Compound Interest

Percentage Accurate: 27.9% → 98.7%

Time: 21.2s

Alternatives: 24

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: 27.9% 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: 98.7% accurate, 0.3× speedup

Math

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

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 (- INFINITY))
     (* 100.0 (* (/ n i) t_1))
     (if (<= t_2 0.0)
       (/ (* n 100.0) (/ i (expm1 (* n (log1p (/ i n))))))
       (if (<= t_2 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (/ (* n 100.0) (+ 1.0 (* (/ i 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 <= -((double) INFINITY)) {
		tmp = 100.0 * ((n / i) * t_1);
	} else if (t_2 <= 0.0) {
		tmp = (n * 100.0) / (i / expm1((n * log1p((i / n)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + ((i / 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 <= -Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((n / i) * t_1);
	} else if (t_2 <= 0.0) {
		tmp = (n * 100.0) / (i / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + ((i / 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 <= -math.inf:
		tmp = 100.0 * ((n / i) * t_1)
	elif t_2 <= 0.0:
		tmp = (n * 100.0) / (i / math.expm1((n * math.log1p((i / n)))))
	elif t_2 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = (n * 100.0) / (1.0 + ((i / 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 <= Float64(-Inf))
		tmp = Float64(100.0 * Float64(Float64(n / i) * t_1));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(n * 100.0) / Float64(i / expm1(Float64(n * log1p(Float64(i / n))))));
	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(Float64(i / 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, (-Infinity)], N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], 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$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[(N[(i / n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. clear-num 100.0%

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

      3. sub-neg 100.0%

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

      4. div-inv 100.0%

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

      5. clear-num 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-rgt-out 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

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

   1. Initial program 27.0%

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

      1. *-commutative 27.0%

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

      2. associate-/r/ 27.0%

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

      3. sub-neg 27.0%

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

      4. metadata-eval 27.0%

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

      5. associate-*r* 27.0%

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

      6. *-commutative 27.0%

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

      7. clear-num 27.0%

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

      8. un-div-inv 27.0%

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

      9. metadata-eval 27.0%

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

      10. sub-neg 27.0%

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

      11. pow-to-exp 27.0%

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

      12. expm1-def 39.9%

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

      13. add-log-exp 27.0%

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

      14. pow-to-exp 27.0%

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

      15. log-pow 39.9%

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

      16. log1p-udef 99.1%

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

   3. Applied egg-rr 99.1%

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

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

   1. Initial program 98.6%

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

      1. div-sub 98.8%

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

      2. clear-num 98.8%

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

      3. sub-neg 98.8%

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

      4. div-inv 98.8%

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

      5. clear-num 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

   5. Simplified 98.8%

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

   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/ 2.0%

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

      3. sub-neg 2.0%

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

      4. metadata-eval 2.0%

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

      5. associate-*r* 2.0%

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

      6. *-commutative 2.0%

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

      7. clear-num 2.0%

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

      8. un-div-inv 2.0%

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

      9. metadata-eval 2.0%

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

      10. sub-neg 2.0%

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

      11. pow-to-exp 2.0%

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

      12. expm1-def 2.0%

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

      13. add-log-exp 2.0%

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

      14. pow-to-exp 2.0%

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

      15. log-pow 2.0%

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

      16. log1p-udef 2.0%

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

   3. Applied egg-rr 2.0%

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

   4. Taylor expanded in i around 0 100.0%

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

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

   7. Taylor expanded in n around 0 100.0%

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

4. Final simplification 99.3%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

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

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 -1e-42)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* 100.0 (* (/ n i) (expm1 (* n (log1p (/ i n))))))
       (if (<= t_1 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (/ (* n 100.0) (+ 1.0 (* (/ i 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 <= -1e-42) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * ((n / 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 / 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 <= -1e-42) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * ((n / i) * Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + ((i / 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 <= -1e-42:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = 100.0 * ((n / 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 / 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 <= -1e-42)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(Float64(n / 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(Float64(i / 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, -1e-42], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(n / i), $MachinePrecision] * 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[(N[(i / n), $MachinePrecision] * 0.5), $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)) < -1.00000000000000004e-42

   1. Initial program 99.9%

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

   if -1.00000000000000004e-42 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 25.6%

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

      1. clear-num 25.6%

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

      2. associate-/r/ 25.6%

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

      3. clear-num 25.6%

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

      4. pow-to-exp 25.6%

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

      5. expm1-def 38.7%

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

      6. add-log-exp 25.6%

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

      7. pow-to-exp 25.6%

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

      8. log-pow 38.7%

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

      9. log1p-udef 97.8%

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

   3. Applied egg-rr 97.8%

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

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

   1. Initial program 98.6%

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

      1. div-sub 98.8%

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

      2. clear-num 98.8%

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

      3. sub-neg 98.8%

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

      4. div-inv 98.8%

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

      5. clear-num 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

   5. Simplified 98.8%

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

   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/ 2.0%

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

      3. sub-neg 2.0%

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

      4. metadata-eval 2.0%

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

      5. associate-*r* 2.0%

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

      6. *-commutative 2.0%

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

      7. clear-num 2.0%

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

      8. un-div-inv 2.0%

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

      9. metadata-eval 2.0%

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

      10. sub-neg 2.0%

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

      11. pow-to-exp 2.0%

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

      12. expm1-def 2.0%

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

      13. add-log-exp 2.0%

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

      14. pow-to-exp 2.0%

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

      15. log-pow 2.0%

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

      16. log1p-udef 2.0%

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

   3. Applied egg-rr 2.0%

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

   4. Taylor expanded in i around 0 100.0%

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

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

   7. Taylor expanded in n around 0 100.0%

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

4. Final simplification 98.5%

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

Alternative 3: 98.6% accurate, 0.3× speedup

Math

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

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 -1e-42)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* 100.0 (* n (/ (expm1 (* n (log1p (/ i n)))) i)))
       (if (<= t_1 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (/ (* n 100.0) (+ 1.0 (* (/ i 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 <= -1e-42) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (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 / 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 <= -1e-42) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (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 / 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 <= -1e-42:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = 100.0 * (n * (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 / 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 <= -1e-42)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n * 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(Float64(i / 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, -1e-42], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n * 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[(N[(i / n), $MachinePrecision] * 0.5), $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)) < -1.00000000000000004e-42

   1. Initial program 99.9%

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

   if -1.00000000000000004e-42 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 25.6%

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

      1. div-inv 25.6%

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

      2. associate-/r/ 25.6%

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

      3. associate-*l* 25.6%

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

      4. div-inv 25.6%

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

      5. pow-to-exp 25.6%

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

      6. expm1-def 38.7%

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

      7. add-log-exp 25.6%

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

      8. pow-to-exp 25.6%

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

      9. log-pow 38.7%

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

      10. log1p-udef 99.0%

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

   3. Applied egg-rr 99.0%

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

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

   1. Initial program 98.6%

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

      1. div-sub 98.8%

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

      2. clear-num 98.8%

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

      3. sub-neg 98.8%

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

      4. div-inv 98.8%

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

      5. clear-num 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

   5. Simplified 98.8%

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

   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/ 2.0%

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

      3. sub-neg 2.0%

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

      4. metadata-eval 2.0%

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

      5. associate-*r* 2.0%

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

      6. *-commutative 2.0%

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

      7. clear-num 2.0%

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

      8. un-div-inv 2.0%

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

      9. metadata-eval 2.0%

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

      10. sub-neg 2.0%

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

      11. pow-to-exp 2.0%

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

      12. expm1-def 2.0%

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

      13. add-log-exp 2.0%

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

      14. pow-to-exp 2.0%

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

      15. log-pow 2.0%

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

      16. log1p-udef 2.0%

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

   3. Applied egg-rr 2.0%

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

   4. Taylor expanded in i around 0 100.0%

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

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

   7. Taylor expanded in n around 0 100.0%

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

4. Final simplification 99.3%

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

Alternative 4: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-94}:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \end{array} \end{array} \]

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 -5e-94)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* 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 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 <= -5e-94) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		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 / 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 <= -5e-94) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		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 / 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 <= -5e-94:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		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 / 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 <= -5e-94)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		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(Float64(i / 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, -5e-94], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], 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[(N[(i / n), $MachinePrecision] * 0.5), $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)) < -4.9999999999999995e-94

   1. Initial program 99.7%

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

   if -4.9999999999999995e-94 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 23.7%

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

      1. *-commutative 23.7%

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

      2. associate-/r/ 23.8%

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

      3. sub-neg 23.8%

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

      4. metadata-eval 23.8%

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

      5. associate-*r* 23.7%

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

      6. *-commutative 23.7%

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

      7. clear-num 23.7%

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

      8. un-div-inv 23.8%

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

      9. metadata-eval 23.8%

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

      10. sub-neg 23.8%

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

      11. pow-to-exp 23.8%

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

      12. expm1-def 37.2%

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

      13. add-log-exp 23.8%

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

      14. pow-to-exp 23.8%

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

      15. log-pow 37.2%

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

      16. log1p-udef 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. div-inv 99.0%

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

      2. clear-num 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. associate-*l* 99.1%

