Cubic critical, narrow range

Percentage Accurate: 54.9% → 97.0%

Time: 16.1s

Alternatives: 13

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left({b}^{2} - {\left(-b\right)}^{2}\right) - \left(a \cdot c\right) \cdot 3}{b + \sqrt{{b}^{2} + \left(a \cdot c\right) \cdot -3}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (- (- (pow b 2.0) (pow (- b) 2.0)) (* (* a c) 3.0))
   (+ b (sqrt (+ (pow b 2.0) (* (* a c) -3.0)))))
  (pow (pow (* a 3.0) 3.0) 0.3333333333333333)))

C

double code(double a, double b, double c) {
	return (((pow(b, 2.0) - pow(-b, 2.0)) - ((a * c) * 3.0)) / (b + sqrt((pow(b, 2.0) + ((a * c) * -3.0))))) / pow(pow((a * 3.0), 3.0), 0.3333333333333333);
}

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 ** 2.0d0) - (-b ** 2.0d0)) - ((a * c) * 3.0d0)) / (b + sqrt(((b ** 2.0d0) + ((a * c) * (-3.0d0)))))) / (((a * 3.0d0) ** 3.0d0) ** 0.3333333333333333d0)
end function

Java

public static double code(double a, double b, double c) {
	return (((Math.pow(b, 2.0) - Math.pow(-b, 2.0)) - ((a * c) * 3.0)) / (b + Math.sqrt((Math.pow(b, 2.0) + ((a * c) * -3.0))))) / Math.pow(Math.pow((a * 3.0), 3.0), 0.3333333333333333);
}

Python

def code(a, b, c):
	return (((math.pow(b, 2.0) - math.pow(-b, 2.0)) - ((a * c) * 3.0)) / (b + math.sqrt((math.pow(b, 2.0) + ((a * c) * -3.0))))) / math.pow(math.pow((a * 3.0), 3.0), 0.3333333333333333)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64((b ^ 2.0) - (Float64(-b) ^ 2.0)) - Float64(Float64(a * c) * 3.0)) / Float64(b + sqrt(Float64((b ^ 2.0) + Float64(Float64(a * c) * -3.0))))) / ((Float64(a * 3.0) ^ 3.0) ^ 0.3333333333333333))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((b ^ 2.0) - (-b ^ 2.0)) - ((a * c) * 3.0)) / (b + sqrt(((b ^ 2.0) + ((a * c) * -3.0))))) / (((a * 3.0) ^ 3.0) ^ 0.3333333333333333);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[N[(a * 3.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

   1. log1p-expm1-u 41.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-undefine 40.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

4. Applied egg-rr 40.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
5. Step-by-step derivation

   1. add-cbrt-cube 40.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\sqrt[3]{\left(\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right) \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right) \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}}} \]

   2. pow1/3 40.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left(\left(\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right) \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right) \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right)}^{0.3333333333333333}}} \]

   3. log1p-define 40.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left(\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)} \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right) \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right)}^{0.3333333333333333}} \]

   4. log1p-expm1-u 40.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left(\left(\color{blue}{\left(3 \cdot a\right)} \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right) \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right)}^{0.3333333333333333}} \]

   5. log1p-define 40.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left(\left(\left(3 \cdot a\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}\right) \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right)}^{0.3333333333333333}} \]

   6. log1p-expm1-u 40.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left(\left(\left(3 \cdot a\right) \cdot \color{blue}{\left(3 \cdot a\right)}\right) \cdot \log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)\right)}^{0.3333333333333333}} \]

   7. log1p-define 41.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left(\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}\right)}^{0.3333333333333333}} \]

   8. log1p-expm1-u 55.2%

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

   9. pow3 55.2%

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

   10. *-commutative 55.2%

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

6. Applied egg-rr 55.2%

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

   1. flip-+ 55.2%

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

   2. pow2 55.2%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   4. pow2 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. associate-*l* 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. pow2 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. associate-*l* 56.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

8. Applied egg-rr 56.7%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]
9. Step-by-step derivation

   1. associate--r- 97.2%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   2. *-commutative 97.2%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{\left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   3. cancel-sign-sub-inv 97.2%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-3\right) \cdot \left(a \cdot c\right)}}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   4. metadata-eval 97.2%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{{b}^{2} + \color{blue}{-3} \cdot \left(a \cdot c\right)}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

