Cubic critical, wide range

Percentage Accurate: 33.3% → 81.6%

Time: 19.5s

Alternatives: 11

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 33.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{+25} \lor \neg \left(b \leq 2.95 \cdot 10^{+27}\right) \land b \leq 9.4 \cdot 10^{+27}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{4} \cdot 1.265625}{a \cdot {b}^{7}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= b 8.6e+25) (and (not (<= b 2.95e+27)) (<= b 9.4e+27)))
   (+
    (* -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
       (/
        (+
         (* 5.0625 (* (pow a 4.0) (pow c 4.0)))
         (* (pow (* a c) 4.0) 1.265625))
        (* a (pow b 7.0)))))))
   (/ (log (exp (* a (* (/ c b) -1.5)))) (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if ((b <= 8.6e+25) || (!(b <= 2.95e+27) && (b <= 9.4e+27))) {
		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 * (((5.0625 * (pow(a, 4.0) * pow(c, 4.0))) + (pow((a * c), 4.0) * 1.265625)) / (a * pow(b, 7.0))))));
	} else {
		tmp = log(exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((b <= 8.6d+25) .or. (.not. (b <= 2.95d+27)) .and. (b <= 9.4d+27)) then
        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) * (((5.0625d0 * ((a ** 4.0d0) * (c ** 4.0d0))) + (((a * c) ** 4.0d0) * 1.265625d0)) / (a * (b ** 7.0d0))))))
    else
        tmp = log(exp((a * ((c / b) * (-1.5d0))))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if ((b <= 8.6e+25) || (!(b <= 2.95e+27) && (b <= 9.4e+27))) {
		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 * (((5.0625 * (Math.pow(a, 4.0) * Math.pow(c, 4.0))) + (Math.pow((a * c), 4.0) * 1.265625)) / (a * Math.pow(b, 7.0))))));
	} else {
		tmp = Math.log(Math.exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if (b <= 8.6e+25) or (not (b <= 2.95e+27) and (b <= 9.4e+27)):
		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 * (((5.0625 * (math.pow(a, 4.0) * math.pow(c, 4.0))) + (math.pow((a * c), 4.0) * 1.265625)) / (a * math.pow(b, 7.0))))))
	else:
		tmp = math.log(math.exp((a * ((c / b) * -1.5)))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((b <= 8.6e+25) || (!(b <= 2.95e+27) && (b <= 9.4e+27)))
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64(Float64(5.0625 * Float64((a ^ 4.0) * (c ^ 4.0))) + Float64((Float64(a * c) ^ 4.0) * 1.265625)) / Float64(a * (b ^ 7.0)))))));
	else
		tmp = Float64(log(exp(Float64(a * Float64(Float64(c / b) * -1.5)))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if ((b <= 8.6e+25) || (~((b <= 2.95e+27)) && (b <= 9.4e+27)))
		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 * (((5.0625 * ((a ^ 4.0) * (c ^ 4.0))) + (((a * c) ^ 4.0) * 1.265625)) / (a * (b ^ 7.0))))));
	else
		tmp = log(exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[b, 8.6e+25], And[N[Not[LessEqual[b, 2.95e+27]], $MachinePrecision], LessEqual[b, 9.4e+27]]], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(5.0625 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 1.265625), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[Exp[N[(a * N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.59999999999999996e25 or 2.95000000000000022e27 < b < 9.39999999999999952e27

   1. Initial program 29.1%

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

   3. Taylor expanded in b around inf 88.2%

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

      1. *-commutative 88.2%

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

      2. unpow-prod-down 88.2%

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

      3. pow-prod-down 88.2%

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

      4. pow-pow 88.2%

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

      5. metadata-eval 88.2%

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

      6. metadata-eval 88.2%

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

   5. Applied egg-rr 88.2%

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

   if 8.59999999999999996e25 < b < 2.95000000000000022e27 or 9.39999999999999952e27 < b