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

   7. Simplified 99.1%

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

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

   1. Initial program 98.6%

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

      1. div-sub 98.8%

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

      2. clear-num 98.8%

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

      3. sub-neg 98.8%

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

      4. div-inv 98.8%

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

      5. clear-num 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

   5. Simplified 98.8%

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

   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/ 2.0%

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

      3. sub-neg 2.0%

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

      4. metadata-eval 2.0%

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

      5. associate-*r* 2.0%

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

      6. *-commutative 2.0%

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

      7. clear-num 2.0%

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

      8. un-div-inv 2.0%

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

      9. metadata-eval 2.0%

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

      10. sub-neg 2.0%

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

      11. pow-to-exp 2.0%

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

      12. expm1-def 2.0%

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

      13. add-log-exp 2.0%

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

      14. pow-to-exp 2.0%

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

      15. log-pow 2.0%

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

      16. log1p-udef 2.0%

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

   3. Applied egg-rr 2.0%

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

   4. Taylor expanded in i around 0 100.0%

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

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

   7. Taylor expanded in n around 0 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \end{array} \]

Alternative 5: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot t_1\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \end{array} \end{array} \]

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-24)
     (* 100.0 (* (/ n i) t_1))
     (if (<= t_2 0.0)
       (* n (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 i)))
       (if (<= t_2 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (/ (* n 100.0) (+ 1.0 (* (/ i 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-24) {
		tmp = 100.0 * ((n / i) * t_1);
	} else if (t_2 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) * (100.0 / i));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = (n * 100.0) / (1.0 + ((i / 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-24) {
		tmp = 100.0 * ((n / i) * t_1);
	} else if (t_2 <= 0.0) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) * (100.0 / 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 / 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-24:
		tmp = 100.0 * ((n / i) * t_1)
	elif t_2 <= 0.0:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) * (100.0 / i))
	elif t_2 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = (n * 100.0) / (1.0 + ((i / 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-24)
		tmp = Float64(100.0 * Float64(Float64(n / i) * t_1));
	elseif (t_2 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / 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(Float64(i / 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-24], N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / 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[(N[(i / n), $MachinePrecision] * 0.5), $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)) < -4.9999999999999998e-24

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. clear-num 100.0%

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

      3. sub-neg 100.0%

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

      4. div-inv 100.0%

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

      5. clear-num 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-rgt-out 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -4.9999999999999998e-24 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 26.5%

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

      1. clear-num 26.5%

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

      2. associate-/r/ 26.5%

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

      3. clear-num 26.5%

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

      4. pow-to-exp 26.5%

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

      5. expm1-def 39.4%

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

      6. add-log-exp 26.5%

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

      7. pow-to-exp 26.5%

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

      8. log-pow 39.4%

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

      9. log1p-udef 97.8%

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

   3. Applied egg-rr 97.8%

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

      1. expm1-log1p-u 80.5%

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

      2. expm1-udef 43.9%

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

   5. Applied egg-rr 43.9%

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

      1. expm1-def 80.5%

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

      2. expm1-log1p 97.8%

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

      3. associate-/r/ 99.0%

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

      4. associate-*r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*r/ 99.1%

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

      7. associate-/r/ 99.1%

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

   7. Simplified 99.1%

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

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

   1. Initial program 98.6%

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

      1. div-sub 98.8%

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

      2. clear-num 98.8%

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

      3. sub-neg 98.8%

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

      4. div-inv 98.8%

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

      5. clear-num 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

   5. Simplified 98.8%

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

   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/ 2.0%

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

      3. sub-neg 2.0%

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

      4. metadata-eval 2.0%

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

      5. associate-*r* 2.0%

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

      6. *-commutative 2.0%

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

      7. clear-num 2.0%

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

      8. un-div-inv 2.0%

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

      9. metadata-eval 2.0%

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

      10. sub-neg 2.0%

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

      11. pow-to-exp 2.0%

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

      12. expm1-def 2.0%

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

      13. add-log-exp 2.0%

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

      14. pow-to-exp 2.0%

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

      15. log-pow 2.0%

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

      16. log1p-udef 2.0%

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

   3. Applied egg-rr 2.0%

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

   4. Taylor expanded in i around 0 100.0%

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

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

   7. Taylor expanded in n around 0 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \end{array} \]

Alternative 6: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2050:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;n \cdot \frac{1}{\mathsf{fma}\left(i, \frac{0.5}{n} + -0.5, 1\right) \cdot 0.01}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2050.0)
   (/ (* n 100.0) (/ i (expm1 i)))
   (if (<= n 0.58)
     (* n (/ 1.0 (* (fma i (+ (/ 0.5 n) -0.5) 1.0) 0.01)))
     (* n (* 100.0 (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2050.0) {
		tmp = (n * 100.0) / (i / expm1(i));
	} else if (n <= 0.58) {
		tmp = n * (1.0 / (fma(i, ((0.5 / n) + -0.5), 1.0) * 0.01));
	} else {
		tmp = n * (100.0 * (expm1(i) / i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2050.0)
		tmp = Float64(Float64(n * 100.0) / Float64(i / expm1(i)));
	elseif (n <= 0.58)
		tmp = Float64(n * Float64(1.0 / Float64(fma(i, Float64(Float64(0.5 / n) + -0.5), 1.0) * 0.01)));
	else
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2050.0], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.58], N[(n * N[(1.0 / N[(N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 0.01), $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 < -2050

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-/l* 38.8%

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

      3. expm1-def 87.8%

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

   4. Simplified 87.8%

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

      1. associate-*l/ 87.8%

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

   6. Applied egg-rr 87.8%

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

   if -2050 < n < 0.57999999999999996

   1. Initial program 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-/r/ 35.6%

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

      3. sub-neg 35.6%

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

      4. metadata-eval 35.6%

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

      5. associate-*r* 35.6%

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

      6. *-commutative 35.6%

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

      7. clear-num 35.6%

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

      8. un-div-inv 35.6%

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

      9. metadata-eval 35.6%

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

      10. sub-neg 35.6%

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

      11. pow-to-exp 35.6%

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

      12. expm1-def 57.1%

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

      13. add-log-exp 35.6%

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

      14. pow-to-exp 35.6%

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

      15. log-pow 57.1%

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

      16. log1p-udef 90.9%

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

   3. Applied egg-rr 90.9%

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

   4. Taylor expanded in i around 0 80.4%

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

      1. sub-neg 80.4%

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

      2. associate-*r/ 80.4%

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

      3. metadata-eval 80.4%

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

      4. metadata-eval 80.4%

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

   6. Simplified 80.4%

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

      1. associate-/l* 80.4%

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

      2. div-inv 80.4%

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

      3. div-inv 80.4%

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

      4. +-commutative 80.4%

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

      5. fma-def 80.4%

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

      6. metadata-eval 80.4%

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

   8. Applied egg-rr 80.4%

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

   if 0.57999999999999996 < n

   1. Initial program 29.1%

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

   2. Taylor expanded in n around inf 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/l* 44.9%

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

      3. expm1-def 92.6%

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

   4. Simplified 92.6%

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

      1. associate-*l/ 92.7%

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

   6. Applied egg-rr 92.7%

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

   7. Taylor expanded in n around 0 45.0%

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

      1. expm1-def 92.6%

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

      2. associate-/l* 92.6%

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

      3. associate-*r/ 92.7%

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

      4. associate-*l/ 92.6%

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

      5. *-commutative 92.6%

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

      6. associate-/r/ 92.6%

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

   9. Simplified 92.6%

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

   10. Taylor expanded in i around inf 45.0%

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

       1. expm1-def 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2050:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;n \cdot \frac{1}{\mathsf{fma}\left(i, \frac{0.5}{n} + -0.5, 1\right) \cdot 0.01}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 7: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2050:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{1}{\mathsf{fma}\left(i, \frac{0.5}{n} + -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2050.0)
   (/ (* n 100.0) (/ i (expm1 i)))
   (if (<= n 0.58)
     (* (* n 100.0) (/ 1.0 (fma i (+ (/ 0.5 n) -0.5) 1.0)))
     (* n (* 100.0 (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2050.0) {
		tmp = (n * 100.0) / (i / expm1(i));
	} else if (n <= 0.58) {
		tmp = (n * 100.0) * (1.0 / fma(i, ((0.5 / n) + -0.5), 1.0));
	} else {
		tmp = n * (100.0 * (expm1(i) / i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2050.0)
		tmp = Float64(Float64(n * 100.0) / Float64(i / expm1(i)));
	elseif (n <= 0.58)
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 / fma(i, Float64(Float64(0.5 / n) + -0.5), 1.0)));
	else
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2050.0], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.58], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 / N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $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 < -2050