10. Simplified 97.2%

    \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{{b}^{2} + -3 \cdot \left(a \cdot c\right)}}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

11. Final simplification 97.2%

    \[\leadsto \frac{\frac{\left({b}^{2} - {\left(-b\right)}^{2}\right) - \left(a \cdot c\right) \cdot 3}{b + \sqrt{{b}^{2} + \left(a \cdot c\right) \cdot -3}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]
12. Add Preprocessing

Alternative 2: 92.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot 3 \leq 0.0342 \lor \neg \left(a \cdot 3 \leq 385\right):\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {\left(-b\right)}^{2}\right) - \left(a \cdot c\right) \cdot 3}{b + \sqrt{{b}^{2} + \left(a \cdot c\right) \cdot -3}}}{\log \left(1 + \mathsf{expm1}\left(a \cdot 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= (* a 3.0) 0.0342) (not (<= (* a 3.0) 385.0)))
   (*
    c
    (-
     (*
      c
      (*
       a
       (+
        (*
         a
         (+
          (* -1.0546875 (/ (* a (pow c 2.0)) (pow b 7.0)))
          (* -0.5625 (/ c (pow b 5.0)))))
        (* 0.375 (/ -1.0 (pow b 3.0))))))
     (/ 0.5 b)))
   (/
    (/
     (- (- (pow b 2.0) (pow (- b) 2.0)) (* (* a c) 3.0))
     (+ b (sqrt (+ (pow b 2.0) (* (* a c) -3.0)))))
    (log (+ 1.0 (expm1 (* a 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((a * 3.0) <= 0.0342) || !((a * 3.0) <= 385.0)) {
		tmp = c * ((c * (a * ((a * ((-1.0546875 * ((a * pow(c, 2.0)) / pow(b, 7.0))) + (-0.5625 * (c / pow(b, 5.0))))) + (0.375 * (-1.0 / pow(b, 3.0)))))) - (0.5 / b));
	} else {
		tmp = (((pow(b, 2.0) - pow(-b, 2.0)) - ((a * c) * 3.0)) / (b + sqrt((pow(b, 2.0) + ((a * c) * -3.0))))) / log((1.0 + expm1((a * 3.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((a * 3.0) <= 0.0342) || !((a * 3.0) <= 385.0)) {
		tmp = c * ((c * (a * ((a * ((-1.0546875 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 7.0))) + (-0.5625 * (c / Math.pow(b, 5.0))))) + (0.375 * (-1.0 / Math.pow(b, 3.0)))))) - (0.5 / b));
	} else {
		tmp = (((Math.pow(b, 2.0) - Math.pow(-b, 2.0)) - ((a * c) * 3.0)) / (b + Math.sqrt((Math.pow(b, 2.0) + ((a * c) * -3.0))))) / Math.log((1.0 + Math.expm1((a * 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((a * 3.0) <= 0.0342) or not ((a * 3.0) <= 385.0):
		tmp = c * ((c * (a * ((a * ((-1.0546875 * ((a * math.pow(c, 2.0)) / math.pow(b, 7.0))) + (-0.5625 * (c / math.pow(b, 5.0))))) + (0.375 * (-1.0 / math.pow(b, 3.0)))))) - (0.5 / b))
	else:
		tmp = (((math.pow(b, 2.0) - math.pow(-b, 2.0)) - ((a * c) * 3.0)) / (b + math.sqrt((math.pow(b, 2.0) + ((a * c) * -3.0))))) / math.log((1.0 + math.expm1((a * 3.0))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((Float64(a * 3.0) <= 0.0342) || !(Float64(a * 3.0) <= 385.0))
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(a * Float64(Float64(-1.0546875 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 7.0))) + Float64(-0.5625 * Float64(c / (b ^ 5.0))))) + Float64(0.375 * Float64(-1.0 / (b ^ 3.0)))))) - Float64(0.5 / b)));
	else
		tmp = Float64(Float64(Float64(Float64((b ^ 2.0) - (Float64(-b) ^ 2.0)) - Float64(Float64(a * c) * 3.0)) / Float64(b + sqrt(Float64((b ^ 2.0) + Float64(Float64(a * c) * -3.0))))) / log(Float64(1.0 + expm1(Float64(a * 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[N[(a * 3.0), $MachinePrecision], 0.0342], N[Not[LessEqual[N[(a * 3.0), $MachinePrecision], 385.0]], $MachinePrecision]], N[(c * N[(N[(c * N[(a * N[(N[(a * N[(N[(-1.0546875 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[N[(1.0 + N[(Exp[N[(a * 3.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 3 binary64) a) < 0.0342000000000000012 or 385 < (*.f64 #s(literal 3 binary64) a)

