Compound Interest

Percentage Accurate: 28.5% → 98.6%

Time: 21.3s

Alternatives: 16

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: 28.5% 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: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 10^{-212}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 (- INFINITY))
     (* t_1 100.0)
     (if (<= t_1 1e-212)
       (/ (* n 100.0) (/ i (expm1 (* n (log1p (/ i n))))))
       (if (<= t_1 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5)))))))))

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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 1e-212) {
		tmp = (n * 100.0) / (i / expm1((n * log1p((i / n)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	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 tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 1e-212) {
		tmp = (n * 100.0) / (i / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_1 * 100.0
	elif t_1 <= 1e-212:
		tmp = (n * 100.0) / (i / math.expm1((n * math.log1p((i / n)))))
	elif t_1 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	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))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 1e-212)
		tmp = Float64(Float64(n * 100.0) / Float64(i / expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 1e-212], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $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%

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

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 9.99999999999999954e-213

   1. Initial program 19.4%

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

      1. *-commutative 19.4%

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

      2. associate-/r/ 19.4%

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

      3. sub-neg 19.4%

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

      4. metadata-eval 19.4%

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

      5. associate-*r* 19.5%

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

      6. *-commutative 19.5%

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

      7. clear-num 19.5%

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

      8. un-div-inv 19.5%

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

      9. metadata-eval 19.5%

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

      10. sub-neg 19.5%

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

      11. pow-to-exp 19.5%

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

      12. expm1-def 33.3%

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

      13. *-commutative 33.3%

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

      14. log1p-udef 99.6%

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

   3. Applied egg-rr 99.6%

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

   if 9.99999999999999954e-213 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.5%

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

      1. div-sub 99.5%

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

      2. clear-num 99.7%

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

      3. sub-neg 99.7%

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

      4. div-inv 99.7%

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

      5. clear-num 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. sub-neg 99.7%

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

   5. Simplified 99.7%

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

   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. Step-by-step derivation

      1. *-commutative 0.0%

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

      2. associate-/r/ 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. associate-*r* 1.9%

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

      6. *-commutative 1.9%

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

      7. clear-num 1.9%

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

      8. un-div-inv 1.9%

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

      9. metadata-eval 1.9%

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

      10. sub-neg 1.9%

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

      11. pow-to-exp 1.9%

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

      12. expm1-def 1.9%

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

      13. *-commutative 1.9%

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

      14. log1p-udef 1.9%

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

   3. Applied egg-rr 1.9%

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

   4. Taylor expanded in i around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-212}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 2: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-160}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{t_1}{i}\\ \mathbf{elif}\;t_2 \leq 10^{-212}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -5e-160)
     (* (* n 100.0) (/ t_1 i))
     (if (<= t_2 1e-212)
       (* (expm1 (* n (log1p (/ i n)))) (* 100.0 (/ n i)))
       (if (<= t_2 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -5e-160) {
		tmp = (n * 100.0) * (t_1 / i);
	} else if (t_2 <= 1e-212) {
		tmp = expm1((n * log1p((i / n)))) * (100.0 * (n / i));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	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;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -5e-160) {
		tmp = (n * 100.0) * (t_1 / i);
	} else if (t_2 <= 1e-212) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) * (100.0 * (n / i));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = t_0 + -1.0
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -5e-160:
		tmp = (n * 100.0) * (t_1 / i)
	elif t_2 <= 1e-212:
		tmp = math.expm1((n * math.log1p((i / n)))) * (100.0 * (n / i))
	elif t_2 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -5e-160)
		tmp = Float64(Float64(n * 100.0) * Float64(t_1 / i));
	elseif (t_2 <= 1e-212)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 * Float64(n / i)));
	elseif (t_2 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	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[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-160], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-212], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $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)) < -4.99999999999999994e-160

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r/ 99.6%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   if -4.99999999999999994e-160 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 9.99999999999999954e-213

   1. Initial program 16.2%

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

      1. associate-*r/ 16.2%

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

      2. sub-neg 16.2%

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

      3. distribute-lft-in 16.2%

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

      4. fma-def 16.2%

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

      5. metadata-eval 16.2%

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

      6. metadata-eval 16.2%

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

   3. Simplified 16.2%

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

      1. fma-udef 16.2%

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

      2. metadata-eval 16.2%

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

      3. distribute-lft-in 16.2%

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

      4. metadata-eval 16.2%

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

      5. sub-neg 16.2%

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

      6. associate-*r/ 16.2%

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

      7. *-commutative 16.2%

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

      8. div-inv 16.2%

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

      9. clear-num 16.2%

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

      10. associate-*l* 16.2%

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

      11. pow-to-exp 16.2%

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

      12. expm1-def 30.6%

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

      13. *-commutative 30.6%

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

      14. log1p-udef 98.9%

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

   5. Applied egg-rr 98.9%

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

   if 9.99999999999999954e-213 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.5%

