Compound Interest

Percentage Accurate: 28.1% → 99.6%

Time: 38.2s

Alternatives: 25

Speedup: 38.0×

• could not determine a ground truth

  1. i = -2.0000000000000001e231
  2. n = 1.99999999999999997e77

Specification

Math

\[\begin{array}{l} \\ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 28.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := \frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (/ (+ (* t_0 100.0) -100.0) (/ i n))))
   (if (<= t_1 -50000000.0)
     t_2
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/
           n
           (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0)))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = ((t_0 * 100.0) + -100.0) / (i / n);
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = ((t_0 * 100.0) + -100.0) / (i / n);
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	t_2 = ((t_0 * 100.0) + -100.0) / (i / n)
	tmp = 0
	if t_1 <= -50000000.0:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], t$95$2, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -5e7 or 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 99.9%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-def 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   if -5e7 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 20.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 20.3%

         \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      2. add-exp-log 20.3%

         \[\leadsto 100 \cdot \frac{1 \cdot \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right)}{\frac{i}{n}} \]

      3. expm1-def 20.3%

         \[\leadsto 100 \cdot \frac{1 \cdot \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)}}{\frac{i}{n}} \]

      4. log-pow 36.2%

         \[\leadsto 100 \cdot \frac{1 \cdot \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      5. log1p-udef 99.7%

         \[\leadsto 100 \cdot \frac{1 \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. *-lft-identity 99.7%

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   6. Simplified 99.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 1.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 1.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 1.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 73.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -50000000:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;100 \cdot \frac{t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{t\_0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (/ i n) n) -1.0))
        (t_1 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_1 (- INFINITY))
     (* 100.0 (/ t_0 (/ i n)))
     (if (<= t_1 0.0)
       (/ 1.0 (/ (/ i (expm1 i)) (* n 100.0)))
       (if (<= t_1 INFINITY)
         (* (* n 100.0) (/ t_0 i))
         (*
          100.0
          (/
           n
           (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0)))))))))))

C

double code(double i, double n) {
	double t_0 = pow((i / n), n) + -1.0;
	double t_1 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 100.0 * (t_0 / (i / n));
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / ((i / expm1(i)) / (n * 100.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (n * 100.0) * (t_0 / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((i / n), n) + -1.0;
	double t_1 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (t_0 / (i / n));
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / ((i / Math.expm1(i)) / (n * 100.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (n * 100.0) * (t_0 / i);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((i / n), n) + -1.0
	t_1 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 100.0 * (t_0 / (i / n))
	elif t_1 <= 0.0:
		tmp = 1.0 / ((i / math.expm1(i)) / (n * 100.0))
	elif t_1 <= math.inf:
		tmp = (n * 100.0) * (t_0 / i)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(i / n) ^ n) + -1.0)
	t_1 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(100.0 * Float64(t_0 / Float64(i / n)));
	elseif (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(i / expm1(i)) / Float64(n * 100.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(n * 100.0) * Float64(t_0 / i));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(100.0 * N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -inf.0

   1. Initial program 100.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf 100.0%

      \[\leadsto 100 \cdot \frac{{\color{blue}{\left(\frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 21.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 41.7%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 41.7%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 41.7%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 78.5%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 78.5%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 79.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 79.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 99.7%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 99.7%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 99.7%

      \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n}\right)}}^{n} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 1.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 1.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 1.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 73.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := \frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (/ (+ (* t_0 100.0) -100.0) (/ i n))))
   (if (<= t_1 -4e-5)
     t_2
     (if (<= t_1 0.0)
       (/ 1.0 (/ (/ i (expm1 i)) (* n 100.0)))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/
           n
           (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0)))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = ((t_0 * 100.0) + -100.0) / (i / n);
	double tmp;
	if (t_1 <= -4e-5) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / ((i / expm1(i)) / (n * 100.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = ((t_0 * 100.0) + -100.0) / (i / n);
	double tmp;
	if (t_1 <= -4e-5) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / ((i / Math.expm1(i)) / (n * 100.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	t_2 = ((t_0 * 100.0) + -100.0) / (i / n)
	tmp = 0
	if t_1 <= -4e-5:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = 1.0 / ((i / math.expm1(i)) / (n * 100.0))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -4e-5)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(i / expm1(i)) / Float64(n * 100.0)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-5], t$95$2, If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.00000000000000033e-5 or 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 99.9%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-def 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   if -4.00000000000000033e-5 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 19.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 40.3%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 40.3%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 40.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 78.0%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 78.0%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 78.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 1.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 1.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 1.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 73.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := \frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (/ (+ (* t_0 100.0) -100.0) (/ i n))))
   (if (<= t_1 -2e-42)
     t_2
     (if (<= t_1 0.0)
       (* 100.0 (* (expm1 (* n (log1p (/ i n)))) (/ n i)))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/
           n
           (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0)))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = ((t_0 * 100.0) + -100.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-42) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) * (n / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = ((t_0 * 100.0) + -100.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-42) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) * (n / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	t_2 = ((t_0 * 100.0) + -100.0) / (i / n)
	tmp = 0
	if t_1 <= -2e-42:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) * (n / i))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-42)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(n / i)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-42], t$95$2, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2.00000000000000008e-42 or 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 99.9%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-def 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   if -2.00000000000000008e-42 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 17.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 17.7%

         \[\leadsto 100 \cdot \color{blue}{\frac{1}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]

      2. associate-/r/ 17.7%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{\frac{i}{n}} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)} \]

      3. clear-num 17.1%

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{n}{i}} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \]

      4. add-exp-log 17.1%

         \[\leadsto 100 \cdot \left(\frac{n}{i} \cdot \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right)\right) \]

