Cubic critical, narrow range

Percentage Accurate: 55.0% → 92.2%

Time: 20.3s

Alternatives: 10

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% 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: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;b \leq 0.37:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\frac{-0.16666666666666666}{a} \cdot \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right)}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 3.0)))))
   (if (<= b 0.37)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (/
         (* (/ -0.16666666666666666 a) (* (pow (* c a) 4.0) 6.328125))
         (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 3.0));
	double tmp;
	if (b <= 0.37) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (((-0.16666666666666666 / a) * (pow((c * a), 4.0) * 6.328125)) / pow(b, 7.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (c * (a * 3.0d0))
    if (b <= 0.37d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (a * 3.0d0)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((((-0.16666666666666666d0) / a) * (((c * a) ** 4.0d0) * 6.328125d0)) / (b ** 7.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (c * (a * 3.0));
	double tmp;
	if (b <= 0.37) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (((-0.16666666666666666 / a) * (Math.pow((c * a), 4.0) * 6.328125)) / Math.pow(b, 7.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (c * (a * 3.0))
	tmp = 0
	if b <= 0.37:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (a * 3.0)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (((-0.16666666666666666 / a) * (math.pow((c * a), 4.0) * 6.328125)) / math.pow(b, 7.0))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 3.0)))
	tmp = 0.0
	if (b <= 0.37)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(Float64(Float64(-0.16666666666666666 / a) * Float64((Float64(c * a) ^ 4.0) * 6.328125)) / (b ^ 7.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 3.0));
	tmp = 0.0;
	if (b <= 0.37)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (((-0.16666666666666666 / a) * (((c * a) ^ 4.0) * 6.328125)) / (b ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.37], 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[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.16666666666666666 / a), $MachinePrecision] * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.37

   1. Initial program 88.2%

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

      1. add-log-exp 65.9%

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

      2. exp-prod 58.1%

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

      3. *-commutative 58.1%

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

      4. exp-prod 57.8%

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

   4. Applied egg-rr 57.8%

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

      1. log-pow 70.8%

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

   6. Simplified 70.8%

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

      1. flip-+ 70.2%

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

      2. pow2 70.2%

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

      3. add-sqr-sqrt 70.2%

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

      4. pow2 70.2%

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

      5. pow-exp 71.9%

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

      6. add-log-exp 76.0%

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

      7. pow2 76.0%

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

      8. pow-exp 76.7%

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

      9. add-log-exp 88.8%

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

   8. Applied egg-rr 88.8%

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

   if 0.37 < b

   1. Initial program 49.0%

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

   3. Taylor expanded in b around inf 93.5%

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

   4. Taylor expanded in c around 0 93.5%

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

      1. associate-*r/ 93.5%

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

   6. Simplified 93.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.37:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 3\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\frac{-0.16666666666666666}{a} \cdot \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right)}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 3.0)))))
   (if (<= b 0.35)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 3.0));
	double tmp;
	if (b <= 0.35) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

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 = (b ** 2.0d0) - (c * (a * 3.0d0))
    if (b <= 0.35d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (a * 3.0d0)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-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 t_0 = Math.pow(b, 2.0) - (c * (a * 3.0));
	double tmp;
	if (b <= 0.35) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-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):
	t_0 = math.pow(b, 2.0) - (c * (a * 3.0))
	tmp = 0
	if b <= 0.35:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (a * 3.0)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-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)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 3.0)))
	tmp = 0.0
	if (b <= 0.35)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 3.0));
	tmp = 0.0;
	if (b <= 0.35)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.35], 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[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.34999999999999998

   1. Initial program 88.2%

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

      1. add-log-exp 65.9%

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

      2. exp-prod 58.1%

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

      3. *-commutative 58.1%

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

      4. exp-prod 57.8%

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

   4. Applied egg-rr 57.8%

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

      1. log-pow 70.8%

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

   6. Simplified 70.8%

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

      1. flip-+ 70.2%

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

      2. pow2 70.2%

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

      3. add-sqr-sqrt 70.2%

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

      4. pow2 70.2%

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

      5. pow-exp 71.9%

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

      6. add-log-exp 76.0%

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

      7. pow2 76.0%

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

      8. pow-exp 76.7%

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

      9. add-log-exp 88.8%

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

   8. Applied egg-rr 88.8%

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

   if 0.34999999999999998 < b

   1. Initial program 49.0%

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

   3. Taylor expanded in b around inf 91.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 3\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 3\right)\\ \mathbf{if}\;b \leq 62:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 3.0)))))
   (if (<= b 62.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 3.0));
	double tmp;
	if (b <= 62.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