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

      1. div-sub 99.5%

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

      2. clear-num 99.7%

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

      3. sub-neg 99.7%

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

      4. div-inv 99.7%

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

      5. clear-num 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. sub-neg 99.7%

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

   5. Simplified 99.7%

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

   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. Step-by-step derivation

      1. *-commutative 0.0%

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

      2. associate-/r/ 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. associate-*r* 1.9%

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

      6. *-commutative 1.9%

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

      7. clear-num 1.9%

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

      8. un-div-inv 1.9%

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

      9. metadata-eval 1.9%

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

      10. sub-neg 1.9%

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

      11. pow-to-exp 1.9%

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

      12. expm1-def 1.9%

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

      13. *-commutative 1.9%

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

      14. log1p-udef 1.9%

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

   3. Applied egg-rr 1.9%

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

   4. Taylor expanded in i around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-160}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-212}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 3: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-83}:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 10^{-212}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -2e-83)
     (* t_1 100.0)
     (if (<= t_1 1e-212)
       (* n (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) i)))
       (if (<= t_1 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5)))))))))

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 tmp;
	if (t_1 <= -2e-83) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 1e-212) {
		tmp = n * (100.0 * (expm1((n * log1p((i / n)))) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	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 tmp;
	if (t_1 <= -2e-83) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 1e-212) {
		tmp = n * (100.0 * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -2e-83:
		tmp = t_1 * 100.0
	elif t_1 <= 1e-212:
		tmp = n * (100.0 * (math.expm1((n * math.log1p((i / n)))) / i))
	elif t_1 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	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))
	tmp = 0.0
	if (t_1 <= -2e-83)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 1e-212)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	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]}, If[LessEqual[t$95$1, -2e-83], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 1e-212], N[(n * N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $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)) < -2.0000000000000001e-83

   1. Initial program 100.0%

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

   if -2.0000000000000001e-83 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 9.99999999999999954e-213

   1. Initial program 17.1%

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

      1. associate-*r/ 17.1%

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

      2. sub-neg 17.1%

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

      3. distribute-lft-in 17.1%

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

      4. fma-def 17.1%

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

      5. metadata-eval 17.1%

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

      6. metadata-eval 17.1%

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

   3. Simplified 17.1%

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

      1. fma-udef 17.1%

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

      2. metadata-eval 17.1%

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

      3. distribute-lft-in 17.1%

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

      4. metadata-eval 17.1%

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

      5. sub-neg 17.1%

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

      6. associate-*r/ 17.1%

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

      7. *-commutative 17.1%

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

      8. associate-/r/ 17.1%

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

      9. sub-neg 17.1%

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

      10. metadata-eval 17.1%

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

      11. associate-*r* 17.1%

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

      12. *-commutative 17.1%

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

      13. associate-*r* 17.1%

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

   5. Applied egg-rr 99.5%

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

   if 9.99999999999999954e-213 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.5%

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

      1. div-sub 99.5%

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

      2. clear-num 99.7%

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

      3. sub-neg 99.7%

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

      4. div-inv 99.7%

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

      5. clear-num 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. sub-neg 99.7%

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

   5. Simplified 99.7%

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

   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. Step-by-step derivation

      1. *-commutative 0.0%

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

      2. associate-/r/ 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. associate-*r* 1.9%

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

      6. *-commutative 1.9%

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

      7. clear-num 1.9%

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

      8. un-div-inv 1.9%

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

      9. metadata-eval 1.9%

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

      10. sub-neg 1.9%

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

      11. pow-to-exp 1.9%

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

      12. expm1-def 1.9%

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

      13. *-commutative 1.9%

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

      14. log1p-udef 1.9%

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

   3. Applied egg-rr 1.9%

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

   4. Taylor expanded in i around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-83}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-212}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 4: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.45 \cdot 10^{-6} \lor \neg \left(i \leq 6.1 \cdot 10^{+15}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.45e-6) (not (<= i 6.1e+15)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.45e-6) || !(i <= 6.1e+15)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.45e-6) || !(i <= 6.1e+15)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.45e-6) or not (i <= 6.1e+15):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.45e-6) || !(i <= 6.1e+15))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.45e-6], N[Not[LessEqual[i, 6.1e+15]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.4500000000000001e-6 or 6.1e15 < i