      5. expm1-def 17.1%

         \[\leadsto 100 \cdot \left(\frac{n}{i} \cdot \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)}\right) \]

      6. log-pow 33.6%

         \[\leadsto 100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)\right) \]

      7. log1p-udef 97.8%

         \[\leadsto 100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)\right) \]

   4. Applied egg-rr 97.8%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{n}{i} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 1.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 1.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 1.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 73.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := \frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (/ (+ (* t_0 100.0) -100.0) (/ i n))))
   (if (<= t_1 -2e-42)
     t_2
     (if (<= t_1 0.0)
       (* 100.0 (/ n (/ i (expm1 (* n (log1p (/ i n)))))))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/
           n
           (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0)))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = ((t_0 * 100.0) + -100.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-42) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1((n * log1p((i / n))))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = ((t_0 * 100.0) + -100.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-42) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1((n * Math.log1p((i / n))))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	t_2 = ((t_0 * 100.0) + -100.0) / (i / n)
	tmp = 0
	if t_1 <= -2e-42:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1((n * math.log1p((i / n))))))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-42)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(Float64(n * log1p(Float64(i / n)))))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-42], t$95$2, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2.00000000000000008e-42 or 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 99.9%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \color{blue}{-1}}{\frac{i}{n}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-100}}{\frac{i}{n}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-100\right)}}{\frac{i}{n}} \]

      7. fma-def 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{\frac{i}{n}} \]

      8. metadata-eval 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + -100}}{\frac{i}{n}} \]

      2. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100} + -100}{\frac{i}{n}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}}{\frac{i}{n}} \]

   if -2.00000000000000008e-42 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 17.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 17.7%

         \[\leadsto 100 \cdot \color{blue}{\frac{1}{\frac{\frac{i}{n}}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]

      2. associate-/r/ 17.7%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{\frac{i}{n}} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)} \]

      3. clear-num 17.1%

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{n}{i}} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right) \]

      4. add-exp-log 17.1%

         \[\leadsto 100 \cdot \left(\frac{n}{i} \cdot \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right)\right) \]

      5. expm1-def 17.1%

         \[\leadsto 100 \cdot \left(\frac{n}{i} \cdot \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)}\right) \]

      6. log-pow 33.6%

         \[\leadsto 100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)\right) \]

      7. log1p-udef 97.8%

         \[\leadsto 100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)\right) \]

   4. Applied egg-rr 97.8%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{n}{i} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 87.3%

         \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}} \]

      2. associate-/l* 98.4%

         \[\leadsto 100 \cdot \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   6. Applied egg-rr 98.4%

      \[\leadsto 100 \cdot \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 1.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 1.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 1.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 73.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;100 \cdot \frac{t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{t\_0}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (/ i n) n) -1.0))
        (t_1 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_1 (- INFINITY))
     (* 100.0 (/ t_0 (/ i n)))
     (if (<= t_1 0.0)
       (/ 1.0 (/ (/ i (expm1 i)) (* n 100.0)))
       (if (<= t_1 INFINITY)
         (* (* n 100.0) (/ t_0 i))
         (/ 1.0 (/ (+ 1.0 (* i -0.5)) (* n 100.0))))))))

C

double code(double i, double n) {
	double t_0 = pow((i / n), n) + -1.0;
	double t_1 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 100.0 * (t_0 / (i / n));
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / ((i / expm1(i)) / (n * 100.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (n * 100.0) * (t_0 / i);
	} else {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((i / n), n) + -1.0;
	double t_1 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (t_0 / (i / n));
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / ((i / Math.expm1(i)) / (n * 100.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (n * 100.0) * (t_0 / i);
	} else {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((i / n), n) + -1.0
	t_1 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 100.0 * (t_0 / (i / n))
	elif t_1 <= 0.0:
		tmp = 1.0 / ((i / math.expm1(i)) / (n * 100.0))
	elif t_1 <= math.inf:
		tmp = (n * 100.0) * (t_0 / i)
	else:
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(i / n) ^ n) + -1.0)
	t_1 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(100.0 * Float64(t_0 / Float64(i / n)));
	elseif (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(i / expm1(i)) / Float64(n * 100.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(n * 100.0) * Float64(t_0 / i));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / Float64(n * 100.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(100.0 * N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -inf.0

   1. Initial program 100.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf 100.0%

      \[\leadsto 100 \cdot \frac{{\color{blue}{\left(\frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 21.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 41.7%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 41.7%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 41.7%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 78.5%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 78.5%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 79.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 79.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 99.7%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 99.7%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 99.7%

      \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n}\right)}}^{n} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 1.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 1.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 1.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 73.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 73.6%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 73.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 73.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   8. Taylor expanded in i around 0 99.9%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 + -0.5 \cdot i}}{n \cdot 100}} \]
   9. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   10. Simplified 99.9%

       \[\leadsto \frac{1}{\frac{\color{blue}{1 + i \cdot -0.5}}{n \cdot 100}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{-213}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -2.8e-213)
     t_0
     (if (<= n 2.7e-245)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 1.75e-37) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -2.8e-213) {
		tmp = t_0;
	} else if (n <= 2.7e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 1.75e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -2.8e-213) {
		tmp = t_0;
	} else if (n <= 2.7e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 1.75e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -2.8e-213:
		tmp = t_0
	elif n <= 2.7e-245:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 1.75e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -2.8e-213)
		tmp = t_0;
	elseif (n <= 2.7e-245)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 1.75e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.8e-213], t$95$0, If[LessEqual[n, 2.7e-245], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.8e-213 or 1.7500000000000001e-37 < n