   1. Initial program 65.7%

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

   3. Taylor expanded in b around inf 38.6%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 38.6%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

   5. Simplified 38.6%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 32.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      2. log1p-udef 79.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      3. associate-*r/ 79.1%

         \[\leadsto \frac{\log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}}\right)\right)}{3 \cdot a} \]

   7. Applied egg-rr 79.1%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. add-exp-log 79.1%

         \[\leadsto \frac{\log \color{blue}{\left(e^{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}\right)}}{3 \cdot a} \]

      2. log1p-def 79.1%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}\right)}{3 \cdot a} \]

      3. log1p-expm1-u 79.1%

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

      4. div-inv 79.1%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\left(-1.5 \cdot a\right) \cdot \frac{1}{\frac{b}{c}}}}\right)}{3 \cdot a} \]

      5. clear-num 79.1%

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

      6. *-commutative 79.1%

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

      7. associate-*l* 79.1%

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

   9. Applied egg-rr 79.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{+25} \lor \neg \left(b \leq 2.95 \cdot 10^{+27}\right) \land b \leq 9.4 \cdot 10^{+27}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{4} \cdot 1.265625}{a \cdot {b}^{7}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+25} \lor \neg \left(b \leq 2.95 \cdot 10^{+27}\right) \land b \leq 1.7 \cdot 10^{+28}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= b 6.5e+25) (and (not (<= b 2.95e+27)) (<= b 1.7e+28)))
   (+
    (* -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)))))
   (/ (log (exp (* a (* (/ c b) -1.5)))) (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if ((b <= 6.5e+25) || (!(b <= 2.95e+27) && (b <= 1.7e+28))) {
		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))));
	} else {
		tmp = log(exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((b <= 6.5d+25) .or. (.not. (b <= 2.95d+27)) .and. (b <= 1.7d+28)) then
        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))))
    else
        tmp = log(exp((a * ((c / b) * (-1.5d0))))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if ((b <= 6.5e+25) || (!(b <= 2.95e+27) && (b <= 1.7e+28))) {
		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))));
	} else {
		tmp = Math.log(Math.exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if (b <= 6.5e+25) or (not (b <= 2.95e+27) and (b <= 1.7e+28)):
		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))))
	else:
		tmp = math.log(math.exp((a * ((c / b) * -1.5)))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((b <= 6.5e+25) || (!(b <= 2.95e+27) && (b <= 1.7e+28)))
		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)))));
	else
		tmp = Float64(log(exp(Float64(a * Float64(Float64(c / b) * -1.5)))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if ((b <= 6.5e+25) || (~((b <= 2.95e+27)) && (b <= 1.7e+28)))
		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))));
	else
		tmp = log(exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[b, 6.5e+25], And[N[Not[LessEqual[b, 2.95e+27]], $MachinePrecision], LessEqual[b, 1.7e+28]]], 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], N[(N[Log[N[Exp[N[(a * N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.50000000000000005e25 or 2.95000000000000022e27 < b < 1.7e28

   1. Initial program 29.1%

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

   3. Taylor expanded in b around inf 87.5%

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

   if 6.50000000000000005e25 < b < 2.95000000000000022e27 or 1.7e28 < b

   1. Initial program 65.7%

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

   3. Taylor expanded in b around inf 38.6%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 38.6%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

   5. Simplified 38.6%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 32.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      2. log1p-udef 79.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      3. associate-*r/ 79.1%

         \[\leadsto \frac{\log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}}\right)\right)}{3 \cdot a} \]

   7. Applied egg-rr 79.1%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. add-exp-log 79.1%

         \[\leadsto \frac{\log \color{blue}{\left(e^{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}\right)}}{3 \cdot a} \]

      2. log1p-def 79.1%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}\right)}{3 \cdot a} \]

      3. log1p-expm1-u 79.1%

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

      4. div-inv 79.1%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\left(-1.5 \cdot a\right) \cdot \frac{1}{\frac{b}{c}}}}\right)}{3 \cdot a} \]