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-/l* 38.8%

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

      3. expm1-def 87.8%

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

   4. Simplified 87.8%

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

      1. associate-*l/ 87.8%

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

   6. Applied egg-rr 87.8%

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

   if -2050 < n < 0.57999999999999996

   1. Initial program 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-/r/ 35.6%

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

      3. sub-neg 35.6%

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

      4. metadata-eval 35.6%

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

      5. associate-*r* 35.6%

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

      6. *-commutative 35.6%

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

      7. clear-num 35.6%

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

      8. un-div-inv 35.6%

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

      9. metadata-eval 35.6%

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

      10. sub-neg 35.6%

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

      11. pow-to-exp 35.6%

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

      12. expm1-def 57.1%

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

      13. add-log-exp 35.6%

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

      14. pow-to-exp 35.6%

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

      15. log-pow 57.1%

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

      16. log1p-udef 90.9%

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

   3. Applied egg-rr 90.9%

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

   4. Taylor expanded in i around 0 80.4%

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

      1. sub-neg 80.4%

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

      2. associate-*r/ 80.4%

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

      3. metadata-eval 80.4%

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

      4. metadata-eval 80.4%

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

   6. Simplified 80.4%

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

      1. div-inv 80.4%

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

      2. *-commutative 80.4%

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

      3. +-commutative 80.4%

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

      4. fma-def 80.4%

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

   8. Applied egg-rr 80.4%

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

   if 0.57999999999999996 < n

   1. Initial program 29.1%

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

   2. Taylor expanded in n around inf 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/l* 44.9%

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

      3. expm1-def 92.6%

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

   4. Simplified 92.6%

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

      1. associate-*l/ 92.7%

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

   6. Applied egg-rr 92.7%

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

   7. Taylor expanded in n around 0 45.0%

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

      1. expm1-def 92.6%

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

      2. associate-/l* 92.6%

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

      3. associate-*r/ 92.7%

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

      4. associate-*l/ 92.6%

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

      5. *-commutative 92.6%

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

      6. associate-/r/ 92.6%

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

   9. Simplified 92.6%

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

   10. Taylor expanded in i around inf 45.0%

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

       1. expm1-def 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2050:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{1}{\mathsf{fma}\left(i, \frac{0.5}{n} + -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 8: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -2.1 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 1.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{+229}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 6.3 \cdot 10^{+255}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -2.1e-15)
     t_0
     (if (<= i 1.6e-20)
       (/ (* n 100.0) (+ 1.0 (* (/ i n) 0.5)))
       (if (<= i 6.8e+229)
         t_0
         (if (<= i 6.3e+255)
           (/ 100.0 (/ (+ 1.0 (* i -0.5)) n))
           (* 100.0 (/ (* i n) i))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -2.1e-15) {
		tmp = t_0;
	} else if (i <= 1.6e-20) {
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5));
	} else if (i <= 6.8e+229) {
		tmp = t_0;
	} else if (i <= 6.3e+255) {
		tmp = 100.0 / ((1.0 + (i * -0.5)) / n);
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -2.1e-15) {
		tmp = t_0;
	} else if (i <= 1.6e-20) {
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5));
	} else if (i <= 6.8e+229) {
		tmp = t_0;
	} else if (i <= 6.3e+255) {
		tmp = 100.0 / ((1.0 + (i * -0.5)) / n);
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -2.1e-15:
		tmp = t_0
	elif i <= 1.6e-20:
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5))
	elif i <= 6.8e+229:
		tmp = t_0
	elif i <= 6.3e+255:
		tmp = 100.0 / ((1.0 + (i * -0.5)) / n)
	else:
		tmp = 100.0 * ((i * n) / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -2.1e-15)
		tmp = t_0;
	elseif (i <= 1.6e-20)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(Float64(i / n) * 0.5)));
	elseif (i <= 6.8e+229)
		tmp = t_0;
	elseif (i <= 6.3e+255)
		tmp = Float64(100.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / n));
	else
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.1e-15], t$95$0, If[LessEqual[i, 1.6e-20], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(N[(i / n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 6.8e+229], t$95$0, If[LessEqual[i, 6.3e+255], N[(100.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -2.09999999999999981e-15 or 1.59999999999999985e-20 < i < 6.8000000000000002e229

   1. Initial program 55.0%

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

   2. Taylor expanded in n around inf 68.2%

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

      1. expm1-def 69.2%

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

   4. Simplified 69.2%

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

   if -2.09999999999999981e-15 < i < 1.59999999999999985e-20

   1. Initial program 8.3%

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

      1. *-commutative 8.3%

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

      2. associate-/r/ 9.0%

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

      3. sub-neg 9.0%

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

      4. metadata-eval 9.0%

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

      5. associate-*r* 9.0%

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

      6. *-commutative 9.0%

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

      7. clear-num 9.0%

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

      8. un-div-inv 9.0%

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

      9. metadata-eval 9.0%

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

      10. sub-neg 9.0%

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

      11. pow-to-exp 9.0%

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

      12. expm1-def 18.5%

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

      13. add-log-exp 9.0%

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

      14. pow-to-exp 9.0%

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

      15. log-pow 18.5%

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

      16. log1p-udef 65.1%

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

   3. Applied egg-rr 65.1%

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

   4. Taylor expanded in i around 0 92.3%

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

      1. sub-neg 92.3%

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

      2. associate-*r/ 92.3%

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

      3. metadata-eval 92.3%

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

      4. metadata-eval 92.3%

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

   6. Simplified 92.3%

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

   7. Taylor expanded in n around 0 92.3%

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

   if 6.8000000000000002e229 < i < 6.3e255

   1. Initial program 33.1%

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

   2. Taylor expanded in n around inf 1.4%

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

      1. *-commutative 1.4%

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

      2. associate-/l* 1.4%

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

      3. expm1-def 1.4%

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

   4. Simplified 1.4%

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

      1. *-commutative 1.4%

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

      2. clear-num 1.4%

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

      3. un-div-inv 1.4%

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

   6. Applied egg-rr 1.4%

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

   7. Taylor expanded in i around 0 72.9%

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

      1. *-commutative 54.5%

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

   9. Simplified 72.9%

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

   if 6.3e255 < i

   1. Initial program 83.3%

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

      1. div-sub 83.3%

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

      2. clear-num 83.3%

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

      3. sub-neg 83.3%

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

      4. div-inv 83.3%

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

      5. clear-num 83.3%

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

   3. Applied egg-rr 83.3%

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

      1. sub-neg 83.3%

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

   5. Simplified 83.3%

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

      1. associate-*r/ 83.6%

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

      2. sub-div 83.6%

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

   7. Applied egg-rr 83.6%

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

   8. Taylor expanded in i around 0 76.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.1 \cdot 10^{-15}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{+229}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 6.3 \cdot 10^{+255}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \]

Alternative 9: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2050 \lor \neg \left(n \leq 0.58\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2050.0) (not (<= n 0.58)))
   (* 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 <= -2050.0) || !(n <= 0.58)) {
		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 <= -2050.0) || !(n <= 0.58)) {
		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 <= -2050.0) or not (n <= 0.58):
		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 <= -2050.0) || !(n <= 0.58))
		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, -2050.0], N[Not[LessEqual[n, 0.58]], $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 < -2050 or 0.57999999999999996 < n

   1. Initial program 29.1%

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

   2. Taylor expanded in n around inf 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-/l* 42.0%

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

      3. expm1-def 90.3%

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

   4. Simplified 90.3%

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

      1. associate-*l/ 90.4%

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

   6. Applied egg-rr 90.4%

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

   7. Taylor expanded in n around 0 42.1%

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

      1. expm1-def 90.3%

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

      2. associate-/l* 90.3%

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

      3. associate-*r/ 90.4%

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

      4. associate-*l/ 90.3%

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

      5. *-commutative 90.3%

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

      6. associate-/r/ 89.7%

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

   9. Simplified 89.7%

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

   10. Taylor expanded in i around inf 42.0%

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

       1. expm1-def 90.3%

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

   12. Simplified 90.3%

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

   if -2050 < n < 0.57999999999999996

   1. Initial program 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-/r/ 35.6%

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

      3. sub-neg 35.6%

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

      4. metadata-eval 35.6%

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

      5. associate-*r* 35.6%

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

      6. *-commutative 35.6%

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

      7. clear-num 35.6%

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

      8. un-div-inv 35.6%

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

      9. metadata-eval 35.6%

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

      10. sub-neg 35.6%

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

      11. pow-to-exp 35.6%

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

      12. expm1-def 57.1%

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

      13. add-log-exp 35.6%

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

      14. pow-to-exp 35.6%

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

      15. log-pow 57.1%

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

      16. log1p-udef 90.9%

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

   3. Applied egg-rr 90.9%

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

   4. Taylor expanded in i around 0 80.4%

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

      1. sub-neg 80.4%

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

      2. associate-*r/ 80.4%

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

      3. metadata-eval 80.4%

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

      4. metadata-eval 80.4%

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

   6. Simplified 80.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2050 \lor \neg \left(n \leq 0.58\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 10: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2050:\\ \;\;\;\;100 \cdot \left(n \cdot t_0\right)\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2050.0)
     (* 100.0 (* n t_0))
     (if (<= n 0.58)
       (/ (* 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 <= -2050.0) {
		tmp = 100.0 * (n * t_0);
	} else if (n <= 0.58) {
		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 <= -2050.0) {
		tmp = 100.0 * (n * t_0);
	} else if (n <= 0.58) {
		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 <= -2050.0:
		tmp = 100.0 * (n * t_0)
	elif n <= 0.58:
		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 <= -2050.0)
		tmp = Float64(100.0 * Float64(n * t_0));
	elseif (n <= 0.58)
		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, -2050.0], N[(100.0 * N[(n * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.58], 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 < -2050