   1. Initial program 29.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 35.3%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 35.3%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 35.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 81.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 81.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   if -2.8e-213 < n < 2.69999999999999989e-245

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 2.69999999999999989e-245 < n < 1.7500000000000001e-37

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i}{\mathsf{expm1}\left(i\right)}\\ \mathbf{if}\;n \leq -6 \cdot 10^{-214}:\\ \;\;\;\;100 \cdot \frac{n}{t\_0}\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ i (expm1 i))))
   (if (<= n -6e-214)
     (* 100.0 (/ n t_0))
     (if (<= n 1.9e-245)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.8e-37) (* 100.0 (/ i (/ i n))) (/ (* n 100.0) t_0))))))

C

double code(double i, double n) {
	double t_0 = i / expm1(i);
	double tmp;
	if (n <= -6e-214) {
		tmp = 100.0 * (n / t_0);
	} else if (n <= 1.9e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) / t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = i / Math.expm1(i);
	double tmp;
	if (n <= -6e-214) {
		tmp = 100.0 * (n / t_0);
	} else if (n <= 1.9e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) / t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = i / math.expm1(i)
	tmp = 0
	if n <= -6e-214:
		tmp = 100.0 * (n / t_0)
	elif n <= 1.9e-245:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) / t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(i / expm1(i))
	tmp = 0.0
	if (n <= -6e-214)
		tmp = Float64(100.0 * Float64(n / t_0));
	elseif (n <= 1.9e-245)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) / t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6e-214], N[(100.0 * N[(n / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.9e-245], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.99999999999999989e-214

   1. Initial program 34.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 34.2%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 34.2%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 34.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 74.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   if -5.99999999999999989e-214 < n < 1.9e-245

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 1.9e-245 < n < 2.8000000000000001e-37

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 2.8000000000000001e-37 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 36.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 92.2%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   7. Applied egg-rr 92.2%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-214}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i}{\mathsf{expm1}\left(i\right)}\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \frac{0.01}{n}}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ i (expm1 i))))
   (if (<= n -4.2e-213)
     (/ 1.0 (* t_0 (/ 0.01 n)))
     (if (<= n 3.1e-245)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 5e-38) (* 100.0 (/ i (/ i n))) (/ (* n 100.0) t_0))))))

C

double code(double i, double n) {
	double t_0 = i / expm1(i);
	double tmp;
	if (n <= -4.2e-213) {
		tmp = 1.0 / (t_0 * (0.01 / n));
	} else if (n <= 3.1e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 5e-38) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) / t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = i / Math.expm1(i);
	double tmp;
	if (n <= -4.2e-213) {
		tmp = 1.0 / (t_0 * (0.01 / n));
	} else if (n <= 3.1e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 5e-38) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) / t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = i / math.expm1(i)
	tmp = 0
	if n <= -4.2e-213:
		tmp = 1.0 / (t_0 * (0.01 / n))
	elif n <= 3.1e-245:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 5e-38:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) / t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(i / expm1(i))
	tmp = 0.0
	if (n <= -4.2e-213)
		tmp = Float64(1.0 / Float64(t_0 * Float64(0.01 / n)));
	elseif (n <= 3.1e-245)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 5e-38)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) / t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.2e-213], N[(1.0 / N[(t$95$0 * N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.1e-245], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5e-38], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.1999999999999997e-213

   1. Initial program 34.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 34.2%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 34.2%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 34.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 74.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 74.1%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 74.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 74.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. div-inv 74.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{1}{n \cdot 100}}} \]

      2. *-commutative 74.5%

         \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{1}{\color{blue}{100 \cdot n}}} \]

      3. associate-/r* 74.4%

         \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \color{blue}{\frac{\frac{1}{100}}{n}}} \]

      4. metadata-eval 74.4%

         \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{\color{blue}{0.01}}{n}} \]

   9. Applied egg-rr 74.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{0.01}{n}}} \]

   if -4.1999999999999997e-213 < n < 3.10000000000000003e-245

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 3.10000000000000003e-245 < n < 5.00000000000000033e-38

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 5.00000000000000033e-38 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 36.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 92.2%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   7. Applied egg-rr 92.2%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{0.01}{n}}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i}{\mathsf{expm1}\left(i\right)}\\ \mathbf{if}\;n \leq -5.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ i (expm1 i))))
   (if (<= n -5.8e-214)
     (/ 1.0 (/ t_0 (* n 100.0)))
     (if (<= n 2.4e-245)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.8e-37) (* 100.0 (/ i (/ i n))) (/ (* n 100.0) t_0))))))

C

double code(double i, double n) {
	double t_0 = i / expm1(i);
	double tmp;
	if (n <= -5.8e-214) {
		tmp = 1.0 / (t_0 / (n * 100.0));
	} else if (n <= 2.4e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) / t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = i / Math.expm1(i);
	double tmp;
	if (n <= -5.8e-214) {
		tmp = 1.0 / (t_0 / (n * 100.0));
	} else if (n <= 2.4e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) / t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = i / math.expm1(i)
	tmp = 0
	if n <= -5.8e-214:
		tmp = 1.0 / (t_0 / (n * 100.0))
	elif n <= 2.4e-245:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) / t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(i / expm1(i))
	tmp = 0.0
	if (n <= -5.8e-214)
		tmp = Float64(1.0 / Float64(t_0 / Float64(n * 100.0)));
	elseif (n <= 2.4e-245)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) / t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.8e-214], N[(1.0 / N[(t$95$0 / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.4e-245], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.7999999999999997e-214

   1. Initial program 34.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 34.2%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 34.2%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 34.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 74.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 74.1%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 74.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 74.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   if -5.7999999999999997e-214 < n < 2.4e-245