   1. Initial program 45.6%

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

   2. Taylor expanded in n around inf 57.9%

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

      1. expm1-def 57.9%

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

   4. Simplified 57.9%

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

   if -1.4500000000000001e-6 < i < 6.1e15

   1. Initial program 9.1%

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

      1. *-commutative 9.1%

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

      2. associate-/r/ 9.7%

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

      3. sub-neg 9.7%

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

      4. metadata-eval 9.7%

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

      5. associate-*r* 9.7%

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

      6. *-commutative 9.7%

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

      7. clear-num 9.7%

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

      8. un-div-inv 9.7%

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

      9. metadata-eval 9.7%

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

      10. sub-neg 9.7%

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

      11. pow-to-exp 9.7%

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

      12. expm1-def 17.8%

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

      13. *-commutative 17.8%

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

      14. log1p-udef 73.0%

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

   3. Applied egg-rr 73.0%

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

   4. Taylor expanded in i around 0 93.5%

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

      1. sub-neg 93.5%

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

      2. associate-*r/ 93.5%

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

      3. metadata-eval 93.5%

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

      4. metadata-eval 93.5%

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

   6. Simplified 93.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.45 \cdot 10^{-6} \lor \neg \left(i \leq 6.1 \cdot 10^{+15}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 5: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+27} \lor \neg \left(n \leq 1\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -7.5e+27) (not (<= n 1.0)))
   (* n (* 100.0 (/ (expm1 i) i)))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -7.5e+27) || !(n <= 1.0)) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -7.5e+27) || !(n <= 1.0)) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -7.5e+27) or not (n <= 1.0):
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -7.5e+27) || !(n <= 1.0))
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -7.5e+27], N[Not[LessEqual[n, 1.0]], $MachinePrecision]], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7.5000000000000002e27 or 1 < n

   1. Initial program 23.5%

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

      1. associate-*r/ 23.5%

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

      2. sub-neg 23.5%

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

      3. distribute-lft-in 23.5%

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

      4. fma-def 23.5%

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

      5. metadata-eval 23.5%

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

      6. metadata-eval 23.5%

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

   3. Simplified 23.5%

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

      1. fma-udef 23.5%

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

      2. metadata-eval 23.5%

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

      3. distribute-lft-in 23.5%

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

      4. metadata-eval 23.5%

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

      5. sub-neg 23.5%

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

      6. associate-*r/ 23.5%

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

      7. *-commutative 23.5%

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

      8. associate-/r/ 24.0%

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

      9. sub-neg 24.0%

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

      10. metadata-eval 24.0%

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

      11. associate-*r* 24.1%

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

      12. *-commutative 24.1%

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

      13. associate-*r* 24.1%

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

   5. Applied egg-rr 62.8%

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

   6. Taylor expanded in n around inf 90.4%

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

   if -7.5000000000000002e27 < n < 1

   1. Initial program 24.9%

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

      1. *-commutative 24.9%

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

      2. associate-/r/ 25.1%

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

      3. sub-neg 25.1%

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

      4. metadata-eval 25.1%

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

      5. associate-*r* 25.1%

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

      6. *-commutative 25.1%

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

      7. clear-num 25.1%

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

      8. un-div-inv 25.1%

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

      9. metadata-eval 25.1%

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

      10. sub-neg 25.1%

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

      11. pow-to-exp 24.2%

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

      12. expm1-def 46.9%

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

      13. *-commutative 46.9%

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

      14. log1p-udef 90.7%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in i around 0 78.5%

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

      1. sub-neg 78.5%

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

      2. associate-*r/ 78.5%

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

      3. metadata-eval 78.5%

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

      4. metadata-eval 78.5%

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

   6. Simplified 78.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+27} \lor \neg \left(n \leq 1\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 6: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.5e+27)
   (* 100.0 (/ n (/ i (expm1 i))))
   (if (<= n 1.0)
     (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))
     (* n (* 100.0 (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.5e+27) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (n <= 1.0) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = n * (100.0 * (expm1(i) / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.5e+27) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (n <= 1.0) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.5e+27:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif n <= 1.0:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	else:
		tmp = n * (100.0 * (math.expm1(i) / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.5e+27)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (n <= 1.0)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	else
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.5e+27], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.0], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.5000000000000002e27

   1. Initial program 24.4%

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

   2. Taylor expanded in n around inf 33.9%

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

      1. *-commutative 33.9%

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

      2. associate-/l* 33.9%

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

      3. expm1-def 88.3%

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

   4. Simplified 88.3%

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

   if -7.5000000000000002e27 < n < 1

   1. Initial program 24.9%

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

      1. *-commutative 24.9%

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

      2. associate-/r/ 25.1%

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

      3. sub-neg 25.1%

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

      4. metadata-eval 25.1%

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

      5. associate-*r* 25.1%

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

      6. *-commutative 25.1%

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

      7. clear-num 25.1%

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

      8. un-div-inv 25.1%

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

      9. metadata-eval 25.1%

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

      10. sub-neg 25.1%

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

      11. pow-to-exp 24.2%

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

      12. expm1-def 46.9%

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

      13. *-commutative 46.9%

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

      14. log1p-udef 90.7%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in i around 0 78.5%