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-/l* 38.8%

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

      3. expm1-def 87.8%

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

   4. Simplified 87.8%

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

   5. Taylor expanded in n around 0 38.8%

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

      1. expm1-def 87.7%

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

      2. associate-*r/ 87.8%

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

   7. Simplified 87.8%

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

   if -2050 < n < 0.57999999999999996

   1. Initial program 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-/r/ 35.6%

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

      3. sub-neg 35.6%

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

      4. metadata-eval 35.6%

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

      5. associate-*r* 35.6%

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

      6. *-commutative 35.6%

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

      7. clear-num 35.6%

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

      8. un-div-inv 35.6%

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

      9. metadata-eval 35.6%

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

      10. sub-neg 35.6%

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

      11. pow-to-exp 35.6%

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

      12. expm1-def 57.1%

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

      13. add-log-exp 35.6%

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

      14. pow-to-exp 35.6%

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

      15. log-pow 57.1%

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

      16. log1p-udef 90.9%

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

   3. Applied egg-rr 90.9%

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

   4. Taylor expanded in i around 0 80.4%

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

      1. sub-neg 80.4%

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

      2. associate-*r/ 80.4%

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

      3. metadata-eval 80.4%

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

      4. metadata-eval 80.4%

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

   6. Simplified 80.4%

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

   if 0.57999999999999996 < n

   1. Initial program 29.1%

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

   2. Taylor expanded in n around inf 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/l* 44.9%

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

      3. expm1-def 92.6%

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

   4. Simplified 92.6%

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

      1. associate-*l/ 92.7%

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

   6. Applied egg-rr 92.7%

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

   7. Taylor expanded in n around 0 45.0%

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

      1. expm1-def 92.6%

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

      2. associate-/l* 92.6%

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

      3. associate-*r/ 92.7%

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

      4. associate-*l/ 92.6%

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

      5. *-commutative 92.6%

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

      6. associate-/r/ 92.6%

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

   9. Simplified 92.6%

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

   10. Taylor expanded in i around inf 45.0%

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

       1. expm1-def 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2050:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 11: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2050:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2050.0)
   (/ (* n 100.0) (/ i (expm1 i)))
   (if (<= n 0.58)
     (/ (* 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 <= -2050.0) {
		tmp = (n * 100.0) / (i / expm1(i));
	} else if (n <= 0.58) {
		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 <= -2050.0) {
		tmp = (n * 100.0) / (i / Math.expm1(i));
	} else if (n <= 0.58) {
		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 <= -2050.0:
		tmp = (n * 100.0) / (i / math.expm1(i))
	elif n <= 0.58:
		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 <= -2050.0)
		tmp = Float64(Float64(n * 100.0) / Float64(i / expm1(i)));
	elseif (n <= 0.58)
		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, -2050.0], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.58], 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 < -2050

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-/l* 38.8%

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

      3. expm1-def 87.8%

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

   4. Simplified 87.8%

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

      1. associate-*l/ 87.8%

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

   6. Applied egg-rr 87.8%

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

   if -2050 < n < 0.57999999999999996

   1. Initial program 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-/r/ 35.6%

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

      3. sub-neg 35.6%

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

      4. metadata-eval 35.6%

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

      5. associate-*r* 35.6%

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

      6. *-commutative 35.6%

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

      7. clear-num 35.6%

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

      8. un-div-inv 35.6%

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

      9. metadata-eval 35.6%

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

      10. sub-neg 35.6%

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

      11. pow-to-exp 35.6%

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

      12. expm1-def 57.1%

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

      13. add-log-exp 35.6%

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

      14. pow-to-exp 35.6%

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

      15. log-pow 57.1%

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

      16. log1p-udef 90.9%

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

   3. Applied egg-rr 90.9%

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

   4. Taylor expanded in i around 0 80.4%

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

      1. sub-neg 80.4%

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

      2. associate-*r/ 80.4%

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

      3. metadata-eval 80.4%

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

      4. metadata-eval 80.4%

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

   6. Simplified 80.4%

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

   if 0.57999999999999996 < n

   1. Initial program 29.1%

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

   2. Taylor expanded in n around inf 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/l* 44.9%

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

      3. expm1-def 92.6%

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

   4. Simplified 92.6%

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

      1. associate-*l/ 92.7%

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

   6. Applied egg-rr 92.7%

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

   7. Taylor expanded in n around 0 45.0%

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

      1. expm1-def 92.6%

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

      2. associate-/l* 92.6%

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

      3. associate-*r/ 92.7%

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

      4. associate-*l/ 92.6%

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

      5. *-commutative 92.6%

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

      6. associate-/r/ 92.6%

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

   9. Simplified 92.6%

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

   10. Taylor expanded in i around inf 45.0%

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

       1. expm1-def 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2050:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 12: 65.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 200 \cdot \frac{n \cdot n}{i}\\ \mathbf{if}\;i \leq -1600000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 5.7 \cdot 10^{-188}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 29:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{+260} \lor \neg \left(i \leq 8.5 \cdot 10^{+283}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 200.0 (/ (* n n) i))))
   (if (<= i -1600000000000.0)
     t_0
     (if (<= i 5.7e-188)
       (* n 100.0)
       (if (<= i 29.0)
         (* 100.0 (/ (* i n) i))
         (if (or (<= i 1.35e+260) (not (<= i 8.5e+283)))
           t_0
           (* (* i n) 50.0)))))))

C

double code(double i, double n) {
	double t_0 = 200.0 * ((n * n) / i);
	double tmp;
	if (i <= -1600000000000.0) {
		tmp = t_0;
	} else if (i <= 5.7e-188) {
		tmp = n * 100.0;
	} else if (i <= 29.0) {
		tmp = 100.0 * ((i * n) / i);
	} else if ((i <= 1.35e+260) || !(i <= 8.5e+283)) {
		tmp = t_0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

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 = 200.0d0 * ((n * n) / i)
    if (i <= (-1600000000000.0d0)) then
        tmp = t_0
    else if (i <= 5.7d-188) then
        tmp = n * 100.0d0
    else if (i <= 29.0d0) then
        tmp = 100.0d0 * ((i * n) / i)
    else if ((i <= 1.35d+260) .or. (.not. (i <= 8.5d+283))) then
        tmp = t_0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 200.0 * ((n * n) / i);
	double tmp;
	if (i <= -1600000000000.0) {
		tmp = t_0;
	} else if (i <= 5.7e-188) {
		tmp = n * 100.0;
	} else if (i <= 29.0) {
		tmp = 100.0 * ((i * n) / i);
	} else if ((i <= 1.35e+260) || !(i <= 8.5e+283)) {
		tmp = t_0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 200.0 * ((n * n) / i)
	tmp = 0
	if i <= -1600000000000.0:
		tmp = t_0
	elif i <= 5.7e-188:
		tmp = n * 100.0
	elif i <= 29.0:
		tmp = 100.0 * ((i * n) / i)
	elif (i <= 1.35e+260) or not (i <= 8.5e+283):
		tmp = t_0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(200.0 * Float64(Float64(n * n) / i))
	tmp = 0.0
	if (i <= -1600000000000.0)
		tmp = t_0;
	elseif (i <= 5.7e-188)
		tmp = Float64(n * 100.0);
	elseif (i <= 29.0)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif ((i <= 1.35e+260) || !(i <= 8.5e+283))
		tmp = t_0;
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 200.0 * ((n * n) / i);
	tmp = 0.0;
	if (i <= -1600000000000.0)
		tmp = t_0;
	elseif (i <= 5.7e-188)
		tmp = n * 100.0;
	elseif (i <= 29.0)
		tmp = 100.0 * ((i * n) / i);
	elseif ((i <= 1.35e+260) || ~((i <= 8.5e+283)))
		tmp = t_0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(200.0 * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1600000000000.0], t$95$0, If[LessEqual[i, 5.7e-188], N[(n * 100.0), $MachinePrecision], If[LessEqual[i, 29.0], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[i, 1.35e+260], N[Not[LessEqual[i, 8.5e+283]], $MachinePrecision]], t$95$0, N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.6e12 or 29 < i < 1.3499999999999999e260 or 8.5000000000000008e283 < i

   1. Initial program 56.1%

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

      1. *-commutative 56.1%

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

      2. associate-/r/ 56.2%

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

      3. sub-neg 56.2%

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

      4. metadata-eval 56.2%

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

      5. associate-*r* 56.2%

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

      6. *-commutative 56.2%

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

      7. clear-num 56.2%

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

      8. un-div-inv 56.2%

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

      9. metadata-eval 56.2%

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

      10. sub-neg 56.2%

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

      11. pow-to-exp 42.6%

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

      12. expm1-def 50.6%

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

      13. add-log-exp 42.6%

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

      14. pow-to-exp 56.2%

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

      15. log-pow 50.6%

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

      16. log1p-udef 78.2%

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

   3. Applied egg-rr 78.2%

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

   4. Taylor expanded in i around 0 36.5%

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

      1. sub-neg 36.5%

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

      2. associate-*r/ 36.5%

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

      3. metadata-eval 36.5%

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

      4. metadata-eval 36.5%

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

   6. Simplified 36.5%

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

   7. Taylor expanded in n around 0 43.2%

      \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]
   8. Step-by-step derivation

      1. unpow2 43.2%

         \[\leadsto 200 \cdot \frac{\color{blue}{n \cdot n}}{i} \]

   9. Simplified 43.2%

      \[\leadsto \color{blue}{200 \cdot \frac{n \cdot n}{i}} \]

   if -1.6e12 < i < 5.70000000000000025e-188

   1. Initial program 4.4%

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

   2. Taylor expanded in i around 0 88.4%

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

      1. *-commutative 88.4%

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

   4. Simplified 88.4%

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

   if 5.70000000000000025e-188 < i < 29

   1. Initial program 18.3%

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

      1. div-sub 18.3%

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

      2. clear-num 16.3%

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

      3. sub-neg 16.3%

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

      4. div-inv 16.3%

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

      5. clear-num 18.3%

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

   3. Applied egg-rr 18.3%

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

      1. sub-neg 18.3%

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

   5. Simplified 18.3%

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

      1. associate-*r/ 18.3%

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

      2. sub-div 18.7%

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

   7. Applied egg-rr 18.7%

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

   8. Taylor expanded in i around 0 82.1%

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

   if 1.3499999999999999e260 < i < 8.5000000000000008e283

   1. Initial program 100.0%

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

   2. Taylor expanded in i around 0 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in i around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{50 \cdot \left(n \cdot i\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1600000000000:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{elif}\;i \leq 5.7 \cdot 10^{-188}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 29:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{+260} \lor \neg \left(i \leq 8.5 \cdot 10^{+283}\right):\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 13: 72.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{+179}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{i + i \cdot \left(i \cdot 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3e+179)
   (* 100.0 (/ (* i n) i))
   (if (<= n 0.58)
     (/ (* n 100.0) (+ 1.0 (* (/ i n) 0.5)))
     (* 100.0 (/ n (/ i (+ i (* i (* i 0.5)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3e+179) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 0.58) {
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5));
	} else {
		tmp = 100.0 * (n / (i / (i + (i * (i * 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 <= (-3d+179)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 0.58d0) then
        tmp = (n * 100.0d0) / (1.0d0 + ((i / n) * 0.5d0))
    else
        tmp = 100.0d0 * (n / (i / (i + (i * (i * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3e+179) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 0.58) {
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5));
	} else {
		tmp = 100.0 * (n / (i / (i + (i * (i * 0.5)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3e+179:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 0.58:
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5))
	else:
		tmp = 100.0 * (n / (i / (i + (i * (i * 0.5)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3e+179)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 0.58)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(Float64(i / n) * 0.5)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(i / Float64(i + Float64(i * Float64(i * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3e+179)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 0.58)
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5));
	else
		tmp = 100.0 * (n / (i / (i + (i * (i * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3e+179], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.58], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(N[(i / n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(i / N[(i + N[(i * N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.9999999999999998e179

