Cubic critical, narrow range

Percentage Accurate: 55.3% → 92.0%

Time: 18.7s

Alternatives: 15

Speedup: 23.2×

• Rival profile vector overflowed, profile may not be complete

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: 55.3% 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: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)\\ \mathbf{if}\;b \leq 0.057:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (pow b 2.0))))
   (if (<= b 0.057)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (+
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
        (*
         a
         (+
          (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
          (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), pow(b, 2.0));
	double tmp;
	if (b <= 0.057) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (b <= 0.057)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.057], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $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[(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 b < 0.0570000000000000021

   1. Initial program 88.6%

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

      1. add-sqr-sqrt 86.7%

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

      2. distribute-rgt-neg-in 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. distribute-rgt-neg-out 86.7%

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

      2. add-sqr-sqrt 88.6%

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

      3. associate-*r* 88.6%

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

      4. flip-+ 87.9%

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

      5. pow2 87.9%

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

      6. add-sqr-sqrt 89.7%

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

      7. cancel-sign-sub-inv 89.7%

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

      8. fma-define 89.2%

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

      9. metadata-eval 89.2%

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

      10. cancel-sign-sub-inv 89.2%

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

      11. fma-define 89.2%

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

      12. metadata-eval 89.2%

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

   6. Applied egg-rr 89.2%

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

      1. unpow2 89.2%

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

      2. sqr-neg 89.2%

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

      3. unpow2 89.2%

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

      4. fma-undefine 89.7%

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

      5. unpow2 89.7%

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

      6. +-commutative 89.7%

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

      7. associate-*r* 89.7%

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

      8. *-commutative 89.7%

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

      9. fma-define 89.7%

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

      10. *-commutative 89.7%

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

      11. fma-undefine 89.7%

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

      12. unpow2 89.7%

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

      13. +-commutative 89.7%

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

      14. associate-*r* 89.7%

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

      15. *-commutative 89.7%

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

      16. fma-define 89.7%

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

      17. *-commutative 89.7%

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

   8. Simplified 89.7%

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

   if 0.0570000000000000021 < b

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in a around 0 94.4%

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

   6. Taylor expanded in c around 0 94.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.057:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)\\ \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{a \cdot -0.5625}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (pow b 2.0))))
   (if (<= b 0.059)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (*
        (pow c 3.0)
        (- (/ (* a -0.5625) (pow b 5.0)) (/ 0.375 (* c (pow b 3.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), pow(b, 2.0));
	double tmp;
	if (b <= 0.059) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * (pow(c, 3.0) * (((a * -0.5625) / pow(b, 5.0)) - (0.375 / (c * pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (b <= 0.059)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64((c ^ 3.0) * Float64(Float64(Float64(a * -0.5625) / (b ^ 5.0)) - Float64(0.375 / Float64(c * (b ^ 3.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.059], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $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[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(N[(a * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.058999999999999997

   1. Initial program 88.6%

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

      1. add-sqr-sqrt 86.7%

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

      2. distribute-rgt-neg-in 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. distribute-rgt-neg-out 86.7%

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

      2. add-sqr-sqrt 88.6%

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

      3. associate-*r* 88.6%

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

      4. flip-+ 87.9%

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

      5. pow2 87.9%

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

      6. add-sqr-sqrt 89.7%

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

      7. cancel-sign-sub-inv 89.7%

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

      8. fma-define 89.2%

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

      9. metadata-eval 89.2%

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

      10. cancel-sign-sub-inv 89.2%

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

      11. fma-define 89.2%

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

      12. metadata-eval 89.2%

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

   6. Applied egg-rr 89.2%

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

      1. unpow2 89.2%

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

      2. sqr-neg 89.2%

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

      3. unpow2 89.2%

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

      4. fma-undefine 89.7%

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

      5. unpow2 89.7%

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

      6. +-commutative 89.7%

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

      7. associate-*r* 89.7%

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

      8. *-commutative 89.7%

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

      9. fma-define 89.7%

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

      10. *-commutative 89.7%

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

      11. fma-undefine 89.7%

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

      12. unpow2 89.7%

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

      13. +-commutative 89.7%

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

      14. associate-*r* 89.7%

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

      15. *-commutative 89.7%

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

      16. fma-define 89.7%

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

      17. *-commutative 89.7%

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

   8. Simplified 89.7%

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

   if 0.058999999999999997 < b

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in a around 0 91.7%

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

   6. Taylor expanded in c around inf 91.7%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \color{blue}{\left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 91.7%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\color{blue}{\frac{-0.5625 \cdot a}{{b}^{5}}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right) \]

      2. associate-*r/ 91.7%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{-0.5625 \cdot a}{{b}^{5}} - \color{blue}{\frac{0.375 \cdot 1}{{b}^{3} \cdot c}}\right)\right) \]

