78.5%

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

      8. fma-def 78.5%

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

      9. metadata-eval 78.5%

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

   5. Applied egg-rr 78.5%

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

   if 32 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in b around inf 89.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 32:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5.5e-5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.5e-5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5.5e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -5.5e-5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $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 3 a) c)))) (*.f64 3 a)) < -5.5000000000000002e-5

   1. Initial program 75.4%

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

      1. +-commutative 75.4%

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

      2. sqr-neg 75.4%

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

      3. unsub-neg 75.4%

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

      4. div-sub 74.5%

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

      5. --rgt-identity 74.5%

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

      6. div-sub 75.4%

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

   3. Simplified 75.5%

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

   if -5.5000000000000002e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 41.8%

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

   3. Taylor expanded in b around inf 76.5%

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

      1. associate-*r/ 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 32:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(a \cdot -3, c, 3 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 32.0)
   (/
    (-
     (sqrt (+ (fma b b (* a (* c -3.0))) (fma (* a -3.0) c (* 3.0 (* a c)))))
     b)
    (* a 3.0))
   (+ (* -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 <= 32.0) {
		tmp = (sqrt((fma(b, b, (a * (c * -3.0))) + fma((a * -3.0), c, (3.0 * (a * c))))) - b) / (a * 3.0);
	} 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 <= 32.0)
		tmp = Float64(Float64(sqrt(Float64(fma(b, b, Float64(a * Float64(c * -3.0))) + fma(Float64(a * -3.0), c, Float64(3.0 * Float64(a * c))))) - b) / Float64(a * 3.0));
	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, 32.0], N[(N[(N[Sqrt[N[(N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -3.0), $MachinePrecision] * c + N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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 < 32

   1. Initial program 77.0%

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

      1. *-commutative 77.0%

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

      2. prod-diff 77.3%

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

      3. distribute-lft-neg-in 77.3%

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

      4. *-commutative 77.3%

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

      5. distribute-rgt-neg-in 77.3%

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

      6. metadata-eval 77.3%

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

      7. associate-*r* 77.3%

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

      8. *-commutative 77.3%

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

      9. *-commutative 77.3%

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

      10. distribute-rgt-neg-in 77.3%

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

      11. metadata-eval 77.3%

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

      12. associate-*l* 77.3%

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

   4. Applied egg-rr 77.3%

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

   if 32 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in b around inf 89.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 32:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(a \cdot -3, c, 3 \cdot \left(a \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 32:\\ \;\;\;\;\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 32.0)
   (* (fma -1.0 b (sqrt (fma b b (* -3.0 (* a c))))) (/ 1.0 (* a 3.0)))
   (+ (* -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 <= 32.0) {
		tmp = fma(-1.0, b, sqrt(fma(b, b, (-3.0 * (a * c))))) * (1.0 / (a * 3.0));
	} 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 <= 32.0)
		tmp = Float64(fma(-1.0, b, sqrt(fma(b, b, Float64(-3.0 * Float64(a * c))))) * Float64(1.0 / Float64(a * 3.0)));
	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, 32.0], N[(N[(-1.0 * b + N[Sqrt[N[(b * b + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $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 < 32

   1. Initial program 77.0%

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

   3. Taylor expanded in a around 0 77.0%

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

      1. div-inv 77.0%

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

      2. neg-mul-1 77.0%

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

      3. fma-def 77.0%

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

      4. cancel-sign-sub-inv 77.0%

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

      5. fma-def 77.3%

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

      6. metadata-eval 77.3%

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

      7. *-commutative 77.3%

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

   5. Applied egg-rr 77.3%

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

   if 32 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in b around inf 89.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 32:\\ \;\;\;\;\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5.5e-5)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.5e-5) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -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 (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-5.5d-5)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.5e-5) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.5e-5:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5.5e-5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.5e-5)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -5.5e-5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $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 3 a) c)))) (*.f64 3 a)) < -5.5000000000000002e-5

   1. Initial program 75.4%

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

   3. Taylor expanded in a around 0 75.4%

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

   if -5.5000000000000002e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 41.8%

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

   3. Taylor expanded in b around inf 76.5%

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

      1. associate-*r/ 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 32:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 32.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (+ (* -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 <= 32.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} 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 <= 32.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	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, 32.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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 < 32

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

      2. sqr-neg 77.0%

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

      3. unsub-neg 77.0%

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

      4. div-sub 76.2%

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

      5. --rgt-identity 76.2%

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

      6. div-sub 77.0%

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

   3. Simplified 77.3%

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

   if 32 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in b around inf 89.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 32:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 56.0%

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

3. Taylor expanded in b around inf 64.5%

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

   1. associate-*r/ 64.5%

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

   2. associate-/l* 64.4%

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

5. Simplified 64.4%

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

   1. associate-/r/ 64.4%

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

7. Applied egg-rr 64.4%

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

8. Final simplification 64.4%

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

Alternative 12: 64.2% accurate, 23.2× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ -0.5 (/ b c)))

C

double code(double a, double b, double c) {
	return -0.5 / (b / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) / (b / c)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 / (b / c);
}

Python

def code(a, b, c):
	return -0.5 / (b / c)

Julia

function code(a, b, c)
	return Float64(-0.5 / Float64(b / c))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in b around inf 64.5%

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

   1. associate-*r/ 64.5%

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

   2. associate-/l* 64.4%

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

5. Simplified 64.4%

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

6. Final simplification 64.4%

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

Alternative 13: 64.3% 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 56.0%

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

3. Taylor expanded in b around inf 64.5%

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

   1. associate-*r/ 64.5%

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

5. Simplified 64.5%

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

6. Final simplification 64.5%

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

Alternative 14: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in b around inf 64.5%

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

   1. associate-*r/ 64.5%

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

   2. associate-/l* 64.4%

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

5. Simplified 64.4%

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

   1. expm1-log1p-u 59.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{\frac{b}{c}}\right)\right)} \]

   2. expm1-udef 48.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\frac{b}{c}}\right)} - 1} \]

   3. associate-/r/ 48.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.5}{b} \cdot c}\right)} - 1 \]

7. Applied egg-rr 48.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{b} \cdot c\right)} - 1} \]

8. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{1} - 1 \]

9. Final simplification 3.2%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(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)))