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

      1. sub-neg 78.5%

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

      2. associate-*r/ 78.5%

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

      3. metadata-eval 78.5%

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

      4. metadata-eval 78.5%

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

   6. Simplified 78.5%

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

   if 1 < n

   1. Initial program 22.4%

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

      1. associate-*r/ 22.5%

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

      2. sub-neg 22.5%

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

      3. distribute-lft-in 22.5%

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

      4. fma-def 22.5%

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

      5. metadata-eval 22.5%

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

      6. metadata-eval 22.5%

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

   3. Simplified 22.5%

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

      1. fma-udef 22.5%

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

      2. metadata-eval 22.5%

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

      3. distribute-lft-in 22.5%

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

      4. metadata-eval 22.5%

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

      5. sub-neg 22.5%

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

      6. associate-*r/ 22.4%

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

      7. *-commutative 22.4%

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

      8. associate-/r/ 23.1%

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

      9. sub-neg 23.1%

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

      10. metadata-eval 23.1%

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

      11. associate-*r* 23.1%

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

      12. *-commutative 23.1%

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

      13. associate-*r* 23.1%

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

   5. Applied egg-rr 63.3%

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

   6. Taylor expanded in n around inf 92.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 7: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot t_0\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -1e+28)
     (* (* n 100.0) t_0)
     (if (<= n 1.0)
       (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))
       (* n (* 100.0 t_0))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -1e+28) {
		tmp = (n * 100.0) * t_0;
	} else if (n <= 1.0) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = n * (100.0 * t_0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -1e+28) {
		tmp = (n * 100.0) * t_0;
	} else if (n <= 1.0) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = n * (100.0 * t_0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -1e+28:
		tmp = (n * 100.0) * t_0
	elif n <= 1.0:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	else:
		tmp = n * (100.0 * t_0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -1e+28)
		tmp = Float64(Float64(n * 100.0) * t_0);
	elseif (n <= 1.0)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	else
		tmp = Float64(n * Float64(100.0 * t_0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1e+28], N[(N[(n * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[n, 1.0], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.99999999999999958e27

   1. Initial program 24.4%

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

      1. *-commutative 24.4%

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

      2. associate-/r/ 24.9%

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

      3. associate-*l* 25.0%

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

      4. sub-neg 25.0%

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

      5. metadata-eval 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in n around inf 34.0%

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

      1. expm1-def 88.3%

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

   6. Simplified 88.3%

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

   if -9.99999999999999958e27 < n < 1

   1. Initial program 24.9%

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

      1. *-commutative 24.9%

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

      2. associate-/r/ 25.1%

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

      3. sub-neg 25.1%

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

      4. metadata-eval 25.1%

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

      5. associate-*r* 25.1%

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

      6. *-commutative 25.1%

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

      7. clear-num 25.1%

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

      8. un-div-inv 25.1%

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

      9. metadata-eval 25.1%

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

      10. sub-neg 25.1%

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

      11. pow-to-exp 24.2%

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

      12. expm1-def 46.9%

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

      13. *-commutative 46.9%

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

      14. log1p-udef 90.7%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in i around 0 78.5%