      3. metadata-eval 91.7%

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

      4. *-commutative 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{a \cdot -0.5625}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)\\ \mathbf{if}\;b \leq 0.057:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{a \cdot -0.5625}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* -3.0 (* c a)))))
   (if (<= b 0.057)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (*
        (pow c 3.0)
        (- (/ (* a -0.5625) (pow b 5.0)) (/ 0.375 (* c (pow b 3.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (-3.0 * (c * a)));
	double tmp;
	if (b <= 0.057) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * (pow(c, 3.0) * (((a * -0.5625) / pow(b, 5.0)) - (0.375 / (c * pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(-3.0 * Float64(c * a)))
	tmp = 0.0
	if (b <= 0.057)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64((c ^ 3.0) * Float64(Float64(Float64(a * -0.5625) / (b ^ 5.0)) - Float64(0.375 / Float64(c * (b ^ 3.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.057], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $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[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(N[(a * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0570000000000000021

   1. Initial program 88.6%

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

      1. add-sqr-sqrt 86.7%

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

      2. distribute-rgt-neg-in 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. distribute-rgt-neg-out 86.7%

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

      2. add-sqr-sqrt 88.6%

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

      3. associate-*r* 88.6%

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

      4. flip-+ 87.9%

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

      5. pow2 87.9%

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

      6. add-sqr-sqrt 89.7%

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

      7. cancel-sign-sub-inv 89.7%

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

      8. fma-define 89.2%

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

      9. metadata-eval 89.2%

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

      10. cancel-sign-sub-inv 89.2%

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

      11. fma-define 89.2%

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

      12. metadata-eval 89.2%

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

   6. Applied egg-rr 89.2%

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

   if 0.0570000000000000021 < b

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in a around 0 91.7%

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

   6. Taylor expanded in c around inf 91.7%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \color{blue}{\left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 91.7%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\color{blue}{\frac{-0.5625 \cdot a}{{b}^{5}}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right) \]

      2. associate-*r/ 91.7%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{-0.5625 \cdot a}{{b}^{5}} - \color{blue}{\frac{0.375 \cdot 1}{{b}^{3} \cdot c}}\right)\right) \]

      3. metadata-eval 91.7%

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

      4. *-commutative 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.057:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{a \cdot -0.5625}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\sqrt{a \cdot 3}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{a \cdot -0.5625}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.059)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (pow (sqrt (* a 3.0)) 2.0))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (*
      (pow c 3.0)
      (- (/ (* a -0.5625) (pow b 5.0)) (/ 0.375 (* c (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.059) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / pow(sqrt((a * 3.0)), 2.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * (pow(c, 3.0) * (((a * -0.5625) / pow(b, 5.0)) - (0.375 / (c * pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.059)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / (sqrt(Float64(a * 3.0)) ^ 2.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64((c ^ 3.0) * Float64(Float64(Float64(a * -0.5625) / (b ^ 5.0)) - Float64(0.375 / Float64(c * (b ^ 3.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.059], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[Sqrt[N[(a * 3.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(N[(a * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.058999999999999997

   1. Initial program 88.6%

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

   3. Simplified 88.7%

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

      1. add-sqr-sqrt 88.9%

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

      2. pow2 88.9%

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

   6. Applied egg-rr 88.9%

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

   if 0.058999999999999997 < b

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in a around 0 91.7%

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

   6. Taylor expanded in c around inf 91.7%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \color{blue}{\left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 91.7%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\color{blue}{\frac{-0.5625 \cdot a}{{b}^{5}}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right) \]

      2. associate-*r/ 91.7%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{-0.5625 \cdot a}{{b}^{5}} - \color{blue}{\frac{0.375 \cdot 1}{{b}^{3} \cdot c}}\right)\right) \]