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 = (b ** 2.0d0) - (c * (a * 3.0d0))
    if (b <= 62.0d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (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 t_0 = Math.pow(b, 2.0) - (c * (a * 3.0));
	double tmp;
	if (b <= 62.0) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (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):
	t_0 = math.pow(b, 2.0) - (c * (a * 3.0))
	tmp = 0
	if b <= 62.0:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (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)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 3.0)))
	tmp = 0.0
	if (b <= 62.0)
		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(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 3.0));
	tmp = 0.0;
	if (b <= 62.0)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (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_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 62.0], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(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 < 62

   1. Initial program 80.3%

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

      1. add-log-exp 71.5%

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

      2. exp-prod 61.0%

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

      3. *-commutative 61.0%

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

      4. exp-prod 60.4%

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

   4. Applied egg-rr 60.4%

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

      1. log-pow 65.6%

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

   6. Simplified 65.6%

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

      1. flip-+ 65.3%

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

      2. pow2 65.3%

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

      3. add-sqr-sqrt 66.0%

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

      4. pow2 66.0%

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

      5. pow-exp 67.3%

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

      6. add-log-exp 70.4%

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

      7. pow2 70.4%

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

      8. pow-exp 70.9%

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

      9. add-log-exp 81.5%

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

   8. Applied egg-rr 81.5%

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

   if 62 < b

   1. Initial program 41.7%

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

   3. Taylor expanded in b around inf 91.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 62:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 3\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)\\ \mathbf{if}\;b \leq 62:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* (* c a) -3.0))))
   (if (<= b 62.0)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, ((c * a) * -3.0));
	double tmp;
	if (b <= 62.0) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(c * a) * -3.0))
	tmp = 0.0
	if (b <= 62.0)
		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(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 62.0], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(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 < 62

   1. Initial program 80.3%

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

      1. add-sqr-sqrt 80.3%

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

      2. difference-of-squares 80.3%

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

      3. associate-*l* 80.2%

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

      4. associate-*l* 80.3%

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

   4. Applied egg-rr 80.3%

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

      1. associate-*r* 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-*r* 80.3%

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

      4. *-commutative 80.3%

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

   6. Simplified 80.3%

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

      1. flip-+ 79.8%

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

   8. Applied egg-rr 81.0%

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

      1. unpow2 81.0%

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

      2. sqr-neg 81.0%

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

      3. unpow2 81.0%

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

      4. difference-of-squares 81.5%

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

      5. rem-square-sqrt 81.5%

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

      6. fma-neg 81.1%

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

      7. associate-*r* 81.0%

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

      8. *-commutative 81.0%

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

      9. distribute-rgt-neg-in 81.0%

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

      10. metadata-eval 81.0%

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

      11. difference-of-squares 81.0%

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

      12. rem-square-sqrt 81.0%

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

   10. Simplified 81.0%

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

   if 62 < b

   1. Initial program 41.7%

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

   3. Taylor expanded in b around inf 91.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 62:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4e-7)
   (/ (* 0.3333333333333333 (- (sqrt (fma b b (* (* c a) -3.0))) b)) a)
   (/ (* 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)) <= -4e-7) {
		tmp = (0.3333333333333333 * (sqrt(fma(b, b, ((c * a) * -3.0))) - b)) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4e-7)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))) - b)) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4e-7], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.9999999999999998e-7

   1. Initial program 73.4%

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

      1. add-log-exp 51.7%

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

      2. exp-prod 47.1%

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

      3. *-commutative 47.1%

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

      4. exp-prod 46.7%

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

   4. Applied egg-rr 46.7%

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

      1. log-pow 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in a around inf 57.5%

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

      1. associate-*r/ 57.5%

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

   9. Simplified 73.5%

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

   if -3.9999999999999998e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 26.8%