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 2.4e-245 < n < 2.8000000000000001e-37

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 2.8000000000000001e-37 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 36.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 92.2%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   7. Applied egg-rr 92.2%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.00165 \lor \neg \left(i \leq 6.1 \cdot 10^{-12}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -0.00165) (not (<= i 6.1e-12)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -0.00165) || !(i <= 6.1e-12)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -0.00165) || !(i <= 6.1e-12)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -0.00165) or not (i <= 6.1e-12):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -0.00165) || !(i <= 6.1e-12))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -0.00165], N[Not[LessEqual[i, 6.1e-12]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -0.00165 or 6.1000000000000003e-12 < i

   1. Initial program 49.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 58.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. expm1-def 58.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   5. Simplified 58.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   if -0.00165 < i < 6.1000000000000003e-12

   1. Initial program 7.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 7.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 7.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 7.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 84.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 84.8%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 84.8%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 84.8%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 84.8%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00165 \lor \neg \left(i \leq 6.1 \cdot 10^{-12}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.00165:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-12}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.00165)
   (* 100.0 (/ (expm1 i) (/ i n)))
   (if (<= i 8.6e-12)
     (* n (+ 100.0 (* i 50.0)))
     (* 100.0 (* (/ n i) (expm1 i))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.00165) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (i <= 8.6e-12) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * ((n / i) * expm1(i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.00165) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (i <= 8.6e-12) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * ((n / i) * Math.expm1(i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.00165:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif i <= 8.6e-12:
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * ((n / i) * math.expm1(i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.00165)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (i <= 8.6e-12)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(Float64(n / i) * expm1(i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.00165], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.6e-12], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -0.00165

   1. Initial program 60.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 75.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{i} - 1}}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. expm1-def 75.6%

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   5. Simplified 75.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]

   if -0.00165 < i < 8.59999999999999971e-12

   1. Initial program 7.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 7.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 7.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 7.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 84.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 84.8%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 84.8%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 84.8%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 84.8%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   if 8.59999999999999971e-12 < i

   1. Initial program 42.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 46.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 46.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 46.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 47.5%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-/r/ 47.2%

         \[\leadsto \color{blue}{\left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)} \cdot 100 \]

   7. Applied egg-rr 47.2%

      \[\leadsto \color{blue}{\left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00165:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.6 \cdot 10^{-12}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 6.9 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 6.9e+118)
   (/ 1.0 (/ (/ i (expm1 i)) (* n 100.0)))
   (* 100.0 (/ (+ (pow (/ i n) n) -1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 6.9e+118) {
		tmp = 1.0 / ((i / expm1(i)) / (n * 100.0));
	} else {
		tmp = 100.0 * ((pow((i / n), n) + -1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 6.9e+118) {
		tmp = 1.0 / ((i / Math.expm1(i)) / (n * 100.0));
	} else {
		tmp = 100.0 * ((Math.pow((i / n), n) + -1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 6.9e+118:
		tmp = 1.0 / ((i / math.expm1(i)) / (n * 100.0))
	else:
		tmp = 100.0 * ((math.pow((i / n), n) + -1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 6.9e+118)
		tmp = Float64(1.0 / Float64(Float64(i / expm1(i)) / Float64(n * 100.0)));
	else
		tmp = Float64(100.0 * Float64(Float64((Float64(i / n) ^ n) + -1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 6.9e+118], N[(1.0 / N[(N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 6.90000000000000003e118

   1. Initial program 20.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 31.2%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 31.2%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 31.2%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 79.4%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 79.3%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 79.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 79.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   if 6.90000000000000003e118 < i

   1. Initial program 62.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf 66.5%

      \[\leadsto 100 \cdot \frac{{\color{blue}{\left(\frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6.9 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 8.3 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e+251)
   (* (/ 1.0 i) (* i (* n 100.0)))
   (if (<= n -5e-214)
     (/ 1.0 (/ (+ 1.0 (* i -0.5)) (* n 100.0)))
     (if (<= n 1.15e-245)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 8.3e-38)
         (* 100.0 (/ i (/ i n)))
         (* 100.0 (+ n (* i (* n (- 0.5 (/ 0.5 n)))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -5e-214) {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	} else if (n <= 1.15e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 8.3e-38) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.4d+251)) then
        tmp = (1.0d0 / i) * (i * (n * 100.0d0))
    else if (n <= (-5d-214)) then
        tmp = 1.0d0 / ((1.0d0 + (i * (-0.5d0))) / (n * 100.0d0))
    else if (n <= 1.15d-245) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (n <= 8.3d-38) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * (n + (i * (n * (0.5d0 - (0.5d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -5e-214) {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	} else if (n <= 1.15e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 8.3e-38) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e+251:
		tmp = (1.0 / i) * (i * (n * 100.0))
	elif n <= -5e-214:
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0))
	elif n <= 1.15e-245:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 8.3e-38:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e+251)
		tmp = Float64(Float64(1.0 / i) * Float64(i * Float64(n * 100.0)));
	elseif (n <= -5e-214)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / Float64(n * 100.0)));
	elseif (n <= 1.15e-245)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 8.3e-38)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(0.5 - Float64(0.5 / n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.4e+251)
		tmp = (1.0 / i) * (i * (n * 100.0));
	elseif (n <= -5e-214)
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	elseif (n <= 1.15e-245)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (n <= 8.3e-38)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e+251], N[(N[(1.0 / i), $MachinePrecision] * N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5e-214], N[(1.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-245], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.3e-38], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(i * N[(n * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.40000000000000011e251