      5. clear-num 79.1%

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

      6. *-commutative 79.1%

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

      7. associate-*l* 79.1%

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

   9. Applied egg-rr 79.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+25} \lor \neg \left(b \leq 2.95 \cdot 10^{+27}\right) \land b \leq 1.7 \cdot 10^{+28}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -2e-21) 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 <= -2e-21) {
		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 <= (-2d-21)) 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 <= -2e-21) {
		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 <= -2e-21:
		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 <= -2e-21)
		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 <= -2e-21)
		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, -2e-21], 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)) < -1.99999999999999982e-21

   1. Initial program 66.2%

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

   if -1.99999999999999982e-21 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 25.3%

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

   3. Taylor expanded in b around inf 78.7%

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

      1. associate-*r/ 78.7%

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

   5. Simplified 78.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot -0.5}{b}\\ \mathbf{if}\;b \leq 9.5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+23}:\\ \;\;\;\;\log \left(1 + \mathsf{expm1}\left(c \cdot \frac{-0.5}{b}\right)\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c -0.5) b)))
   (if (<= b 9.5e+22)
     t_0
     (if (<= b 5.6e+23)
       (log (+ 1.0 (expm1 (* c (/ -0.5 b)))))
       (if (<= b 8.5e+25)
         t_0
         (/ (log (exp (* a (* (/ c b) -1.5)))) (* a 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * -0.5) / b;
	double tmp;
	if (b <= 9.5e+22) {
		tmp = t_0;
	} else if (b <= 5.6e+23) {
		tmp = log((1.0 + expm1((c * (-0.5 / b)))));
	} else if (b <= 8.5e+25) {
		tmp = t_0;
	} else {
		tmp = log(exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * -0.5) / b;
	double tmp;
	if (b <= 9.5e+22) {
		tmp = t_0;
	} else if (b <= 5.6e+23) {
		tmp = Math.log((1.0 + Math.expm1((c * (-0.5 / b)))));
	} else if (b <= 8.5e+25) {
		tmp = t_0;
	} else {
		tmp = Math.log(Math.exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c * -0.5) / b
	tmp = 0
	if b <= 9.5e+22:
		tmp = t_0
	elif b <= 5.6e+23:
		tmp = math.log((1.0 + math.expm1((c * (-0.5 / b)))))
	elif b <= 8.5e+25:
		tmp = t_0
	else:
		tmp = math.log(math.exp((a * ((c / b) * -1.5)))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * -0.5) / b)
	tmp = 0.0
	if (b <= 9.5e+22)
		tmp = t_0;
	elseif (b <= 5.6e+23)
		tmp = log(Float64(1.0 + expm1(Float64(c * Float64(-0.5 / b)))));
	elseif (b <= 8.5e+25)
		tmp = t_0;
	else
		tmp = Float64(log(exp(Float64(a * Float64(Float64(c / b) * -1.5)))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, 9.5e+22], t$95$0, If[LessEqual[b, 5.6e+23], N[Log[N[(1.0 + N[(Exp[N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[b, 8.5e+25], t$95$0, N[(N[Log[N[Exp[N[(a * N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.49999999999999937e22 or 5.6e23 < b < 8.5000000000000007e25

   1. Initial program 27.2%

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

   3. Taylor expanded in b around inf 81.5%

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

      1. associate-*r/ 81.5%

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

   5. Simplified 81.5%

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

   if 9.49999999999999937e22 < b < 5.6e23

   1. Initial program 72.0%

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

   3. Taylor expanded in b around inf 33.8%

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

      1. associate-*r/ 33.8%

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

      2. associate-/l* 33.8%

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

   5. Simplified 33.8%

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

      1. log1p-expm1-u 33.8%

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

      2. log1p-udef 84.7%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-0.5}{\frac{b}{c}}\right)\right)} \]

      3. associate-/r/ 84.7%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{-0.5}{b} \cdot c}\right)\right) \]