   1. Initial program 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in c around 0 91.8%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

   7. Simplified 91.8%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{b \cdot a}\right)\right)\right) - \frac{0.5}{b}\right)} \]

   8. Taylor expanded in a around 0 91.8%

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)} - \frac{0.5}{b}\right) \]

   if 0.0342000000000000012 < (*.f64 #s(literal 3 binary64) a) < 385

   1. Initial program 59.0%

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

      1. log1p-expm1-u 59.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-undefine 59.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   4. Applied egg-rr 59.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
   5. Step-by-step derivation

      1. flip-+ 59.0%

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

      2. pow2 59.0%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 60.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

      4. pow2 60.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

      5. associate-*l* 60.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

      6. pow2 60.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

      7. associate-*l* 60.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Applied egg-rr 60.1%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)} \]
   7. Step-by-step derivation

      1. associate--r- 98.7%

         \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

      2. *-commutative 98.7%

         \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{\left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

      3. cancel-sign-sub-inv 98.7%

         \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-3\right) \cdot \left(a \cdot c\right)}}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

      4. metadata-eval 98.7%

         \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{{b}^{2} + \color{blue}{-3} \cdot \left(a \cdot c\right)}}}{{\left({\left(a \cdot 3\right)}^{3}\right)}^{0.3333333333333333}} \]

   8. Simplified 98.8%

      \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{{b}^{2} + -3 \cdot \left(a \cdot c\right)}}}}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot 3 \leq 0.0342 \lor \neg \left(a \cdot 3 \leq 385\right):\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {\left(-b\right)}^{2}\right) - \left(a \cdot c\right) \cdot 3}{b + \sqrt{{b}^{2} + \left(a \cdot c\right) \cdot -3}}}{\log \left(1 + \mathsf{expm1}\left(a \cdot 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.02)
   (/ (/ (fma -1.0 b (sqrt (fma b b (* (* a c) -3.0)))) a) 3.0)
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))
      (* -0.375 (/ (pow c 2.0) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		tmp = (fma(-1.0, b, sqrt(fma(b, b, ((a * c) * -3.0)))) / a) / 3.0;
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * (pow(c, 2.0) / pow(b, 3.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.02)
		tmp = Float64(Float64(fma(-1.0, b, sqrt(fma(b, b, Float64(Float64(a * c) * -3.0)))) / a) / 3.0);
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(-1.0 * b + N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 81.7%

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

      1. log1p-expm1-u 68.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-undefine 64.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   4. Applied egg-rr 64.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
   5. Step-by-step derivation

      1. log1p-define 68.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-expm1-u 81.7%

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

      3. add-log-exp 66.4%

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

      4. neg-mul-1 66.4%

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

      5. fma-define 66.4%

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

      6. pow2 66.4%

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

      7. associate-*l* 66.3%

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

      8. *-commutative 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. rem-log-exp 81.6%