   1. Initial program 31.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 69.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 69.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 99.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 99.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}}} \]

      2. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{1}{i} \cdot \color{blue}{\left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   10. Taylor expanded in i around 0 64.3%

       \[\leadsto \frac{1}{i} \cdot \left(\color{blue}{i} \cdot \left(n \cdot 100\right)\right) \]

   if -3.40000000000000011e251 < n < -4.9999999999999998e-214

   1. Initial program 34.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 29.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 29.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 29.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 70.4%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 70.4%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 70.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   8. Taylor expanded in i around 0 55.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 + -0.5 \cdot i}}{n \cdot 100}} \]
   9. Step-by-step derivation

      1. *-commutative 54.9%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   10. Simplified 55.4%

       \[\leadsto \frac{1}{\frac{\color{blue}{1 + i \cdot -0.5}}{n \cdot 100}} \]

   if -4.9999999999999998e-214 < n < 1.1500000000000001e-245

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 1.1500000000000001e-245 < n < 8.2999999999999995e-38

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 8.2999999999999995e-38 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 78.6%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 78.6%

         \[\leadsto 100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \]

      2. metadata-eval 78.6%

         \[\leadsto 100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right)\right) \]

   5. Simplified 78.6%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 8.3 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e+251)
   (* (/ 1.0 i) (* i (* n 100.0)))
   (if (<= n -1.65e-214)
     (/ 1.0 (/ (+ 1.0 (* i -0.5)) (* n 100.0)))
     (if (<= n 3.4e-245)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.8e-37)
         (* 100.0 (/ i (/ i n)))
         (* (* n 100.0) (+ 1.0 (* i (- 0.5 (/ 0.5 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -1.65e-214) {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	} else if (n <= 3.4e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.4d+251)) then
        tmp = (1.0d0 / i) * (i * (n * 100.0d0))
    else if (n <= (-1.65d-214)) then
        tmp = 1.0d0 / ((1.0d0 + (i * (-0.5d0))) / (n * 100.0d0))
    else if (n <= 3.4d-245) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (n <= 2.8d-37) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * (0.5d0 - (0.5d0 / n))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -1.65e-214) {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	} else if (n <= 3.4e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e+251:
		tmp = (1.0 / i) * (i * (n * 100.0))
	elif n <= -1.65e-214:
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0))
	elif n <= 3.4e-245:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e+251)
		tmp = Float64(Float64(1.0 / i) * Float64(i * Float64(n * 100.0)));
	elseif (n <= -1.65e-214)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / Float64(n * 100.0)));
	elseif (n <= 3.4e-245)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 - Float64(0.5 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.4e+251)
		tmp = (1.0 / i) * (i * (n * 100.0));
	elseif (n <= -1.65e-214)
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	elseif (n <= 3.4e-245)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (n <= 2.8e-37)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e+251], N[(N[(1.0 / i), $MachinePrecision] * N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.65e-214], N[(1.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.4e-245], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.40000000000000011e251

   1. Initial program 31.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 69.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 69.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 99.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 99.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}}} \]

      2. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{1}{i} \cdot \color{blue}{\left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   10. Taylor expanded in i around 0 64.3%

       \[\leadsto \frac{1}{i} \cdot \left(\color{blue}{i} \cdot \left(n \cdot 100\right)\right) \]

   if -3.40000000000000011e251 < n < -1.6499999999999999e-214

   1. Initial program 34.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 29.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 29.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 29.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 70.4%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 70.4%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 70.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   8. Taylor expanded in i around 0 55.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 + -0.5 \cdot i}}{n \cdot 100}} \]
   9. Step-by-step derivation

      1. *-commutative 54.9%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   10. Simplified 55.4%

       \[\leadsto \frac{1}{\frac{\color{blue}{1 + i \cdot -0.5}}{n \cdot 100}} \]

   if -1.6499999999999999e-214 < n < 3.3999999999999999e-245

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 3.3999999999999999e-245 < n < 2.8000000000000001e-37

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 2.8000000000000001e-37 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 21.2%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 21.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 21.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 21.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 21.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 78.6%

      \[\leadsto \color{blue}{\left(1 + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 78.6%

         \[\leadsto \left(1 + i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot \left(n \cdot 100\right) \]

      2. metadata-eval 78.6%

         \[\leadsto \left(1 + i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot \left(n \cdot 100\right) \]

   7. Simplified 78.6%

      \[\leadsto \color{blue}{\left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)} \cdot \left(n \cdot 100\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-214}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -1.8 \cdot 10^{-216}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e+251)
   (* (/ 1.0 i) (* i (* n 100.0)))
   (if (<= n -1.8e-216)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 1e-245)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.5e-37)
         (* 100.0 (/ i (/ i n)))
         (* n (+ 100.0 (* i 50.0))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -1.8e-216) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.5e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.4d+251)) then
        tmp = (1.0d0 / i) * (i * (n * 100.0d0))
    else if (n <= (-1.8d-216)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1d-245) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (n <= 2.5d-37) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -1.8e-216) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.5e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e+251:
		tmp = (1.0 / i) * (i * (n * 100.0))
	elif n <= -1.8e-216:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1e-245:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.5e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e+251)
		tmp = Float64(Float64(1.0 / i) * Float64(i * Float64(n * 100.0)));
	elseif (n <= -1.8e-216)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1e-245)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.5e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.4e+251)
		tmp = (1.0 / i) * (i * (n * 100.0));
	elseif (n <= -1.8e-216)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1e-245)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (n <= 2.5e-37)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e+251], N[(N[(1.0 / i), $MachinePrecision] * N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.8e-216], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1e-245], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.40000000000000011e251