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

      1. sub-neg 78.5%

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

      2. associate-*r/ 78.5%

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

      3. metadata-eval 78.5%

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

      4. metadata-eval 78.5%

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

   6. Simplified 78.5%

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

   if 1 < n

   1. Initial program 22.4%

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

      1. associate-*r/ 22.5%

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

      2. sub-neg 22.5%

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

      3. distribute-lft-in 22.5%

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

      4. fma-def 22.5%

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

      5. metadata-eval 22.5%

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

      6. metadata-eval 22.5%

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

   3. Simplified 22.5%

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

      1. fma-udef 22.5%

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

      2. metadata-eval 22.5%

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

      3. distribute-lft-in 22.5%

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

      4. metadata-eval 22.5%

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

      5. sub-neg 22.5%

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

      6. associate-*r/ 22.4%

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

      7. *-commutative 22.4%

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

      8. associate-/r/ 23.1%

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

      9. sub-neg 23.1%

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

      10. metadata-eval 23.1%

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

      11. associate-*r* 23.1%

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

      12. *-commutative 23.1%

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

      13. associate-*r* 23.1%

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

   5. Applied egg-rr 63.3%

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

   6. Taylor expanded in n around inf 92.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 8: 73.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{+209} \lor \neg \left(n \leq 2\right):\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\left(0.16666666666666666 + \frac{0.3333333333333333}{n \cdot n}\right) - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9.6e+209) (not (<= n 2.0)))
   (*
    n
    (+
     100.0
     (*
      100.0
      (+
       (* i (- 0.5 (/ 0.5 n)))
       (*
        (* i i)
        (-
         (+ 0.16666666666666666 (/ 0.3333333333333333 (* n n)))
         (/ 0.5 n)))))))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9.6e+209) || !(n <= 2.0)) {
		tmp = n * (100.0 + (100.0 * ((i * (0.5 - (0.5 / n))) + ((i * i) * ((0.16666666666666666 + (0.3333333333333333 / (n * n))) - (0.5 / n))))));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-9.6d+209)) .or. (.not. (n <= 2.0d0))) then
        tmp = n * (100.0d0 + (100.0d0 * ((i * (0.5d0 - (0.5d0 / n))) + ((i * i) * ((0.16666666666666666d0 + (0.3333333333333333d0 / (n * n))) - (0.5d0 / n))))))
    else
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((0.5d0 / n) + (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9.6e+209) || !(n <= 2.0)) {
		tmp = n * (100.0 + (100.0 * ((i * (0.5 - (0.5 / n))) + ((i * i) * ((0.16666666666666666 + (0.3333333333333333 / (n * n))) - (0.5 / n))))));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9.6e+209) or not (n <= 2.0):
		tmp = n * (100.0 + (100.0 * ((i * (0.5 - (0.5 / n))) + ((i * i) * ((0.16666666666666666 + (0.3333333333333333 / (n * n))) - (0.5 / n))))))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9.6e+209) || !(n <= 2.0))
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(Float64(i * Float64(0.5 - Float64(0.5 / n))) + Float64(Float64(i * i) * Float64(Float64(0.16666666666666666 + Float64(0.3333333333333333 / Float64(n * n))) - Float64(0.5 / n)))))));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -9.6e+209) || ~((n <= 2.0)))
		tmp = n * (100.0 + (100.0 * ((i * (0.5 - (0.5 / n))) + ((i * i) * ((0.16666666666666666 + (0.3333333333333333 / (n * n))) - (0.5 / n))))));
	else
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9.6e+209], N[Not[LessEqual[n, 2.0]], $MachinePrecision]], N[(n * N[(100.0 + N[(100.0 * N[(N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(N[(0.16666666666666666 + N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.59999999999999983e209 or 2 < n

   1. Initial program 20.5%

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

      1. associate-*r/ 20.5%

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

      2. sub-neg 20.5%

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

      3. distribute-lft-in 20.5%

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

      4. fma-def 20.5%

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

      5. metadata-eval 20.5%

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

      6. metadata-eval 20.5%

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

   3. Simplified 20.5%

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

      1. fma-udef 20.5%

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

      2. metadata-eval 20.5%

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

      3. distribute-lft-in 20.5%

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

      4. metadata-eval 20.5%

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

      5. sub-neg 20.5%

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

      6. associate-*r/ 20.5%

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

      7. *-commutative 20.5%

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

      8. associate-/r/ 21.2%

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

      9. sub-neg 21.2%

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

      10. metadata-eval 21.2%

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

      11. associate-*r* 21.2%

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

      12. *-commutative 21.2%

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

      13. associate-*r* 21.2%

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

   5. Applied egg-rr 59.3%

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

   6. Taylor expanded in i around 0 74.3%

      \[\leadsto \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \cdot n \]
   7. Step-by-step derivation

      1. distribute-lft-out 74.3%

         \[\leadsto \left(100 + \color{blue}{100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right) + {i}^{2} \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \cdot n \]

      2. associate-*r/ 74.3%

         \[\leadsto \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) + {i}^{2} \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \cdot n \]

      3. metadata-eval 74.3%

         \[\leadsto \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right) + {i}^{2} \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \cdot n \]

      4. unpow2 74.3%

         \[\leadsto \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \cdot n \]

      5. associate-*r/ 74.3%

         \[\leadsto \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\left(0.16666666666666666 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}}\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \cdot n \]

      6. metadata-eval 74.3%

         \[\leadsto \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\left(0.16666666666666666 + \frac{\color{blue}{0.3333333333333333}}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \cdot n \]

      7. unpow2 74.3%

         \[\leadsto \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\left(0.16666666666666666 + \frac{0.3333333333333333}{\color{blue}{n \cdot n}}\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \cdot n \]

      8. associate-*r/ 74.3%

         \[\leadsto \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\left(0.16666666666666666 + \frac{0.3333333333333333}{n \cdot n}\right) - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \cdot n \]

      9. metadata-eval 74.3%

         \[\leadsto \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\left(0.16666666666666666 + \frac{0.3333333333333333}{n \cdot n}\right) - \frac{\color{blue}{0.5}}{n}\right)\right)\right) \cdot n \]

   8. Simplified 74.3%

      \[\leadsto \color{blue}{\left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\left(0.16666666666666666 + \frac{0.3333333333333333}{n \cdot n}\right) - \frac{0.5}{n}\right)\right)\right)} \cdot n \]

   if -9.59999999999999983e209 < n < 2

   1. Initial program 26.3%

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

      1. *-commutative 26.3%

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

      2. associate-/r/ 26.5%

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

      3. sub-neg 26.5%

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

      4. metadata-eval 26.5%

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

      5. associate-*r* 26.5%

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

      6. *-commutative 26.5%

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

      7. clear-num 26.5%

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

      8. un-div-inv 26.5%

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

      9. metadata-eval 26.5%

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

      10. sub-neg 26.5%

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

      11. pow-to-exp 20.9%

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

      12. expm1-def 36.5%

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

      13. *-commutative 36.5%

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

      14. log1p-udef 84.1%

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

   3. Applied egg-rr 84.1%

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

   4. Taylor expanded in i around 0 74.3%

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

      1. sub-neg 74.3%

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

      2. associate-*r/ 74.3%

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

      3. metadata-eval 74.3%

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

      4. metadata-eval 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{+209} \lor \neg \left(n \leq 2\right):\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\left(0.16666666666666666 + \frac{0.3333333333333333}{n \cdot n}\right) - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 9: 70.5% accurate, 7.6× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double tmp;
	if (n <= 1.0) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -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 <= 1.0d0) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((0.5d0 / n) + (-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 <= 1.0) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