   7. Applied egg-rr 84.7%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-0.5}{b} \cdot c\right)\right)} \]

   if 8.5000000000000007e25 < b

   1. Initial program 62.1%

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

   3. Taylor expanded in b around inf 42.9%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 42.8%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

   5. Simplified 42.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 36.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      2. log1p-udef 69.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      3. associate-*r/ 69.9%

         \[\leadsto \frac{\log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}}\right)\right)}{3 \cdot a} \]

   7. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. add-exp-log 69.9%

         \[\leadsto \frac{\log \color{blue}{\left(e^{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}\right)}}{3 \cdot a} \]

      2. log1p-def 69.9%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}\right)}{3 \cdot a} \]

      3. log1p-expm1-u 69.9%

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

      4. div-inv 69.9%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\left(-1.5 \cdot a\right) \cdot \frac{1}{\frac{b}{c}}}}\right)}{3 \cdot a} \]

      5. clear-num 69.9%

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

      6. *-commutative 69.9%

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

      7. associate-*l* 69.9%

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

   9. Applied egg-rr 69.9%

      \[\leadsto \frac{\log \color{blue}{\left(e^{a \cdot \left(-1.5 \cdot \frac{c}{b}\right)}\right)}}{3 \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+23}:\\ \;\;\;\;\log \left(1 + \mathsf{expm1}\left(c \cdot \frac{-0.5}{b}\right)\right)\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+25}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7e+25)
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))
   (/ (log (exp (* a (* (/ c b) -1.5)))) (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7e+25) {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	} else {
		tmp = log(exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7d+25) then
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
    else
        tmp = log(exp((a * ((c / b) * (-1.5d0))))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7e+25) {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
	} else {
		tmp = Math.log(Math.exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7e+25:
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))
	else:
		tmp = math.log(math.exp((a * ((c / b) * -1.5)))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7e+25)
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	else
		tmp = Float64(log(exp(Float64(a * Float64(Float64(c / b) * -1.5)))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7e+25)
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
	else
		tmp = log(exp((a * ((c / b) * -1.5)))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7e+25], 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], N[(N[Log[N[Exp[N[(a * N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.99999999999999999e25

   1. Initial program 28.9%

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

   3. Taylor expanded in b around inf 85.8%

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

   if 6.99999999999999999e25 < b

   1. Initial program 62.1%

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

   3. Taylor expanded in b around inf 42.9%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 42.8%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

   5. Simplified 42.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 36.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      2. log1p-udef 69.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      3. associate-*r/ 69.9%

         \[\leadsto \frac{\log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}}\right)\right)}{3 \cdot a} \]

   7. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. add-exp-log 69.9%

         \[\leadsto \frac{\log \color{blue}{\left(e^{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}\right)}}{3 \cdot a} \]

      2. log1p-def 69.9%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}\right)}{3 \cdot a} \]

      3. log1p-expm1-u 69.9%

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

      4. div-inv 69.9%

         \[\leadsto \frac{\log \left(e^{\color{blue}{\left(-1.5 \cdot a\right) \cdot \frac{1}{\frac{b}{c}}}}\right)}{3 \cdot a} \]

      5. clear-num 69.9%

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

      6. *-commutative 69.9%

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

      7. associate-*l* 69.9%

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

   9. Applied egg-rr 69.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+25}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(1 + a \cdot \left(c \cdot \frac{-1.5}{b}\right)\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e+25)
   (/ (* c -0.5) b)
   (/ (log (+ 1.0 (* a (* c (/ -1.5 b))))) (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e+25) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = log((1.0 + (a * (c * (-1.5 / b))))) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 8d+25) then
        tmp = (c * (-0.5d0)) / b
    else
        tmp = log((1.0d0 + (a * (c * ((-1.5d0) / b))))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e+25) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = Math.log((1.0 + (a * (c * (-1.5 / b))))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e+25:
		tmp = (c * -0.5) / b
	else:
		tmp = math.log((1.0 + (a * (c * (-1.5 / b))))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e+25)
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(log(Float64(1.0 + Float64(a * Float64(c * Float64(-1.5 / b))))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e+25)
		tmp = (c * -0.5) / b;
	else
		tmp = log((1.0 + (a * (c * (-1.5 / b))))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e+25], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(N[Log[N[(1.0 + N[(a * N[(c * N[(-1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000072e25