   1. Initial program 15.4%

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

      1. div-sub 15.4%

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

      2. clear-num 15.6%

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

      3. sub-neg 15.6%

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

      4. div-inv 15.6%

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

      5. clear-num 15.4%

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

   3. Applied egg-rr 15.4%

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

      1. sub-neg 15.4%

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

   5. Simplified 15.4%

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

      1. associate-*r/ 15.4%

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

      2. sub-div 16.2%

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

   7. Applied egg-rr 16.2%

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

   8. Taylor expanded in i around 0 72.0%

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

   if -2.9999999999999998e179 < n < 0.57999999999999996

   1. Initial program 36.9%

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

      1. *-commutative 36.9%

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

      2. associate-/r/ 37.2%

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

      3. sub-neg 37.2%

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

      4. metadata-eval 37.2%

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

      5. associate-*r* 37.2%

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

      6. *-commutative 37.2%

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

      7. clear-num 37.2%

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

      8. un-div-inv 37.3%

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

      9. metadata-eval 37.3%

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

      10. sub-neg 37.3%

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

      11. pow-to-exp 30.8%

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

      12. expm1-def 46.2%

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

      13. add-log-exp 30.8%

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

      14. pow-to-exp 37.3%

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

      15. log-pow 46.2%

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

      16. log1p-udef 78.3%

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

   3. Applied egg-rr 78.3%

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

   4. Taylor expanded in i around 0 72.8%

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

      1. sub-neg 72.8%

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

      2. associate-*r/ 72.8%

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

      3. metadata-eval 72.8%

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

      4. metadata-eval 72.8%

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

   6. Simplified 72.8%

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

   7. Taylor expanded in n around 0 71.3%

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

   if 0.57999999999999996 < n

   1. Initial program 29.1%

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

   2. Taylor expanded in i around 0 56.4%

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

      1. unpow2 56.4%

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

      2. associate-*r/ 56.4%

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

      3. metadata-eval 56.4%

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

   4. Simplified 56.4%

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

   5. Taylor expanded in n around inf 85.6%

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

      1. associate-/l* 77.6%

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

      2. *-commutative 77.6%

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

      3. unpow2 77.6%

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

      4. associate-*l* 77.6%

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

   7. Simplified 77.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{+179}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{i + i \cdot \left(i \cdot 0.5\right)}}\\ \end{array} \]

Alternative 14: 65.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-241}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n)))))
   (if (<= n -1.1e+46)
     (* 100.0 (/ (* i n) i))
     (if (<= n -3.8e-175)
       t_0
       (if (<= n 3.4e-241)
         (* 200.0 (/ (* n n) i))
         (if (<= n 0.58) t_0 (* n (+ 100.0 (* i 50.0)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.1e+46) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -3.8e-175) {
		tmp = t_0;
	} else if (n <= 3.4e-241) {
		tmp = 200.0 * ((n * n) / i);
	} else if (n <= 0.58) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (i / (i / n))
    if (n <= (-1.1d+46)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= (-3.8d-175)) then
        tmp = t_0
    else if (n <= 3.4d-241) then
        tmp = 200.0d0 * ((n * n) / i)
    else if (n <= 0.58d0) then
        tmp = t_0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.1e+46) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -3.8e-175) {
		tmp = t_0;
	} else if (n <= 3.4e-241) {
		tmp = 200.0 * ((n * n) / i);
	} else if (n <= 0.58) {
		tmp = t_0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -1.1e+46:
		tmp = 100.0 * ((i * n) / i)
	elif n <= -3.8e-175:
		tmp = t_0
	elif n <= 3.4e-241:
		tmp = 200.0 * ((n * n) / i)
	elif n <= 0.58:
		tmp = t_0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -1.1e+46)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -3.8e-175)
		tmp = t_0;
	elseif (n <= 3.4e-241)
		tmp = Float64(200.0 * Float64(Float64(n * n) / i));
	elseif (n <= 0.58)
		tmp = t_0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -1.1e+46)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= -3.8e-175)
		tmp = t_0;
	elseif (n <= 3.4e-241)
		tmp = 200.0 * ((n * n) / i);
	elseif (n <= 0.58)
		tmp = t_0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.1e+46], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -3.8e-175], t$95$0, If[LessEqual[n, 3.4e-241], N[(200.0 * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.58], t$95$0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.1e46

   1. Initial program 20.7%

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

      1. div-sub 20.6%

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

      2. clear-num 20.9%

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

      3. sub-neg 20.9%

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

      4. div-inv 20.9%

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

      5. clear-num 20.6%

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

   3. Applied egg-rr 20.6%

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

      1. sub-neg 20.6%

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

   5. Simplified 20.6%

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

      1. associate-*r/ 20.6%

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

      2. sub-div 21.6%

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

   7. Applied egg-rr 21.6%

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

   8. Taylor expanded in i around 0 64.6%

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

   if -1.1e46 < n < -3.8e-175 or 3.3999999999999999e-241 < n < 0.57999999999999996

   1. Initial program 29.7%

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

   2. Taylor expanded in i around 0 59.5%

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

   if -3.8e-175 < n < 3.3999999999999999e-241

   1. Initial program 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-/r/ 65.8%

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

      3. sub-neg 65.8%

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

      4. metadata-eval 65.8%

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

      5. associate-*r* 65.8%

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

      6. *-commutative 65.8%

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

      7. clear-num 65.8%

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

      8. un-div-inv 65.8%

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

      9. metadata-eval 65.8%

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

      10. sub-neg 65.8%

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

      11. pow-to-exp 65.8%

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

      12. expm1-def 78.4%

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

      13. add-log-exp 65.8%

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

      14. pow-to-exp 65.8%

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

      15. log-pow 78.4%

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

      16. log1p-udef 84.0%

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

   3. Applied egg-rr 84.0%

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

   4. Taylor expanded in i around 0 92.0%

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

      1. sub-neg 92.0%

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

      2. associate-*r/ 92.0%

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

      3. metadata-eval 92.0%

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

      4. metadata-eval 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in n around 0 81.7%

      \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]
   8. Step-by-step derivation

      1. unpow2 81.7%

         \[\leadsto 200 \cdot \frac{\color{blue}{n \cdot n}}{i} \]

   9. Simplified 81.7%

      \[\leadsto \color{blue}{200 \cdot \frac{n \cdot n}{i}} \]

   if 0.57999999999999996 < n

   1. Initial program 29.1%

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

   2. Taylor expanded in n around inf 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/l* 44.9%

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

      3. expm1-def 92.6%

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

   4. Simplified 92.6%

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

      1. associate-*l/ 92.7%

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

   6. Applied egg-rr 92.7%

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

   7. Taylor expanded in i around 0 72.1%

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

      1. associate-*r* 72.1%

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

      2. distribute-rgt-out 72.1%

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

   9. Simplified 72.1%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-241}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 15: 65.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + i \cdot -0.5\\ \mathbf{if}\;n \leq -1.22 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{n}{t_0}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-245}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{100}{\frac{t_0}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* i -0.5))))
   (if (<= n -1.22e-175)
     (* 100.0 (/ n t_0))
     (if (<= n 2.8e-245)
       (* 200.0 (/ (* n n) i))
       (if (<= n 7.5e-61) (/ 100.0 (/ t_0 n)) (* n (+ 100.0 (* i 50.0))))))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i * -0.5);
	double tmp;
	if (n <= -1.22e-175) {
		tmp = 100.0 * (n / t_0);
	} else if (n <= 2.8e-245) {
		tmp = 200.0 * ((n * n) / i);
	} else if (n <= 7.5e-61) {
		tmp = 100.0 / (t_0 / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i * (-0.5d0))
    if (n <= (-1.22d-175)) then
        tmp = 100.0d0 * (n / t_0)
    else if (n <= 2.8d-245) then
        tmp = 200.0d0 * ((n * n) / i)
    else if (n <= 7.5d-61) then
        tmp = 100.0d0 / (t_0 / n)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i * -0.5);
	double tmp;
	if (n <= -1.22e-175) {
		tmp = 100.0 * (n / t_0);
	} else if (n <= 2.8e-245) {
		tmp = 200.0 * ((n * n) / i);
	} else if (n <= 7.5e-61) {
		tmp = 100.0 / (t_0 / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 1.0 + (i * -0.5)
	tmp = 0
	if n <= -1.22e-175:
		tmp = 100.0 * (n / t_0)
	elif n <= 2.8e-245:
		tmp = 200.0 * ((n * n) / i)
	elif n <= 7.5e-61:
		tmp = 100.0 / (t_0 / n)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i * -0.5))
	tmp = 0.0
	if (n <= -1.22e-175)
		tmp = Float64(100.0 * Float64(n / t_0));
	elseif (n <= 2.8e-245)
		tmp = Float64(200.0 * Float64(Float64(n * n) / i));
	elseif (n <= 7.5e-61)
		tmp = Float64(100.0 / Float64(t_0 / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i * -0.5);
	tmp = 0.0;
	if (n <= -1.22e-175)
		tmp = 100.0 * (n / t_0);
	elseif (n <= 2.8e-245)
		tmp = 200.0 * ((n * n) / i);
	elseif (n <= 7.5e-61)
		tmp = 100.0 / (t_0 / n);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.22e-175], N[(100.0 * N[(n / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-245], N[(200.0 * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.5e-61], N[(100.0 / N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.2200000000000001e-175