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

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 1.0)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -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 <= 1.0)
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 1.0], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $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 < 1

   1. Initial program 24.7%

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

      1. *-commutative 24.7%

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

      2. associate-/r/ 25.0%

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

      3. sub-neg 25.0%

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

      4. metadata-eval 25.0%

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

      5. associate-*r* 25.0%

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

      6. *-commutative 25.0%

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

      7. clear-num 25.0%

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

      8. un-div-inv 25.0%

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

      9. metadata-eval 25.0%

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

      10. sub-neg 25.0%

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

      11. pow-to-exp 19.7%

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

      12. expm1-def 33.0%

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

      13. *-commutative 33.0%

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

      14. log1p-udef 79.0%

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

   3. Applied egg-rr 79.0%

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

   4. Taylor expanded in i around 0 71.8%

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

      1. sub-neg 71.8%

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

      2. associate-*r/ 71.8%

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

      3. metadata-eval 71.8%

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

      4. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

   if 1 < n

   1. Initial program 22.4%

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

      1. associate-/r/ 23.1%

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

      2. associate-*r* 23.1%

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

      3. *-commutative 23.1%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.1%

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

      7. fma-def 23.1%

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

      8. metadata-eval 23.1%

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

      9. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in i around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. associate-*r/ 72.3%

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

      4. metadata-eval 72.3%

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

   6. Simplified 72.3%

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

   7. Taylor expanded in n around inf 72.3%

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

      1. *-commutative 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 71.9%

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

Alternative 10: 61.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+81} \lor \neg \left(n \leq 1.5\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -7.5e+81) (not (<= n 1.5)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (* i (/ n i)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -7.5e+81) || !(n <= 1.5)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i * (n / i));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-7.5d+81)) .or. (.not. (n <= 1.5d0))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i * (n / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -7.5e+81) || !(n <= 1.5)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i * (n / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -7.5e+81) or not (n <= 1.5):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i * (n / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -7.5e+81) || !(n <= 1.5))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -7.5e+81) || ~((n <= 1.5)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i * (n / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -7.5e+81], N[Not[LessEqual[n, 1.5]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7.49999999999999973e81 or 1.5 < n

   1. Initial program 20.4%

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

      1. associate-/r/ 21.0%

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

      2. associate-*r* 21.0%

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

      3. *-commutative 21.0%

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

      4. associate-*r/ 21.0%

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

      5. sub-neg 21.0%

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

      6. distribute-lft-in 21.0%

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

      7. fma-def 21.0%

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

      8. metadata-eval 21.0%

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

      9. metadata-eval 21.0%

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

   3. Simplified 21.0%

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

   4. Taylor expanded in i around 0 68.2%

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

      1. associate-*r* 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*r/ 68.2%

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

      4. metadata-eval 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in n around inf 68.2%

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

      1. *-commutative 68.2%

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

   9. Simplified 68.2%

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

   if -7.49999999999999973e81 < n < 1.5

   1. Initial program 27.9%

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

   2. Taylor expanded in i around 0 61.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. clear-num 60.8%

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

      2. associate-/r/ 60.9%

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

      3. clear-num 61.0%

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

   4. Applied egg-rr 61.0%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+81} \lor \neg \left(n \leq 1.5\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \end{array} \]

Alternative 11: 61.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+66)
   (/ (* 100.0 (* i n)) i)
   (if (<= n 1.5) (* 100.0 (* i (/ n i))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+66) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 1.5) {
		tmp = 100.0 * (i * (n / i));
	} 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 <= (-2.8d+66)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= 1.5d0) then
        tmp = 100.0d0 * (i * (n / i))
    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 <= -2.8e+66) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 1.5) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.8e+66:
		tmp = (100.0 * (i * n)) / i
	elif n <= 1.5:
		tmp = 100.0 * (i * (n / i))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+66)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= 1.5)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	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 <= -2.8e+66)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= 1.5)
		tmp = 100.0 * (i * (n / i));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+66], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.5], N[(100.0 * N[(i * N[(n / i), $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 < -2.8000000000000001e66

   1. Initial program 19.0%

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

   2. Taylor expanded in i around 0 31.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. unpow2 31.9%

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

      2. associate-*r/ 31.9%

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

      3. metadata-eval 31.9%

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

   4. Simplified 31.9%

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

   5. Taylor expanded in n around inf 64.3%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(i + 0.5 \cdot {i}^{2}\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ 65.6%

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \left(i + 0.5 \cdot {i}^{2}\right)\right)}{i}} \]