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

      2. associate-/r* 81.6%

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

      3. pow2 81.6%

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

      4. cancel-sign-sub-inv 81.6%

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

      5. fma-define 81.8%

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

      6. metadata-eval 81.8%

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

      7. *-commutative 81.8%

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

   8. Applied egg-rr 81.8%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in a around 0 93.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (-
   (*
    c
    (*
     a
     (+
      (*
       a
       (+
        (* -1.0546875 (/ (* a (pow c 2.0)) (pow b 7.0)))
        (* -0.5625 (/ c (pow b 5.0)))))
      (* 0.375 (/ -1.0 (pow b 3.0))))))
   (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((c * (a * ((a * ((-1.0546875 * ((a * pow(c, 2.0)) / pow(b, 7.0))) + (-0.5625 * (c / pow(b, 5.0))))) + (0.375 * (-1.0 / 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 * ((c * (a * ((a * (((-1.0546875d0) * ((a * (c ** 2.0d0)) / (b ** 7.0d0))) + ((-0.5625d0) * (c / (b ** 5.0d0))))) + (0.375d0 * ((-1.0d0) / (b ** 3.0d0)))))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * (a * ((a * ((-1.0546875 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 7.0))) + (-0.5625 * (c / Math.pow(b, 5.0))))) + (0.375 * (-1.0 / Math.pow(b, 3.0)))))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((c * (a * ((a * ((-1.0546875 * ((a * math.pow(c, 2.0)) / math.pow(b, 7.0))) + (-0.5625 * (c / math.pow(b, 5.0))))) + (0.375 * (-1.0 / math.pow(b, 3.0)))))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(a * Float64(Float64(-1.0546875 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 7.0))) + Float64(-0.5625 * Float64(c / (b ^ 5.0))))) + Float64(0.375 * Float64(-1.0 / (b ^ 3.0)))))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (a * ((a * ((-1.0546875 * ((a * (c ^ 2.0)) / (b ^ 7.0))) + (-0.5625 * (c / (b ^ 5.0))))) + (0.375 * (-1.0 / (b ^ 3.0)))))) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(a * N[(N[(a * N[(N[(-1.0546875 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Simplified 55.2%

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

5. Taylor expanded in c around 0 90.6%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
6. Step-by-step derivation

7. Simplified 90.6%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{b \cdot a}\right)\right)\right) - \frac{0.5}{b}\right)} \]

8. Taylor expanded in a around 0 90.6%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)} - \frac{0.5}{b}\right) \]

9. Final simplification 90.6%

   \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]
10. Add Preprocessing

Alternative 5: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.02)
   (/ (/ (fma -1.0 b (sqrt (fma b b (* (* a c) -3.0)))) a) 3.0)
   (*
    c
    (-
     (*
      c
      (*
       a
       (- (* -0.5625 (/ (* a c) (pow b 5.0))) (* 0.375 (/ 1.0 (pow b 3.0))))))
     (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		tmp = (fma(-1.0, b, sqrt(fma(b, b, ((a * c) * -3.0)))) / a) / 3.0;
	} else {
		tmp = c * ((c * (a * ((-0.5625 * ((a * c) / pow(b, 5.0))) - (0.375 * (1.0 / pow(b, 3.0)))))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.02)
		tmp = Float64(Float64(fma(-1.0, b, sqrt(fma(b, b, Float64(Float64(a * c) * -3.0)))) / a) / 3.0);
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * c) / (b ^ 5.0))) - Float64(0.375 * Float64(1.0 / (b ^ 3.0)))))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(-1.0 * b + N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(c * N[(N[(c * N[(a * N[(N[(-0.5625 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 * N[(1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 81.7%

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

      1. log1p-expm1-u 68.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-undefine 64.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   4. Applied egg-rr 64.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
   5. Step-by-step derivation

      1. log1p-define 68.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-expm1-u 81.7%

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

      3. add-log-exp 66.4%

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

      4. neg-mul-1 66.4%

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

      5. fma-define 66.4%

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

      6. pow2 66.4%

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

      7. associate-*l* 66.3%

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

      8. *-commutative 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. rem-log-exp 81.6%

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

      2. associate-/r* 81.6%

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

      3. pow2 81.6%

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

      4. cancel-sign-sub-inv 81.6%

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

      5. fma-define 81.8%

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

      6. metadata-eval 81.8%

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

      7. *-commutative 81.8%

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

   8. Applied egg-rr 81.8%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in c around 0 95.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

   7. Simplified 95.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{b \cdot a}\right)\right)\right) - \frac{0.5}{b}\right)} \]

   8. Taylor expanded in a around 0 93.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.02)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (*
    c
    (-
     (*
      c
      (*
       a
       (- (* -0.5625 (/ (* a c) (pow b 5.0))) (* 0.375 (/ 1.0 (pow b 3.0))))))
     (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((c * (a * ((-0.5625 * ((a * c) / pow(b, 5.0))) - (0.375 * (1.0 / pow(b, 3.0)))))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.02)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * c) / (b ^ 5.0))) - Float64(0.375 * Float64(1.0 / (b ^ 3.0)))))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.02], 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[(c * N[(N[(c * N[(a * N[(N[(-0.5625 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 * N[(1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 81.7%