   1. Initial program 28.6%

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

   2. Taylor expanded in n around inf 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/l* 34.8%

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

      3. expm1-def 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in i around 0 61.1%

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

      1. *-commutative 61.1%

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

   7. Simplified 61.1%

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

   if -1.2200000000000001e-175 < n < 2.8000000000000001e-245

   1. Initial program 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-/r/ 65.8%

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

      3. sub-neg 65.8%

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

      4. metadata-eval 65.8%

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

      5. associate-*r* 65.8%

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

      6. *-commutative 65.8%

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

      7. clear-num 65.8%

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

      8. un-div-inv 65.8%

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

      9. metadata-eval 65.8%

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

      10. sub-neg 65.8%

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

      11. pow-to-exp 65.8%

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

      12. expm1-def 78.4%

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

      13. add-log-exp 65.8%

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

      14. pow-to-exp 65.8%

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

      15. log-pow 78.4%

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

      16. log1p-udef 84.0%

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

   3. Applied egg-rr 84.0%

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

   4. Taylor expanded in i around 0 92.0%

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

      1. sub-neg 92.0%

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

      2. associate-*r/ 92.0%

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

      3. metadata-eval 92.0%

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

      4. metadata-eval 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in n around 0 81.7%

      \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]
   8. Step-by-step derivation

      1. unpow2 81.7%

         \[\leadsto 200 \cdot \frac{\color{blue}{n \cdot n}}{i} \]

   9. Simplified 81.7%

      \[\leadsto \color{blue}{200 \cdot \frac{n \cdot n}{i}} \]

   if 2.8000000000000001e-245 < n < 7.50000000000000047e-61

   1. Initial program 19.4%

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

   2. Taylor expanded in n around inf 4.4%

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

      1. *-commutative 4.4%

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

      2. associate-/l* 4.4%

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

      3. expm1-def 47.2%

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

   4. Simplified 47.2%

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

      1. *-commutative 47.2%

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

      2. clear-num 47.2%

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

      3. un-div-inv 47.2%

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

   6. Applied egg-rr 47.2%

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

   7. Taylor expanded in i around 0 63.3%

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

      1. *-commutative 59.1%

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

   9. Simplified 63.3%

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

   if 7.50000000000000047e-61 < n

   1. Initial program 27.1%

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

   2. Taylor expanded in n around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-/l* 38.8%

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

      3. expm1-def 87.6%

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

   4. Simplified 87.6%

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

      1. associate-*l/ 87.6%

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

   6. Applied egg-rr 87.6%

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

   7. Taylor expanded in i around 0 70.2%

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

      1. associate-*r* 70.2%

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

      2. distribute-rgt-out 70.2%

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

   9. Simplified 70.2%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-245}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 16: 65.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + i \cdot -0.5\\ \mathbf{if}\;n \leq -6 \cdot 10^{-170}:\\ \;\;\;\;100 \cdot \frac{n}{t_0}\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-245}:\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{\frac{0.5}{n} + -0.5}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{100}{\frac{t_0}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* i -0.5))))
   (if (<= n -6e-170)
     (* 100.0 (/ n t_0))
     (if (<= n 6.5e-245)
       (* 100.0 (/ (/ n i) (+ (/ 0.5 n) -0.5)))
       (if (<= n 7.5e-61) (/ 100.0 (/ t_0 n)) (* n (+ 100.0 (* i 50.0))))))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i * -0.5);
	double tmp;
	if (n <= -6e-170) {
		tmp = 100.0 * (n / t_0);
	} else if (n <= 6.5e-245) {
		tmp = 100.0 * ((n / i) / ((0.5 / n) + -0.5));
	} else if (n <= 7.5e-61) {
		tmp = 100.0 / (t_0 / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i * (-0.5d0))
    if (n <= (-6d-170)) then
        tmp = 100.0d0 * (n / t_0)
    else if (n <= 6.5d-245) then
        tmp = 100.0d0 * ((n / i) / ((0.5d0 / n) + (-0.5d0)))
    else if (n <= 7.5d-61) then
        tmp = 100.0d0 / (t_0 / n)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i * -0.5);
	double tmp;
	if (n <= -6e-170) {
		tmp = 100.0 * (n / t_0);
	} else if (n <= 6.5e-245) {
		tmp = 100.0 * ((n / i) / ((0.5 / n) + -0.5));
	} else if (n <= 7.5e-61) {
		tmp = 100.0 / (t_0 / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 1.0 + (i * -0.5)
	tmp = 0
	if n <= -6e-170:
		tmp = 100.0 * (n / t_0)
	elif n <= 6.5e-245:
		tmp = 100.0 * ((n / i) / ((0.5 / n) + -0.5))
	elif n <= 7.5e-61:
		tmp = 100.0 / (t_0 / n)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i * -0.5))
	tmp = 0.0
	if (n <= -6e-170)
		tmp = Float64(100.0 * Float64(n / t_0));
	elseif (n <= 6.5e-245)
		tmp = Float64(100.0 * Float64(Float64(n / i) / Float64(Float64(0.5 / n) + -0.5)));
	elseif (n <= 7.5e-61)
		tmp = Float64(100.0 / Float64(t_0 / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i * -0.5);
	tmp = 0.0;
	if (n <= -6e-170)
		tmp = 100.0 * (n / t_0);
	elseif (n <= 6.5e-245)
		tmp = 100.0 * ((n / i) / ((0.5 / n) + -0.5));
	elseif (n <= 7.5e-61)
		tmp = 100.0 / (t_0 / n);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6e-170], N[(100.0 * N[(n / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6.5e-245], N[(100.0 * N[(N[(n / i), $MachinePrecision] / N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.5e-61], N[(100.0 / N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.00000000000000027e-170

   1. Initial program 28.6%

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

   2. Taylor expanded in n around inf 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/l* 34.8%

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

      3. expm1-def 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in i around 0 61.1%

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

      1. *-commutative 61.1%

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

   7. Simplified 61.1%

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

   if -6.00000000000000027e-170 < n < 6.5000000000000004e-245

   1. Initial program 65.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 65.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. sub-neg 65.8%

         \[\leadsto \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \cdot 100 \]

      4. metadata-eval 65.8%

         \[\leadsto \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \cdot 100 \]

      5. associate-*r* 65.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]

      6. *-commutative 65.8%

         \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}} \]

      7. clear-num 65.8%

         \[\leadsto \left(n \cdot 100\right) \cdot \color{blue}{\frac{1}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

      8. un-div-inv 65.8%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

      9. metadata-eval 65.8%

         \[\leadsto \frac{n \cdot 100}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}}} \]

      10. sub-neg 65.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]

      11. pow-to-exp 65.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}} \]

      12. expm1-def 78.4%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}} \]

      13. add-log-exp 65.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\color{blue}{\log \left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}\right)}} \]

      14. pow-to-exp 65.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}\right)}} \]

      15. log-pow 78.4%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}} \]

      16. log1p-udef 84.0%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}} \]

   3. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   4. Taylor expanded in i around 0 92.0%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 92.0%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}} \]

      2. associate-*r/ 92.0%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)} \]

      3. metadata-eval 92.0%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)} \]

      4. metadata-eval 92.0%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)} \]

   6. Simplified 92.0%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]

   7. Taylor expanded in i around inf 82.4%

      \[\leadsto \color{blue}{100 \cdot \frac{n}{i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 82.4%

         \[\leadsto 100 \cdot \color{blue}{\frac{\frac{n}{i}}{0.5 \cdot \frac{1}{n} - 0.5}} \]

      2. sub-neg 82.4%

         \[\leadsto 100 \cdot \frac{\frac{n}{i}}{\color{blue}{0.5 \cdot \frac{1}{n} + \left(-0.5\right)}} \]

      3. metadata-eval 82.4%

         \[\leadsto 100 \cdot \frac{\frac{n}{i}}{0.5 \cdot \frac{1}{n} + \color{blue}{-0.5}} \]

      4. associate-*r/ 82.4%

         \[\leadsto 100 \cdot \frac{\frac{n}{i}}{\color{blue}{\frac{0.5 \cdot 1}{n}} + -0.5} \]

      5. metadata-eval 82.4%

         \[\leadsto 100 \cdot \frac{\frac{n}{i}}{\frac{\color{blue}{0.5}}{n} + -0.5} \]

      6. +-commutative 82.4%

         \[\leadsto 100 \cdot \frac{\frac{n}{i}}{\color{blue}{-0.5 + \frac{0.5}{n}}} \]

   9. Simplified 82.4%

      \[\leadsto \color{blue}{100 \cdot \frac{\frac{n}{i}}{-0.5 + \frac{0.5}{n}}} \]

   if 6.5000000000000004e-245 < n < 7.50000000000000047e-61

   1. Initial program 19.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 4.4%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 4.4%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 4.4%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 47.2%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 47.2%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   5. Step-by-step derivation