   1. Initial program 28.9%

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

   3. Taylor expanded in b around inf 79.8%

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

      1. associate-*r/ 79.8%

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

   5. Simplified 79.8%

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

   if 8.00000000000000072e25 < b

   1. Initial program 62.1%

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

   3. Taylor expanded in b around inf 42.9%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 42.8%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

   5. Simplified 42.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 36.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      2. log1p-udef 69.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(-1.5 \cdot \frac{a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

      3. associate-*r/ 69.9%

         \[\leadsto \frac{\log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}}\right)\right)}{3 \cdot a} \]

   7. Applied egg-rr 69.9%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)}}{3 \cdot a} \]

   8. Taylor expanded in a around 0 64.0%

      \[\leadsto \frac{\log \color{blue}{\left(1 + -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   9. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{\log \color{blue}{\left(-1.5 \cdot \frac{a \cdot c}{b} + 1\right)}}{3 \cdot a} \]

      2. associate-*r/ 64.0%

         \[\leadsto \frac{\log \left(-1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)} + 1\right)}{3 \cdot a} \]

      3. associate-*r* 64.0%

         \[\leadsto \frac{\log \left(\color{blue}{\left(-1.5 \cdot a\right) \cdot \frac{c}{b}} + 1\right)}{3 \cdot a} \]

      4. *-commutative 64.0%

         \[\leadsto \frac{\log \left(\color{blue}{\left(a \cdot -1.5\right)} \cdot \frac{c}{b} + 1\right)}{3 \cdot a} \]

      5. associate-*r* 64.0%

         \[\leadsto \frac{\log \left(\color{blue}{a \cdot \left(-1.5 \cdot \frac{c}{b}\right)} + 1\right)}{3 \cdot a} \]

      6. associate-*r/ 64.0%

         \[\leadsto \frac{\log \left(a \cdot \color{blue}{\frac{-1.5 \cdot c}{b}} + 1\right)}{3 \cdot a} \]

      7. *-commutative 64.0%

         \[\leadsto \frac{\log \left(a \cdot \frac{\color{blue}{c \cdot -1.5}}{b} + 1\right)}{3 \cdot a} \]

      8. associate-*r/ 64.0%

         \[\leadsto \frac{\log \left(a \cdot \color{blue}{\left(c \cdot \frac{-1.5}{b}\right)} + 1\right)}{3 \cdot a} \]

   10. Simplified 64.0%

       \[\leadsto \frac{\log \color{blue}{\left(a \cdot \left(c \cdot \frac{-1.5}{b}\right) + 1\right)}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(1 + a \cdot \left(c \cdot \frac{-1.5}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(\frac{a \cdot -1.5}{\frac{b}{c}} - b\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.2e+25)
   (/ (* c -0.5) b)
   (/ (+ b (- (/ (* a -1.5) (/ b c)) b)) (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2e+25) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = (b + (((a * -1.5) / (b / c)) - b)) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 8.2d+25) then
        tmp = (c * (-0.5d0)) / b
    else
        tmp = (b + (((a * (-1.5d0)) / (b / c)) - b)) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.2e+25) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = (b + (((a * -1.5) / (b / c)) - b)) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.2e+25:
		tmp = (c * -0.5) / b
	else:
		tmp = (b + (((a * -1.5) / (b / c)) - b)) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.2e+25)
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(Float64(b + Float64(Float64(Float64(a * -1.5) / Float64(b / c)) - b)) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.2e+25)
		tmp = (c * -0.5) / b;
	else
		tmp = (b + (((a * -1.5) / (b / c)) - b)) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.2e+25], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(N[(b + N[(N[(N[(a * -1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.19999999999999933e25