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

   3. Simplified 81.8%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in c around 0 95.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

   7. Simplified 95.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{b \cdot a}\right)\right)\right) - \frac{0.5}{b}\right)} \]

   8. Taylor expanded in a around 0 93.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.02)
   (/ (- (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 (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.02) {
		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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.02)
		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[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.02], 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 81.7%

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

   3. Simplified 81.8%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in c around 0 88.3%

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

   6. Taylor expanded in b around inf 88.8%

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

      1. +-commutative 88.8%

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

      2. fma-define 88.8%

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

      3. associate-/l* 88.8%

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

      4. unpow2 88.8%

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

      5. unpow2 88.8%

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

      6. times-frac 88.8%

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

      7. unpow2 88.8%

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

      8. *-commutative 88.8%

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

   8. Simplified 88.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.02)
     t_0
     (/ (fma -0.375 (* a (pow (/ c b) 2.0)) (* c -0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0;
	} else {
		tmp = fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = t_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_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], t$95$0, 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 81.7%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in c around 0 88.3%

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

   6. Taylor expanded in b around inf 88.8%

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

      1. +-commutative 88.8%

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

      2. fma-define 88.8%

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

      3. associate-/l* 88.8%

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

      4. unpow2 88.8%

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

      5. unpow2 88.8%

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

      6. times-frac 88.8%

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

      7. unpow2 88.8%

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

      8. *-commutative 88.8%

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

   8. Simplified 88.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.02)
     t_0
     (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0;
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.02d0)) then
        tmp = t_0
    else
        tmp = c * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0;
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_0
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = t_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)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_0;
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], t$95$0, 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0200000000000000004

   1. Initial program 81.7%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in c around 0 88.6%

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

      1. associate-/l* 88.6%

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

      2. associate-*r/ 88.6%

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

      3. metadata-eval 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.112:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.112)
   (/ (- (sqrt (- (* b b) (* (* a c) 3.0))) 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 <= 0.112) {
		tmp = (sqrt(((b * b) - ((a * c) * 3.0))) - 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 <= 0.112d0) then
        tmp = (sqrt(((b * b) - ((a * c) * 3.0d0))) - 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 <= 0.112) {
		tmp = (Math.sqrt(((b * b) - ((a * c) * 3.0))) - 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 <= 0.112:
		tmp = (math.sqrt(((b * b) - ((a * c) * 3.0))) - 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 <= 0.112)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 3.0))) - 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 <= 0.112)
		tmp = (sqrt(((b * b) - ((a * c) * 3.0))) - 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, 0.112], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 3.0), $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 < 0.112000000000000002

   1. Initial program 82.7%

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

      1. sqr-neg 82.7%

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

      2. sqr-neg 82.7%

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

      3. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   if 0.112000000000000002 < b

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in c around 0 83.4%

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

      1. associate-/l* 83.4%

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

      2. associate-*r/ 83.4%

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

      3. metadata-eval 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.112:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 55.3%

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

3. Simplified 55.2%

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

5. Taylor expanded in c around 0 81.5%

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

   1. associate-/l* 81.5%

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

   2. associate-*r/ 81.5%

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

   3. metadata-eval 81.5%

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

7. Simplified 81.5%

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

Alternative 12: 64.8% 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 55.3%

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

3. Simplified 55.2%

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

5. Taylor expanded in b around inf 64.5%

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

Alternative 13: 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 55.3%

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

   1. log1p-expm1-u 41.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-undefine 40.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

4. Applied egg-rr 40.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
5. Step-by-step derivation

   1. log1p-define 41.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-expm1-u 55.3%

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

   3. add-log-exp 50.1%

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

   4. neg-mul-1 50.1%

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

   5. fma-define 50.1%

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

   6. pow2 50.1%

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

   7. associate-*l* 50.1%

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

   8. *-commutative 50.1%

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

6. Applied egg-rr 50.1%

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

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
8. 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%

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

9. Simplified 3.2%

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

10. Taylor expanded in a around 0 3.2%

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

Reproduce

herbie shell --seed 2024181 
(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)))