   1. Initial program 31.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 69.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 69.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 99.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 99.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}}} \]

      2. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{1}{i} \cdot \color{blue}{\left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   10. Taylor expanded in i around 0 64.3%

       \[\leadsto \frac{1}{i} \cdot \left(\color{blue}{i} \cdot \left(n \cdot 100\right)\right) \]

   if -3.40000000000000011e251 < n < -1.7999999999999999e-216

   1. Initial program 34.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 29.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 29.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 29.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 70.4%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 54.9%

      \[\leadsto \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \cdot 100 \]
   7. Step-by-step derivation

      1. *-commutative 54.9%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   8. Simplified 54.9%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]

   if -1.7999999999999999e-216 < n < 9.9999999999999993e-246

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 9.9999999999999993e-246 < n < 2.4999999999999999e-37

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 2.4999999999999999e-37 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 36.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 78.3%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 78.3%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 78.3%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 78.3%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -1.8 \cdot 10^{-216}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.9 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -2.35 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{\left(1 + i \cdot -0.5\right) \cdot \frac{0.01}{n}}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.9e+251)
   (* (/ 1.0 i) (* i (* n 100.0)))
   (if (<= n -2.35e-213)
     (/ 1.0 (* (+ 1.0 (* i -0.5)) (/ 0.01 n)))
     (if (<= n 1.5e-245)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.8e-37)
         (* 100.0 (/ i (/ i n)))
         (* n (+ 100.0 (* i 50.0))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.9e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -2.35e-213) {
		tmp = 1.0 / ((1.0 + (i * -0.5)) * (0.01 / n));
	} else if (n <= 1.5e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.9d+251)) then
        tmp = (1.0d0 / i) * (i * (n * 100.0d0))
    else if (n <= (-2.35d-213)) then
        tmp = 1.0d0 / ((1.0d0 + (i * (-0.5d0))) * (0.01d0 / n))
    else if (n <= 1.5d-245) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (n <= 2.8d-37) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.9e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -2.35e-213) {
		tmp = 1.0 / ((1.0 + (i * -0.5)) * (0.01 / n));
	} else if (n <= 1.5e-245) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.9e+251:
		tmp = (1.0 / i) * (i * (n * 100.0))
	elif n <= -2.35e-213:
		tmp = 1.0 / ((1.0 + (i * -0.5)) * (0.01 / n))
	elif n <= 1.5e-245:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.9e+251)
		tmp = Float64(Float64(1.0 / i) * Float64(i * Float64(n * 100.0)));
	elseif (n <= -2.35e-213)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(i * -0.5)) * Float64(0.01 / n)));
	elseif (n <= 1.5e-245)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.9e+251)
		tmp = (1.0 / i) * (i * (n * 100.0));
	elseif (n <= -2.35e-213)
		tmp = 1.0 / ((1.0 + (i * -0.5)) * (0.01 / n));
	elseif (n <= 1.5e-245)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (n <= 2.8e-37)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.9e+251], N[(N[(1.0 / i), $MachinePrecision] * N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.35e-213], N[(1.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] * N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5e-245], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.89999999999999976e251

   1. Initial program 31.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 69.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 69.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 99.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 99.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}}} \]

      2. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{1}{i} \cdot \color{blue}{\left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   10. Taylor expanded in i around 0 64.3%

       \[\leadsto \frac{1}{i} \cdot \left(\color{blue}{i} \cdot \left(n \cdot 100\right)\right) \]

   if -3.89999999999999976e251 < n < -2.35e-213

   1. Initial program 34.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 29.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 29.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 29.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 70.4%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 70.4%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 70.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. div-inv 70.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{1}{n \cdot 100}}} \]

      2. *-commutative 70.8%

         \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{1}{\color{blue}{100 \cdot n}}} \]

      3. associate-/r* 70.7%

         \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \color{blue}{\frac{\frac{1}{100}}{n}}} \]

      4. metadata-eval 70.7%

         \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{\color{blue}{0.01}}{n}} \]

   9. Applied egg-rr 70.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{i}{\mathsf{expm1}\left(i\right)} \cdot \frac{0.01}{n}}} \]

   10. Taylor expanded in i around 0 55.3%

       \[\leadsto \frac{1}{\color{blue}{\left(1 + -0.5 \cdot i\right)} \cdot \frac{0.01}{n}} \]
   11. Step-by-step derivation

       1. *-commutative 54.9%

          \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   12. Simplified 55.3%

       \[\leadsto \frac{1}{\color{blue}{\left(1 + i \cdot -0.5\right)} \cdot \frac{0.01}{n}} \]

   if -2.35e-213 < n < 1.5000000000000001e-245

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 1.5000000000000001e-245 < n < 2.8000000000000001e-37