      2. associate-*r* 65.9%

         \[\leadsto \frac{\color{blue}{\left(100 \cdot n\right) \cdot \left(i + 0.5 \cdot {i}^{2}\right)}}{i} \]

      3. +-commutative 65.9%

         \[\leadsto \frac{\left(100 \cdot n\right) \cdot \color{blue}{\left(0.5 \cdot {i}^{2} + i\right)}}{i} \]

      4. fma-def 65.9%

         \[\leadsto \frac{\left(100 \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, {i}^{2}, i\right)}}{i} \]

      5. unpow2 65.9%

         \[\leadsto \frac{\left(100 \cdot n\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{i \cdot i}, i\right)}{i} \]

   7. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\left(100 \cdot n\right) \cdot \mathsf{fma}\left(0.5, i \cdot i, i\right)}{i}} \]

   8. Taylor expanded in i around 0 63.7%

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

      1. *-commutative 63.7%

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

   10. Simplified 63.7%

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

   if -2.8000000000000001e66 < n < 1.5

   1. Initial program 27.5%

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

   2. Taylor expanded in i around 0 61.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. clear-num 60.9%

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

      2. associate-/r/ 61.1%

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

      3. clear-num 61.1%

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

   4. Applied egg-rr 61.1%

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

   if 1.5 < n

   1. Initial program 22.4%

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

      1. associate-/r/ 23.1%

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

      2. associate-*r* 23.1%

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

      3. *-commutative 23.1%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.1%

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

      7. fma-def 23.1%

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

      8. metadata-eval 23.1%

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

      9. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in i around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. associate-*r/ 72.3%

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

      4. metadata-eval 72.3%

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

   6. Simplified 72.3%

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

   7. Taylor expanded in n around inf 72.3%

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

      1. *-commutative 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 12: 61.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100\right)}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+85)
   (/ (* i (* n 100.0)) i)
   (if (<= n 1.5) (* 100.0 (* i (/ n i))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+85) {
		tmp = (i * (n * 100.0)) / i;
	} else if (n <= 1.5) {
		tmp = 100.0 * (i * (n / i));
	} 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 <= (-2.8d+85)) then
        tmp = (i * (n * 100.0d0)) / i
    else if (n <= 1.5d0) then
        tmp = 100.0d0 * (i * (n / i))
    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 <= -2.8e+85) {
		tmp = (i * (n * 100.0)) / i;
	} else if (n <= 1.5) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.8e+85:
		tmp = (i * (n * 100.0)) / i
	elif n <= 1.5:
		tmp = 100.0 * (i * (n / i))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+85)
		tmp = Float64(Float64(i * Float64(n * 100.0)) / i);
	elseif (n <= 1.5)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	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 <= -2.8e+85)
		tmp = (i * (n * 100.0)) / i;
	elseif (n <= 1.5)
		tmp = 100.0 * (i * (n / i));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+85], N[(N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.5], N[(100.0 * N[(i * N[(n / i), $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 < -2.7999999999999999e85

   1. Initial program 18.2%

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

   2. Taylor expanded in i around 0 31.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. unpow2 31.8%

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

      2. associate-*r/ 31.8%

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

      3. metadata-eval 31.8%

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

   4. Simplified 31.8%

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

   5. Taylor expanded in n around inf 65.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(i + 0.5 \cdot {i}^{2}\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \left(i + 0.5 \cdot {i}^{2}\right)\right)}{i}} \]

      2. associate-*r* 67.5%

         \[\leadsto \frac{\color{blue}{\left(100 \cdot n\right) \cdot \left(i + 0.5 \cdot {i}^{2}\right)}}{i} \]

      3. +-commutative 67.5%

         \[\leadsto \frac{\left(100 \cdot n\right) \cdot \color{blue}{\left(0.5 \cdot {i}^{2} + i\right)}}{i} \]

      4. fma-def 67.5%

         \[\leadsto \frac{\left(100 \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, {i}^{2}, i\right)}}{i} \]

      5. unpow2 67.5%

         \[\leadsto \frac{\left(100 \cdot n\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{i \cdot i}, i\right)}{i} \]

   7. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{\left(100 \cdot n\right) \cdot \mathsf{fma}\left(0.5, i \cdot i, i\right)}{i}} \]

   8. Taylor expanded in i around 0 65.5%

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

   if -2.7999999999999999e85 < n < 1.5

   1. Initial program 27.7%

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

   2. Taylor expanded in i around 0 60.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. clear-num 60.4%

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

      2. associate-/r/ 60.5%

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

      3. clear-num 60.6%

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

   4. Applied egg-rr 60.6%

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

   if 1.5 < n

   1. Initial program 22.4%

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

      1. associate-/r/ 23.1%

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

      2. associate-*r* 23.1%

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

      3. *-commutative 23.1%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.1%

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

      7. fma-def 23.1%

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

      8. metadata-eval 23.1%

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

      9. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in i around 0 72.3%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. associate-*r/ 72.3%