      1. *-commutative 47.2%

         \[\leadsto \color{blue}{100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      2. clear-num 47.2%

         \[\leadsto 100 \cdot \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n}}} \]

      3. un-div-inv 47.2%

         \[\leadsto \color{blue}{\frac{100}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n}}} \]

   6. Applied egg-rr 47.2%

      \[\leadsto \color{blue}{\frac{100}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n}}} \]

   7. Taylor expanded in i around 0 63.3%

      \[\leadsto \frac{100}{\frac{\color{blue}{1 + -0.5 \cdot i}}{n}} \]
   8. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   9. Simplified 63.3%

      \[\leadsto \frac{100}{\frac{\color{blue}{1 + i \cdot -0.5}}{n}} \]

   if 7.50000000000000047e-61 < n

   1. Initial program 27.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 38.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 38.8%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 38.8%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 87.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 87.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   5. Step-by-step derivation

      1. associate-*l/ 87.6%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   6. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   7. Taylor expanded in i around 0 70.2%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   8. Step-by-step derivation

      1. associate-*r* 70.2%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 70.2%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   9. Simplified 70.2%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-170}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-245}:\\ \;\;\;\;100 \cdot \frac{\frac{n}{i}}{\frac{0.5}{n} + -0.5}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 17: 71.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+182}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.55e+182)
   (* 100.0 (/ (* i n) i))
   (if (<= n 0.58)
     (/ (* n 100.0) (+ 1.0 (* (/ i n) 0.5)))
     (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.55e+182) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 0.58) {
		tmp = (n * 100.0) / (1.0 + ((i / 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 <= (-1.55d+182)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 0.58d0) then
        tmp = (n * 100.0d0) / (1.0d0 + ((i / 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 <= -1.55e+182) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 0.58) {
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.55e+182:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 0.58:
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.55e+182)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 0.58)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(Float64(i / 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 <= -1.55e+182)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 0.58)
		tmp = (n * 100.0) / (1.0 + ((i / n) * 0.5));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.55e+182], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.58], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(N[(i / n), $MachinePrecision] * 0.5), $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 < -1.54999999999999998e182

   1. Initial program 15.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. div-sub 15.4%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      2. clear-num 15.6%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      3. sub-neg 15.6%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(-\frac{n}{i}\right)\right)} \]

      4. div-inv 15.6%

         \[\leadsto 100 \cdot \left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{1}{\frac{i}{n}}} + \left(-\frac{n}{i}\right)\right) \]

      5. clear-num 15.4%

         \[\leadsto 100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \color{blue}{\frac{n}{i}} + \left(-\frac{n}{i}\right)\right) \]

   3. Applied egg-rr 15.4%

      \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} + \left(-\frac{n}{i}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg 15.4%

         \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)} \]

   5. Simplified 15.4%

      \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 15.4%

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot n}{i}} - \frac{n}{i}\right) \]

      2. sub-div 16.2%

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot n - n}{i}} \]

   7. Applied egg-rr 16.2%

      \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot n - n}{i}} \]

   8. Taylor expanded in i around 0 72.0%

      \[\leadsto 100 \cdot \frac{\color{blue}{i \cdot n}}{i} \]

   if -1.54999999999999998e182 < n < 0.57999999999999996

   1. Initial program 36.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 36.9%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 37.2%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. sub-neg 37.2%

         \[\leadsto \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \cdot 100 \]

      4. metadata-eval 37.2%

         \[\leadsto \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \cdot 100 \]

      5. associate-*r* 37.2%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]

      6. *-commutative 37.2%

         \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}} \]

      7. clear-num 37.2%

         \[\leadsto \left(n \cdot 100\right) \cdot \color{blue}{\frac{1}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

      8. un-div-inv 37.3%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

      9. metadata-eval 37.3%

         \[\leadsto \frac{n \cdot 100}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}}} \]

      10. sub-neg 37.3%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]

      11. pow-to-exp 30.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}} \]

      12. expm1-def 46.2%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}} \]

      13. add-log-exp 30.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\color{blue}{\log \left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}\right)}} \]

      14. pow-to-exp 37.3%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}\right)}} \]

      15. log-pow 46.2%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}} \]

      16. log1p-udef 78.3%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}} \]

   3. Applied egg-rr 78.3%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   4. Taylor expanded in i around 0 72.8%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 72.8%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}} \]

      2. associate-*r/ 72.8%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)} \]

      3. metadata-eval 72.8%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)} \]

      4. metadata-eval 72.8%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)} \]

   6. Simplified 72.8%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]

   7. Taylor expanded in n around 0 71.3%

      \[\leadsto \frac{n \cdot 100}{1 + \color{blue}{0.5 \cdot \frac{i}{n}}} \]

   if 0.57999999999999996 < n

   1. Initial program 29.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 45.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 45.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 44.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   5. Step-by-step derivation

      1. associate-*l/ 92.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   6. Applied egg-rr 92.7%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   7. Taylor expanded in i around 0 72.1%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   8. Step-by-step derivation

      1. associate-*r* 72.1%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 72.1%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   9. Simplified 72.1%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+182}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + \frac{i}{n} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 18: 72.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{i + i \cdot \left(i \cdot 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 0.58)
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))
   (* 100.0 (/ n (/ i (+ i (* i (* i 0.5))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 0.58) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = 100.0 * (n / (i / (i + (i * (i * 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 <= 0.58d0) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((0.5d0 / n) + (-0.5d0))))
    else
        tmp = 100.0d0 * (n / (i / (i + (i * (i * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 0.58) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = 100.0 * (n / (i / (i + (i * (i * 0.5)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 0.58:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	else:
		tmp = 100.0 * (n / (i / (i + (i * (i * 0.5)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 0.58)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	else
		tmp = Float64(100.0 * Float64(n / Float64(i / Float64(i + Float64(i * Float64(i * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 0.58)
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	else
		tmp = 100.0 * (n / (i / (i + (i * (i * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 0.58], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(i / N[(i + N[(i * N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.57999999999999996

   1. Initial program 32.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 32.7%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 33.1%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. sub-neg 33.1%

         \[\leadsto \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \cdot 100 \]

      4. metadata-eval 33.1%

         \[\leadsto \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \cdot 100 \]

      5. associate-*r* 33.1%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]

      6. *-commutative 33.1%

         \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}} \]

      7. clear-num 33.1%

         \[\leadsto \left(n \cdot 100\right) \cdot \color{blue}{\frac{1}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

      8. un-div-inv 33.2%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

      9. metadata-eval 33.2%

         \[\leadsto \frac{n \cdot 100}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}}} \]

      10. sub-neg 33.2%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]

      11. pow-to-exp 28.0%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}} \]

      12. expm1-def 40.3%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}} \]

      13. add-log-exp 28.0%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\color{blue}{\log \left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}\right)}} \]

      14. pow-to-exp 33.2%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}\right)}} \]

      15. log-pow 40.3%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}} \]

      16. log1p-udef 73.5%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}} \]

   3. Applied egg-rr 73.5%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   4. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 70.5%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}} \]

      2. associate-*r/ 70.5%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)} \]

      3. metadata-eval 70.5%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)} \]

      4. metadata-eval 70.5%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)} \]

   6. Simplified 70.5%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]

   if 0.57999999999999996 < n

   1. Initial program 29.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 56.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. unpow2 56.4%

         \[\leadsto 100 \cdot \frac{i + \color{blue}{\left(i \cdot i\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{\frac{i}{n}} \]

      2. associate-*r/ 56.4%

         \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)}{\frac{i}{n}} \]

      3. metadata-eval 56.4%

         \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)}{\frac{i}{n}} \]

   4. Simplified 56.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf 85.6%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(i + 0.5 \cdot {i}^{2}\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* 77.6%

         \[\leadsto 100 \cdot \color{blue}{\frac{n}{\frac{i}{i + 0.5 \cdot {i}^{2}}}} \]

      2. *-commutative 77.6%

         \[\leadsto 100 \cdot \frac{n}{\frac{i}{i + \color{blue}{{i}^{2} \cdot 0.5}}} \]

      3. unpow2 77.6%

         \[\leadsto 100 \cdot \frac{n}{\frac{i}{i + \color{blue}{\left(i \cdot i\right)} \cdot 0.5}} \]

      4. associate-*l* 77.6%

         \[\leadsto 100 \cdot \frac{n}{\frac{i}{i + \color{blue}{i \cdot \left(i \cdot 0.5\right)}}} \]

   7. Simplified 77.6%

      \[\leadsto 100 \cdot \color{blue}{\frac{n}{\frac{i}{i + i \cdot \left(i \cdot 0.5\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 0.58:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{i + i \cdot \left(i \cdot 0.5\right)}}\\ \end{array} \]

Alternative 19: 65.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-248}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.48:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.45e-175)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 4.8e-248)
     (* 200.0 (/ (* n n) i))
     (if (<= n 0.48) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.45e-175) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.8e-248) {
		tmp = 200.0 * ((n * n) / i);
	} else if (n <= 0.48) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.45d-175)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 4.8d-248) then
        tmp = 200.0d0 * ((n * n) / i)
    else if (n <= 0.48d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.45e-175) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.8e-248) {
		tmp = 200.0 * ((n * n) / i);
	} else if (n <= 0.48) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.45e-175:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 4.8e-248:
		tmp = 200.0 * ((n * n) / i)
	elif n <= 0.48:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.45e-175)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 4.8e-248)
		tmp = Float64(200.0 * Float64(Float64(n * n) / i));
	elseif (n <= 0.48)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.45e-175)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 4.8e-248)
		tmp = 200.0 * ((n * n) / i);
	elseif (n <= 0.48)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.45e-175], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.8e-248], N[(200.0 * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.48], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.44999999999999999e-175