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 2.8000000000000001e-37 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 36.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 78.3%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 78.3%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 78.3%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 78.3%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.9 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -2.35 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{\left(1 + i \cdot -0.5\right) \cdot \frac{0.01}{n}}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-245}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -1.15 \cdot 10^{-215}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-246}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e+251)
   (* (/ 1.0 i) (* i (* n 100.0)))
   (if (<= n -1.15e-215)
     (/ 1.0 (/ (+ 1.0 (* i -0.5)) (* n 100.0)))
     (if (<= n 3.3e-246)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.8e-37)
         (* 100.0 (/ i (/ i n)))
         (* n (+ 100.0 (* i 50.0))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -1.15e-215) {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	} else if (n <= 3.3e-246) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.4d+251)) then
        tmp = (1.0d0 / i) * (i * (n * 100.0d0))
    else if (n <= (-1.15d-215)) then
        tmp = 1.0d0 / ((1.0d0 + (i * (-0.5d0))) / (n * 100.0d0))
    else if (n <= 3.3d-246) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (n <= 2.8d-37) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= -1.15e-215) {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	} else if (n <= 3.3e-246) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e+251:
		tmp = (1.0 / i) * (i * (n * 100.0))
	elif n <= -1.15e-215:
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0))
	elif n <= 3.3e-246:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e+251)
		tmp = Float64(Float64(1.0 / i) * Float64(i * Float64(n * 100.0)));
	elseif (n <= -1.15e-215)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / Float64(n * 100.0)));
	elseif (n <= 3.3e-246)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.4e+251)
		tmp = (1.0 / i) * (i * (n * 100.0));
	elseif (n <= -1.15e-215)
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	elseif (n <= 3.3e-246)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (n <= 2.8e-37)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e+251], N[(N[(1.0 / i), $MachinePrecision] * N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.15e-215], N[(1.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.3e-246], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.40000000000000011e251

   1. Initial program 31.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 69.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 69.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 99.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 99.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}}} \]

      2. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{1}{i} \cdot \color{blue}{\left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   10. Taylor expanded in i around 0 64.3%

       \[\leadsto \frac{1}{i} \cdot \left(\color{blue}{i} \cdot \left(n \cdot 100\right)\right) \]

   if -3.40000000000000011e251 < n < -1.15e-215

   1. Initial program 34.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 29.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 29.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 29.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 70.4%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 70.4%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 70.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   8. Taylor expanded in i around 0 55.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 + -0.5 \cdot i}}{n \cdot 100}} \]
   9. Step-by-step derivation

      1. *-commutative 54.9%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   10. Simplified 55.4%

       \[\leadsto \frac{1}{\frac{\color{blue}{1 + i \cdot -0.5}}{n \cdot 100}} \]

   if -1.15e-215 < n < 3.3000000000000001e-246

   1. Initial program 66.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 63.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. associate-*l* 63.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)} \]

      4. sub-neg 63.8%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot \left(n \cdot 100\right) \]

      5. metadata-eval 63.8%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot \left(n \cdot 100\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{\color{blue}{1} + -1}{i} \cdot \left(n \cdot 100\right) \]

   if 3.3000000000000001e-246 < n < 2.8000000000000001e-37

   1. Initial program 3.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 2.8000000000000001e-37 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 36.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 78.3%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 78.3%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 78.3%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 78.3%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq -1.15 \cdot 10^{-215}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-246}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq 0.054:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e+251)
   (* (/ 1.0 i) (* i (* n 100.0)))
   (if (<= n 0.054)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= 0.054) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.4d+251)) then
        tmp = (1.0d0 / i) * (i * (n * 100.0d0))
    else if (n <= 0.054d0) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= 0.054) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e+251:
		tmp = (1.0 / i) * (i * (n * 100.0))
	elif n <= 0.054:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e+251)
		tmp = Float64(Float64(1.0 / i) * Float64(i * Float64(n * 100.0)));
	elseif (n <= 0.054)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.4e+251)
		tmp = (1.0 / i) * (i * (n * 100.0));
	elseif (n <= 0.054)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e+251], N[(N[(1.0 / i), $MachinePrecision] * N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.054], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.40000000000000011e251

   1. Initial program 31.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 69.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 69.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 99.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 99.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}}} \]

      2. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{1}{i} \cdot \color{blue}{\left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   10. Taylor expanded in i around 0 64.3%

       \[\leadsto \frac{1}{i} \cdot \left(\color{blue}{i} \cdot \left(n \cdot 100\right)\right) \]

   if -3.40000000000000011e251 < n < 0.0539999999999999994

   1. Initial program 31.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 26.4%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 26.4%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 26.4%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 58.9%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 57.4%

      \[\leadsto \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \cdot 100 \]
   7. Step-by-step derivation

      1. *-commutative 57.4%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   8. Simplified 57.4%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]

   if 0.0539999999999999994 < n

   1. Initial program 21.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.8%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 37.8%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.3%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.3%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 78.1%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 78.1%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 78.1%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 78.1%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq 0.054:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+56} \lor \neg \left(n \leq 2.8 \cdot 10^{-37}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.2e+56) (not (<= n 2.8e-37)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.2e+56) || !(n <= 2.8e-37)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-4.2d+56)) .or. (.not. (n <= 2.8d-37))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.2e+56) || !(n <= 2.8e-37)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.2e+56) or not (n <= 2.8e-37):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.2e+56) || !(n <= 2.8e-37))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -4.2e+56) || ~((n <= 2.8e-37)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.2e+56], N[Not[LessEqual[n, 2.8e-37]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.20000000000000034e56 or 2.8000000000000001e-37 < n

   1. Initial program 26.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.8%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 37.8%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 88.1%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 67.3%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 67.3%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 67.3%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 67.3%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   if -4.20000000000000034e56 < n < 2.8000000000000001e-37

   1. Initial program 31.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 55.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+56} \lor \neg \left(n \leq 2.8 \cdot 10^{-37}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 62.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.3e+251)
   (* (/ 1.0 i) (* i (* n 100.0)))
   (if (<= n 1.45e-37)
     (/ n (+ 0.01 (* i -0.005)))
     (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.3e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= 1.45e-37) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.3d+251)) then
        tmp = (1.0d0 / i) * (i * (n * 100.0d0))
    else if (n <= 1.45d-37) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.3e+251) {
		tmp = (1.0 / i) * (i * (n * 100.0));
	} else if (n <= 1.45e-37) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.3e+251:
		tmp = (1.0 / i) * (i * (n * 100.0))
	elif n <= 1.45e-37:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.3e+251)
		tmp = Float64(Float64(1.0 / i) * Float64(i * Float64(n * 100.0)));
	elseif (n <= 1.45e-37)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.3e+251)
		tmp = (1.0 / i) * (i * (n * 100.0));
	elseif (n <= 1.45e-37)
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.3e+251], N[(N[(1.0 / i), $MachinePrecision] * N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.45e-37], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.3e251