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

      4. metadata-eval 72.3%

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

   6. Simplified 72.3%

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

   7. Taylor expanded in n around inf 72.3%

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

      1. *-commutative 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100\right)}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13: 57.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+135}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 4.8 \cdot 10^{+25}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.2e+135)
   (* 100.0 (* i (/ n i)))
   (if (<= i 4.8e+25) (* n 100.0) (* 50.0 (* i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.2e+135) {
		tmp = 100.0 * (i * (n / i));
	} else if (i <= 4.8e+25) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (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 (i <= (-2.2d+135)) then
        tmp = 100.0d0 * (i * (n / i))
    else if (i <= 4.8d+25) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.2e+135) {
		tmp = 100.0 * (i * (n / i));
	} else if (i <= 4.8e+25) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.2e+135:
		tmp = 100.0 * (i * (n / i))
	elif i <= 4.8e+25:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.2e+135)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	elseif (i <= 4.8e+25)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.2e+135)
		tmp = 100.0 * (i * (n / i));
	elseif (i <= 4.8e+25)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.2e+135], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.8e+25], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.1999999999999999e135

   1. Initial program 69.5%

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

   2. Taylor expanded in i around 0 26.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. clear-num 26.4%

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

      2. associate-/r/ 26.4%

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

      3. clear-num 26.4%

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

   4. Applied egg-rr 26.4%

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

   if -2.1999999999999999e135 < i < 4.79999999999999992e25

   1. Initial program 11.5%

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

   2. Taylor expanded in i around 0 74.1%

      \[\leadsto \color{blue}{100 \cdot n} \]
   3. Step-by-step derivation

      1. *-commutative 74.1%

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

   4. Simplified 74.1%

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

   if 4.79999999999999992e25 < i

   1. Initial program 46.9%

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

      1. *-commutative 46.9%

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

      2. associate-/r/ 47.2%

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

      3. associate-*l* 47.2%

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

      4. sub-neg 47.2%

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

      5. metadata-eval 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in n around inf 44.7%

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

      1. expm1-def 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in i around 0 33.8%

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

      1. *-commutative 33.8%

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

   9. Simplified 33.8%

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

   10. Taylor expanded in i around inf 33.8%

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

       1. *-commutative 33.8%

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

   12. Simplified 33.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{+135}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 4.8 \cdot 10^{+25}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 14: 54.8% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 4.8 \cdot 10^{+25}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 4.8e+25) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 4.8e+25) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (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 (i <= 4.8d+25) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 4.8e+25) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 4.8e+25:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 4.8e+25)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 4.8e+25)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 4.8e+25], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 4.79999999999999992e25

   1. Initial program 17.6%

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

   2. Taylor expanded in i around 0 66.7%

      \[\leadsto \color{blue}{100 \cdot n} \]
   3. Step-by-step derivation

      1. *-commutative 66.7%

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

   4. Simplified 66.7%

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

   if 4.79999999999999992e25 < i

   1. Initial program 46.9%

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

      1. *-commutative 46.9%

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

      2. associate-/r/ 47.2%

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

      3. associate-*l* 47.2%

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

      4. sub-neg 47.2%

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

      5. metadata-eval 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in n around inf 44.7%

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

      1. expm1-def 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in i around 0 33.8%

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

      1. *-commutative 33.8%

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

   9. Simplified 33.8%

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

   10. Taylor expanded in i around inf 33.8%

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

       1. *-commutative 33.8%

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

   12. Simplified 33.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 4.8 \cdot 10^{+25}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 15: 2.8% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

1. Initial program 24.1%

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

   1. associate-/r/ 24.5%

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

   2. associate-*r* 24.5%

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

   3. *-commutative 24.5%

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

   4. associate-*r/ 24.5%

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

   5. sub-neg 24.5%

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

   6. distribute-lft-in 24.5%

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

   7. fma-def 24.5%

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

   8. metadata-eval 24.5%

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

   9. metadata-eval 24.5%

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

3. Simplified 24.5%

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

4. Taylor expanded in i around 0 59.0%

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

   1. associate-*r* 59.0%

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

   2. *-commutative 59.0%

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

   3. associate-*r/ 59.0%

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

   4. metadata-eval 59.0%

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

6. Simplified 59.0%

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

7. Taylor expanded in n around 0 2.7%

   \[\leadsto \color{blue}{-50 \cdot i} \]
8. Step-by-step derivation

   1. *-commutative 2.7%

      \[\leadsto \color{blue}{i \cdot -50} \]

9. Simplified 2.7%

   \[\leadsto \color{blue}{i \cdot -50} \]

10. Final simplification 2.7%

    \[\leadsto i \cdot -50 \]

Alternative 16: 49.5% 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 24.1%

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

2. Taylor expanded in i around 0 53.0%

   \[\leadsto \color{blue}{100 \cdot n} \]
3. Step-by-step derivation

   1. *-commutative 53.0%

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

4. Simplified 53.0%

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

5. Final simplification 53.0%

   \[\leadsto n \cdot 100 \]

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 2023297 
(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))))