      3. metadata-eval 91.7%

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

      4. *-commutative 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\sqrt{a \cdot 3}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{a \cdot -0.5625}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{a \cdot -0.5625}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.059)
   (/
    (- (sqrt (- (* b b) (* c (* a 3.0)))) b)
    (pow (pow (* a 3.0) 3.0) 0.3333333333333333))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (*
      (pow c 3.0)
      (- (/ (* a -0.5625) (pow b 5.0)) (/ 0.375 (* c (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.059) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / pow(pow((a * 3.0), 3.0), 0.3333333333333333);
	} else {
		tmp = (-0.5 * (c / b)) + (a * (pow(c, 3.0) * (((a * -0.5625) / pow(b, 5.0)) - (0.375 / (c * pow(b, 3.0))))));
	}
	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 <= 0.059d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (((a * 3.0d0) ** 3.0d0) ** 0.3333333333333333d0)
    else
        tmp = ((-0.5d0) * (c / b)) + (a * ((c ** 3.0d0) * (((a * (-0.5625d0)) / (b ** 5.0d0)) - (0.375d0 / (c * (b ** 3.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.059) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / Math.pow(Math.pow((a * 3.0), 3.0), 0.3333333333333333);
	} else {
		tmp = (-0.5 * (c / b)) + (a * (Math.pow(c, 3.0) * (((a * -0.5625) / Math.pow(b, 5.0)) - (0.375 / (c * Math.pow(b, 3.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.059:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / math.pow(math.pow((a * 3.0), 3.0), 0.3333333333333333)
	else:
		tmp = (-0.5 * (c / b)) + (a * (math.pow(c, 3.0) * (((a * -0.5625) / math.pow(b, 5.0)) - (0.375 / (c * math.pow(b, 3.0))))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.059)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / ((Float64(a * 3.0) ^ 3.0) ^ 0.3333333333333333));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64((c ^ 3.0) * Float64(Float64(Float64(a * -0.5625) / (b ^ 5.0)) - Float64(0.375 / Float64(c * (b ^ 3.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.059)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (((a * 3.0) ^ 3.0) ^ 0.3333333333333333);
	else
		tmp = (-0.5 * (c / b)) + (a * ((c ^ 3.0) * (((a * -0.5625) / (b ^ 5.0)) - (0.375 / (c * (b ^ 3.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.059], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[Power[N[(a * 3.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(N[(a * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.058999999999999997

   1. Initial program 88.6%

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

      1. add-cbrt-cube 88.5%

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

      2. pow1/3 88.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left(\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right) \cdot \left(3 \cdot a\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 88.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left(3 \cdot a\right)}^{3}\right)}}^{0.3333333333333333}} \]

   4. Applied egg-rr 88.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]

   if 0.058999999999999997 < b

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in a around 0 91.7%

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

   6. Taylor expanded in c around inf 91.7%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \color{blue}{\left({c}^{3} \cdot \left(-0.5625 \cdot \frac{a}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 91.7%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\color{blue}{\frac{-0.5625 \cdot a}{{b}^{5}}} - 0.375 \cdot \frac{1}{{b}^{3} \cdot c}\right)\right) \]

      2. associate-*r/ 91.7%

         \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{-0.5625 \cdot a}{{b}^{5}} - \color{blue}{\frac{0.375 \cdot 1}{{b}^{3} \cdot c}}\right)\right) \]

      3. metadata-eval 91.7%

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

      4. *-commutative 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left({c}^{3} \cdot \left(\frac{a \cdot -0.5625}{{b}^{5}} - \frac{0.375}{c \cdot {b}^{3}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/
    (- (sqrt (- (* b b) (* c (* a 3.0)))) b)
    (pow (pow (* a 3.0) 3.0) 0.3333333333333333))
   (+ (* -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 <= 0.0875) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / pow(pow((a * 3.0), 3.0), 0.3333333333333333);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	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 <= 0.0875d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (((a * 3.0d0) ** 3.0d0) ** 0.3333333333333333d0)
    else
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / Math.pow(Math.pow((a * 3.0), 3.0), 0.3333333333333333);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.0875:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / math.pow(math.pow((a * 3.0), 3.0), 0.3333333333333333)
	else:
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / ((Float64(a * 3.0) ^ 3.0) ^ 0.3333333333333333));
	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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.0875)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (((a * 3.0) ^ 3.0) ^ 0.3333333333333333);
	else
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[Power[N[(a * 3.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $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 < 0.087499999999999994

   1. Initial program 87.9%

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

      1. add-cbrt-cube 87.9%

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

      2. pow1/3 88.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left(\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right) \cdot \left(3 \cdot a\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 88.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left(3 \cdot a\right)}^{3}\right)}}^{0.3333333333333333}} \]

   4. Applied egg-rr 88.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]

   if 0.087499999999999994 < b

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in a around 0 86.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ 1.0 (* a (/ 3.0 (fma -1.0 b (sqrt (- (pow b 2.0) (* a (* c 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 <= 0.0875) {
		tmp = 1.0 / (a * (3.0 / fma(-1.0, b, sqrt((pow(b, 2.0) - (a * (c * 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 <= 0.0875)
		tmp = Float64(1.0 / Float64(a * Float64(3.0 / fma(-1.0, b, sqrt(Float64((b ^ 2.0) - Float64(a * Float64(c * 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, 0.0875], N[(1.0 / N[(a * N[(3.0 / N[(-1.0 * b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 < 0.087499999999999994

   1. Initial program 87.9%

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

      1. add-cube-cbrt 87.6%

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

      2. pow3 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. rem-cube-cbrt 87.9%

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

      2. clear-num 88.0%

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

      3. inv-pow 88.0%

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

      4. *-commutative 88.0%

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

      5. neg-mul-1 88.0%

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

      6. metadata-eval 88.0%

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

      7. fma-define 88.0%

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

      8. metadata-eval 88.0%

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

      9. pow2 88.0%

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

      10. associate-*l* 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. unpow-1 88.0%