   1. Initial program 28.9%

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

   3. Taylor expanded in b around inf 79.8%

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

      1. associate-*r/ 79.8%

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

   5. Simplified 79.8%

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

   if 8.19999999999999933e25 < b

   1. Initial program 62.1%

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

      1. +-commutative 62.1%

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

      2. sqr-neg 62.1%

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

      3. unsub-neg 62.1%

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

      4. div-sub 62.1%

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

      5. --rgt-identity 62.1%

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

      6. div-sub 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in b around inf 59.4%

      \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate--l+ 59.4%

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

      2. associate-/l* 59.4%

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

      3. associate-*r/ 59.4%

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

   7. Applied egg-rr 59.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(\frac{a \cdot -1.5}{\frac{b}{c}} - b\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+29}:\\ \;\;\;\;\frac{b - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.6e+25)
   (/ (* c -0.5) b)
   (if (<= b 4.7e+29) (/ (- b b) (* a 3.0)) (/ -0.5 (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.6e+25) {
		tmp = (c * -0.5) / b;
	} else if (b <= 4.7e+29) {
		tmp = (b - b) / (a * 3.0);
	} else {
		tmp = -0.5 / (b / c);
	}
	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 <= 8.6d+25) then
        tmp = (c * (-0.5d0)) / b
    else if (b <= 4.7d+29) then
        tmp = (b - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.6e+25) {
		tmp = (c * -0.5) / b;
	} else if (b <= 4.7e+29) {
		tmp = (b - b) / (a * 3.0);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.6e+25:
		tmp = (c * -0.5) / b
	elif b <= 4.7e+29:
		tmp = (b - b) / (a * 3.0)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.6e+25)
		tmp = Float64(Float64(c * -0.5) / b);
	elseif (b <= 4.7e+29)
		tmp = Float64(Float64(b - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.6e+25)
		tmp = (c * -0.5) / b;
	elseif (b <= 4.7e+29)
		tmp = (b - b) / (a * 3.0);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.6e+25], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[b, 4.7e+29], N[(N[(b - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 8.59999999999999996e25

   1. Initial program 28.9%

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

   3. Taylor expanded in b around inf 79.8%

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

      1. associate-*r/ 79.8%

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

   5. Simplified 79.8%

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

   if 8.59999999999999996e25 < b < 4.7000000000000002e29

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. sqr-neg 71.3%

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

      3. unsub-neg 71.3%

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

      4. div-sub 71.3%

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

      5. --rgt-identity 71.3%

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

      6. div-sub 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in b around inf 65.5%

      \[\leadsto \frac{\color{blue}{b} - b}{3 \cdot a} \]

   if 4.7000000000000002e29 < b

   1. Initial program 42.1%

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

   3. Taylor expanded in b around inf 64.0%

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

      1. associate-*r/ 64.0%

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

      2. associate-/l* 64.0%

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

   5. Simplified 64.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+29}:\\ \;\;\;\;\frac{b - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% 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 34.2%

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

3. Taylor expanded in b around inf 73.9%

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

   1. associate-*r/ 73.9%

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

   2. associate-/l* 73.7%

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

5. Simplified 73.7%

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

   1. associate-/r/ 73.6%

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

7. Applied egg-rr 73.6%

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

8. Final simplification 73.6%

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

Alternative 10: 74.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.2%

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

3. Taylor expanded in b around inf 73.9%

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

   1. associate-*r/ 73.9%

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

   2. associate-/l* 73.7%

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

5. Simplified 73.7%

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

6. Final simplification 73.7%

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

Alternative 11: 74.9% 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 34.2%

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

3. Taylor expanded in b around inf 73.9%

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

   1. associate-*r/ 73.9%

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

5. Simplified 73.9%

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

6. Final simplification 73.9%

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

Reproduce

herbie shell --seed 2024031 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))