   1. Initial program 31.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 69.1%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 69.1%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 99.8%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 99.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{i}{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}}} \]

      2. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{1}{i} \cdot \color{blue}{\left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{i} \cdot \left(\mathsf{expm1}\left(i\right) \cdot \left(n \cdot 100\right)\right)} \]

   10. Taylor expanded in i around 0 64.3%

       \[\leadsto \frac{1}{i} \cdot \left(\color{blue}{i} \cdot \left(n \cdot 100\right)\right) \]

   if -4.3e251 < n < 1.45000000000000002e-37

   1. Initial program 31.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 26.7%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 26.7%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 26.7%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 58.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 58.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 58.6%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. associate-/l* 58.4%

         \[\leadsto \color{blue}{\frac{n}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{100}}} \]

   7. Applied egg-rr 58.4%

      \[\leadsto \color{blue}{\frac{n}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{100}}} \]

   8. Taylor expanded in i around 0 56.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]

   if 1.45000000000000002e-37 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 36.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 78.3%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 78.3%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 78.3%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 78.3%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{i} \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 57.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 1.08 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2.0) (not (<= i 1.08e+92))) (* (/ n i) -200.0) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 1.08e+92)) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-2.0d0)) .or. (.not. (i <= 1.08d+92))) then
        tmp = (n / i) * (-200.0d0)
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 1.08e+92)) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2.0) or not (i <= 1.08e+92):
		tmp = (n / i) * -200.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2.0) || !(i <= 1.08e+92))
		tmp = Float64(Float64(n / i) * -200.0);
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -2.0) || ~((i <= 1.08e+92)))
		tmp = (n / i) * -200.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2.0], N[Not[LessEqual[i, 1.08e+92]], $MachinePrecision]], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2 or 1.08e92 < i

   1. Initial program 60.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 56.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 56.8%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 56.7%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 56.7%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 31.1%

      \[\leadsto \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \cdot 100 \]
   7. Step-by-step derivation

      1. *-commutative 31.1%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   8. Simplified 31.1%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]

   9. Taylor expanded in i around inf 31.1%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]

   if -2 < i < 1.08e92

   1. Initial program 7.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 72.9%

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-commutative 72.9%

         \[\leadsto \color{blue}{n \cdot 100} \]

   5. Simplified 72.9%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 1.08 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 58.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 4 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2.0) (not (<= i 4e+90))) (/ -200.0 (/ i n)) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 4e+90)) {
		tmp = -200.0 / (i / n);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-2.0d0)) .or. (.not. (i <= 4d+90))) then
        tmp = (-200.0d0) / (i / n)
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 4e+90)) {
		tmp = -200.0 / (i / n);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2.0) or not (i <= 4e+90):
		tmp = -200.0 / (i / n)
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2.0) || !(i <= 4e+90))
		tmp = Float64(-200.0 / Float64(i / n));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -2.0) || ~((i <= 4e+90)))
		tmp = -200.0 / (i / n);
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2.0], N[Not[LessEqual[i, 4e+90]], $MachinePrecision]], N[(-200.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2 or 3.99999999999999987e90 < i

   1. Initial program 60.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 56.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 56.8%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 56.7%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 56.7%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 31.1%

      \[\leadsto \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \cdot 100 \]
   7. Step-by-step derivation

      1. *-commutative 31.1%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   8. Simplified 31.1%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]

   9. Taylor expanded in i around inf 31.1%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]
   10. Step-by-step derivation

       1. associate-*r/ 31.1%

          \[\leadsto \color{blue}{\frac{-200 \cdot n}{i}} \]

       2. associate-/l* 32.5%

          \[\leadsto \color{blue}{\frac{-200}{\frac{i}{n}}} \]

   11. Simplified 32.5%

       \[\leadsto \color{blue}{\frac{-200}{\frac{i}{n}}} \]

   if -2 < i < 3.99999999999999987e90

   1. Initial program 7.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 72.9%

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-commutative 72.9%

         \[\leadsto \color{blue}{n \cdot 100} \]

   5. Simplified 72.9%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 4 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 62.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 2.2e-37) (/ n (+ 0.01 (* i -0.005))) (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 2.2e-37) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 2.2d-37) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 2.2e-37) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 2.2e-37:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 2.2e-37)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 2.2e-37)
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 2.2e-37], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.20000000000000002e-37

   1. Initial program 31.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 30.5%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 30.5%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 30.5%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 62.3%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   6. Step-by-step derivation

      1. associate-*l/ 62.3%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. associate-/l* 62.2%

         \[\leadsto \color{blue}{\frac{n}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{100}}} \]

   7. Applied egg-rr 62.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{100}}} \]

   8. Taylor expanded in i around 0 55.3%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]

   if 2.20000000000000002e-37 < n

   1. Initial program 21.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 37.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 36.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   5. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   6. Taylor expanded in i around 0 78.3%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* 78.3%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 78.3%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   8. Simplified 78.3%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 49.0% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * 100.0d0
end function

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 28.2%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0 45.8%

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. *-commutative 45.8%

      \[\leadsto \color{blue}{n \cdot 100} \]

5. Simplified 45.8%

   \[\leadsto \color{blue}{n \cdot 100} \]

6. Final simplification 45.8%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer target: 34.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024031 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))