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

   3. Taylor expanded in b around inf 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-*l/ 87.2%

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

   5. Simplified 87.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -4e-7) t_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 <= -4e-7) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-4d-7)) then
        tmp = t_0
    else
        tmp = (c * (-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 <= -4e-7) {
		tmp = t_0;
	} else {
		tmp = (c * -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 <= -4e-7:
		tmp = t_0
	else:
		tmp = (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 <= -4e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -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 <= -4e-7)
		tmp = t_0;
	else
		tmp = (c * -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, -4e-7], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.9999999999999998e-7

   1. Initial program 73.4%

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

   if -3.9999999999999998e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 26.8%

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

   3. Taylor expanded in b around inf 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-*l/ 87.2%

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

   5. Simplified 87.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 62:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 62.0)
   (/ (* 0.3333333333333333 (- (sqrt (fma b b (* (* c a) -3.0))) b)) a)
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 62.0) {
		tmp = (0.3333333333333333 * (sqrt(fma(b, b, ((c * a) * -3.0))) - b)) / a;
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 62.0)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))) - b)) / a);
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 62.0], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $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 < 62

   1. Initial program 80.3%

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

      1. add-log-exp 71.5%

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

      2. exp-prod 61.0%

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

      3. *-commutative 61.0%

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

      4. exp-prod 60.4%

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

   4. Applied egg-rr 60.4%

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

      1. log-pow 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in a around inf 65.6%

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

      1. associate-*r/ 65.6%

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

   9. Simplified 80.3%

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

   if 62 < b

   1. Initial program 41.7%

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

   3. Taylor expanded in b around inf 91.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 62:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. add-log-exp 39.7%

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

   2. exp-prod 31.3%

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

   3. *-commutative 31.3%

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

   4. exp-prod 31.1%

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

4. Applied egg-rr 31.1%

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

   1. log-pow 37.3%

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

6. Simplified 37.3%

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

7. Taylor expanded in b around inf 47.4%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot \log \left({\left(e^{a}\right)}^{3}\right)}{a \cdot b}} \]
8. Step-by-step derivation

   1. *-commutative 47.4%

      \[\leadsto \color{blue}{\frac{c \cdot \log \left({\left(e^{a}\right)}^{3}\right)}{a \cdot b} \cdot -0.16666666666666666} \]

   2. associate-/l* 47.4%

      \[\leadsto \color{blue}{\left(c \cdot \frac{\log \left({\left(e^{a}\right)}^{3}\right)}{a \cdot b}\right)} \cdot -0.16666666666666666 \]

   3. associate-*l* 47.4%

      \[\leadsto \color{blue}{c \cdot \left(\frac{\log \left({\left(e^{a}\right)}^{3}\right)}{a \cdot b} \cdot -0.16666666666666666\right)} \]

   4. exp-prod 48.1%

      \[\leadsto c \cdot \left(\frac{\log \color{blue}{\left(e^{a \cdot 3}\right)}}{a \cdot b} \cdot -0.16666666666666666\right) \]

   5. rem-log-exp 65.4%

      \[\leadsto c \cdot \left(\frac{\color{blue}{a \cdot 3}}{a \cdot b} \cdot -0.16666666666666666\right) \]

9. Simplified 65.4%

   \[\leadsto \color{blue}{c \cdot \left(\frac{a \cdot 3}{a \cdot b} \cdot -0.16666666666666666\right)} \]

10. Taylor expanded in a around 0 65.5%

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

11. Final simplification 65.5%

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

Alternative 9: 64.8% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in b around inf 65.6%

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

   1. *-commutative 65.6%

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

   2. associate-*l/ 65.6%

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

5. Simplified 65.6%

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

6. Final simplification 65.6%

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

Alternative 10: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. add-sqr-sqrt 53.6%

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

   2. difference-of-squares 53.6%

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

   3. associate-*l* 53.6%

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

   4. associate-*l* 53.6%

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

4. Applied egg-rr 53.6%

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

   1. associate-*r* 53.6%

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

   2. *-commutative 53.6%

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

   3. associate-*r* 53.6%

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

   4. *-commutative 53.6%

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

6. Simplified 53.6%

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

7. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024039 
(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)))