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

      2. associate-/l* 88.0%

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

      3. *-commutative 88.0%

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

      4. associate-*r* 88.1%

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

      5. *-commutative 88.1%

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

   8. Simplified 88.1%

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

   if 0.087499999999999994 < b

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in a around 0 86.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{e^{\log \left(a \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (exp (log (* 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 <= 0.0875) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / exp(log((a * 3.0)));
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	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 <= 0.0875d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / exp(log((a * 3.0d0)))
    else
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / Math.exp(Math.log((a * 3.0)));
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.0875:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / math.exp(math.log((a * 3.0)))
	else:
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / exp(log(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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.0875)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / exp(log((a * 3.0)));
	else
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Exp[N[Log[N[(a * 3.0), $MachinePrecision]], $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 < 0.087499999999999994

   1. Initial program 87.9%

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

      1. sqr-neg 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-*l* 88.0%

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

   3. Simplified 88.0%

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

      1. add-exp-log 88.1%

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

   6. Applied egg-rr 88.1%

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

   if 0.087499999999999994 < b

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in a around 0 86.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{e^{\log \left(a \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (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 <= 0.0875) {
		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 <= 0.0875)
		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, 0.0875], 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 < 0.087499999999999994

   1. Initial program 87.9%

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

   3. Simplified 88.1%

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

   if 0.087499999999999994 < b

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in a around 0 86.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], 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[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.087499999999999994

   1. Initial program 87.9%

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

   3. Simplified 88.1%

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

   if 0.087499999999999994 < b

   1. Initial program 51.9%

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

      1. sqr-neg 51.9%

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

      2. sqr-neg 51.9%

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

      3. associate-*l* 51.9%

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

   3. Simplified 51.9%

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

      1. add-exp-log 51.8%

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

   6. Applied egg-rr 51.8%

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

   7. Taylor expanded in b around inf 86.1%

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

      1. +-commutative 86.1%

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

      2. fma-define 86.1%

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

      3. associate-/l* 86.1%

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

      4. unpow2 86.1%

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

      5. unpow2 86.1%

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

      6. times-frac 86.1%

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

      7. unpow1 86.1%

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

      8. pow-plus 86.1%

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

      9. metadata-eval 86.1%

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

      10. *-commutative 86.1%

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

   9. Simplified 86.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0875) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0875)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.375, Float64(a * (Float64(c / b) ^ 2.0)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0875], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.087499999999999994

   1. Initial program 87.9%

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

      1. sqr-neg 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-*l* 88.0%

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

   3. Simplified 88.0%

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

   if 0.087499999999999994 < b

   1. Initial program 51.9%

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

      1. sqr-neg 51.9%

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

      2. sqr-neg 51.9%

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

      3. associate-*l* 51.9%

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

   3. Simplified 51.9%

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

      1. add-exp-log 51.8%

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

   6. Applied egg-rr 51.8%

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

   7. Taylor expanded in b around inf 86.1%

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

      1. +-commutative 86.1%

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

      2. fma-define 86.1%

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

      3. associate-/l* 86.1%

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

      4. unpow2 86.1%

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

      5. unpow2 86.1%

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

      6. times-frac 86.1%

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

      7. unpow1 86.1%

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

      8. pow-plus 86.1%

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

      9. metadata-eval 86.1%

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

      10. *-commutative 86.1%

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

   9. Simplified 86.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0875)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (* 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 <= 0.0875) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.0875d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    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 <= 0.0875) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} 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 <= 0.0875:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	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 <= 0.0875)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	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 <= 0.0875)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	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, 0.0875], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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 < 0.087499999999999994

   1. Initial program 87.9%

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

      1. sqr-neg 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-*l* 88.0%

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

   3. Simplified 88.0%

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

   if 0.087499999999999994 < b

   1. Initial program 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in c around 0 86.0%

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

      1. associate-/l* 86.0%

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

      2. associate-*r/ 86.0%

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

      3. metadata-eval 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0875:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.6% 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 55.4%

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

3. Simplified 55.4%

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

5. Taylor expanded in c around 0 82.6%

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

   1. associate-/l* 82.6%

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

   2. associate-*r/ 82.6%

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

   3. metadata-eval 82.6%

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

7. Simplified 82.6%

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

Alternative 14: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

3. Simplified 55.4%

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

5. Taylor expanded in b around inf 64.9%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Alternative 15: 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 55.4%

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

   1. add-sqr-sqrt 54.0%

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

   2. distribute-rgt-neg-in 54.0%

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

4. Applied egg-rr 54.0%

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

5. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
6. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

7. Simplified 3.2%

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

8. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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)))