   1. Initial program 28.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 34.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 34.8%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 34.8%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 83.1%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   5. Taylor expanded in i around 0 61.1%

      \[\leadsto \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \cdot 100 \]
   6. Step-by-step derivation

      1. *-commutative 61.1%

         \[\leadsto \frac{n}{1 + \color{blue}{i \cdot -0.5}} \cdot 100 \]

   7. Simplified 61.1%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]

   if -1.44999999999999999e-175 < n < 4.80000000000000006e-248

   1. Initial program 65.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} \]

      2. associate-/r/ 65.8%

         \[\leadsto \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \cdot 100 \]

      3. sub-neg 65.8%

         \[\leadsto \left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)}}{i} \cdot n\right) \cdot 100 \]

      4. metadata-eval 65.8%

         \[\leadsto \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{-1}}{i} \cdot n\right) \cdot 100 \]

      5. associate-*r* 65.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)} \]

      6. *-commutative 65.8%

         \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}} \]

      7. clear-num 65.8%

         \[\leadsto \left(n \cdot 100\right) \cdot \color{blue}{\frac{1}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

      8. un-div-inv 65.8%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + -1}}} \]

      9. metadata-eval 65.8%

         \[\leadsto \frac{n \cdot 100}{\frac{i}{{\left(1 + \frac{i}{n}\right)}^{n} + \color{blue}{\left(-1\right)}}} \]

      10. sub-neg 65.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}} \]

      11. pow-to-exp 65.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}} \]

      12. expm1-def 78.4%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}} \]

      13. add-log-exp 65.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\color{blue}{\log \left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}\right)}} \]

      14. pow-to-exp 65.8%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}\right)}} \]

      15. log-pow 78.4%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}} \]

      16. log1p-udef 84.0%

          \[\leadsto \frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}} \]

   3. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

   4. Taylor expanded in i around 0 92.0%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 92.0%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}} \]

      2. associate-*r/ 92.0%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)} \]

      3. metadata-eval 92.0%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)} \]

      4. metadata-eval 92.0%

         \[\leadsto \frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)} \]

   6. Simplified 92.0%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]

   7. Taylor expanded in n around 0 81.7%

      \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]
   8. Step-by-step derivation

      1. unpow2 81.7%

         \[\leadsto 200 \cdot \frac{\color{blue}{n \cdot n}}{i} \]

   9. Simplified 81.7%

      \[\leadsto \color{blue}{200 \cdot \frac{n \cdot n}{i}} \]

   if 4.80000000000000006e-248 < n < 0.47999999999999998

   1. Initial program 17.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 61.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 0.47999999999999998 < n

   1. Initial program 29.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 45.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 45.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 44.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 92.6%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   5. Step-by-step derivation

      1. associate-*l/ 92.7%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   6. Applied egg-rr 92.7%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   7. Taylor expanded in i around 0 72.1%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   8. Step-by-step derivation

      1. associate-*r* 72.1%

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out 72.1%

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   9. Simplified 72.1%

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-248}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.48:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 20: 62.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.36 \cdot 10^{+45} \lor \neg \left(n \leq 3 \cdot 10^{+16}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.36e+45) (not (<= n 3e+16)))
   (* 100.0 (/ (* i n) i))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.36e+45) || !(n <= 3e+16)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.36d+45)) .or. (.not. (n <= 3d+16))) then
        tmp = 100.0d0 * ((i * n) / i)
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.36e+45) || !(n <= 3e+16)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.36e+45) or not (n <= 3e+16):
		tmp = 100.0 * ((i * n) / i)
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.36e+45) || !(n <= 3e+16))
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.36e+45) || ~((n <= 3e+16)))
		tmp = 100.0 * ((i * n) / i);
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.36e+45], N[Not[LessEqual[n, 3e+16]], $MachinePrecision]], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.36e45 or 3e16 < n

   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. div-sub 24.6%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      2. clear-num 24.9%

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      3. sub-neg 24.9%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(-\frac{n}{i}\right)\right)} \]

      4. div-inv 24.9%

         \[\leadsto 100 \cdot \left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{1}{\frac{i}{n}}} + \left(-\frac{n}{i}\right)\right) \]

      5. clear-num 24.6%

         \[\leadsto 100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \color{blue}{\frac{n}{i}} + \left(-\frac{n}{i}\right)\right) \]

   3. Applied egg-rr 24.6%

      \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} + \left(-\frac{n}{i}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg 24.6%

         \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)} \]

   5. Simplified 24.6%

      \[\leadsto 100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 24.6%

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot n}{i}} - \frac{n}{i}\right) \]

      2. sub-div 25.3%

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot n - n}{i}} \]

   7. Applied egg-rr 25.3%

      \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot n - n}{i}} \]

   8. Taylor expanded in i around 0 67.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{i \cdot n}}{i} \]

   if -1.36e45 < n < 3e16

   1. Initial program 40.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 59.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.36 \cdot 10^{+45} \lor \neg \left(n \leq 3 \cdot 10^{+16}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 21: 58.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.8 \cdot 10^{+76}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.8e+76)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 8.2e+92) (* n 100.0) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.8e+76) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 8.2e+92) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 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 (i <= (-2.8d+76)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 8.2d+92) then
        tmp = n * 100.0d0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.8e+76) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 8.2e+92) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.8e+76:
		tmp = 100.0 * (i / (i / n))
	elif i <= 8.2e+92:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.8e+76)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 8.2e+92)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.8e+76)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 8.2e+92)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.8e+76], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.2e+92], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.7999999999999999e76

   1. Initial program 74.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 41.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if -2.7999999999999999e76 < i < 8.20000000000000047e92

   1. Initial program 11.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 70.7%

      \[\leadsto \color{blue}{100 \cdot n} \]
   3. Step-by-step derivation

      1. *-commutative 70.7%

         \[\leadsto \color{blue}{n \cdot 100} \]

   4. Simplified 70.7%

      \[\leadsto \color{blue}{n \cdot 100} \]

   if 8.20000000000000047e92 < i

   1. Initial program 61.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 50.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. unpow2 50.9%

         \[\leadsto 100 \cdot \frac{i + \color{blue}{\left(i \cdot i\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{\frac{i}{n}} \]

      2. associate-*r/ 50.9%

         \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)}{\frac{i}{n}} \]

      3. metadata-eval 50.9%

         \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)}{\frac{i}{n}} \]

   4. Simplified 50.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in i around inf 43.8%

      \[\leadsto \color{blue}{100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 43.8%

         \[\leadsto \color{blue}{\left(100 \cdot i\right) \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)} \]

      2. associate-*r/ 43.8%

         \[\leadsto \left(100 \cdot i\right) \cdot \left(n \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \]

      3. metadata-eval 43.8%

         \[\leadsto \left(100 \cdot i\right) \cdot \left(n \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \]

   7. Simplified 43.8%

      \[\leadsto \color{blue}{\left(100 \cdot i\right) \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)} \]

   8. Taylor expanded in n around inf 44.1%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
   9. Step-by-step derivation

      1. *-commutative 44.1%

         \[\leadsto 50 \cdot \color{blue}{\left(n \cdot i\right)} \]

   10. Simplified 44.1%

       \[\leadsto \color{blue}{50 \cdot \left(n \cdot i\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.8 \cdot 10^{+76}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 22: 55.2% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 8.2e+92) (* n 100.0) (* (* i n) 50.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 8.2e+92) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 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 (i <= 8.2d+92) then
        tmp = n * 100.0d0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 8.2e+92) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 8.2e+92:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 8.2e+92)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 8.2e+92)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 8.2e+92], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 8.20000000000000047e92

   1. Initial program 24.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 57.5%

      \[\leadsto \color{blue}{100 \cdot n} \]
   3. Step-by-step derivation

      1. *-commutative 57.5%

         \[\leadsto \color{blue}{n \cdot 100} \]

   4. Simplified 57.5%

      \[\leadsto \color{blue}{n \cdot 100} \]

   if 8.20000000000000047e92 < i

   1. Initial program 61.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 50.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. unpow2 50.9%

         \[\leadsto 100 \cdot \frac{i + \color{blue}{\left(i \cdot i\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{\frac{i}{n}} \]

      2. associate-*r/ 50.9%

         \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)}{\frac{i}{n}} \]

      3. metadata-eval 50.9%

         \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)}{\frac{i}{n}} \]

   4. Simplified 50.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in i around inf 43.8%

      \[\leadsto \color{blue}{100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 43.8%

         \[\leadsto \color{blue}{\left(100 \cdot i\right) \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)} \]

      2. associate-*r/ 43.8%

         \[\leadsto \left(100 \cdot i\right) \cdot \left(n \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \]

      3. metadata-eval 43.8%

         \[\leadsto \left(100 \cdot i\right) \cdot \left(n \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \]

   7. Simplified 43.8%

      \[\leadsto \color{blue}{\left(100 \cdot i\right) \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)} \]

   8. Taylor expanded in n around inf 44.1%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
   9. Step-by-step derivation

      1. *-commutative 44.1%

         \[\leadsto 50 \cdot \color{blue}{\left(n \cdot i\right)} \]

   10. Simplified 44.1%

       \[\leadsto \color{blue}{50 \cdot \left(n \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 23: 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 31.6%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

2. Taylor expanded in i around 0 38.8%

   \[\leadsto 100 \cdot \frac{\color{blue}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}}{\frac{i}{n}} \]
3. Step-by-step derivation

   1. unpow2 38.8%

      \[\leadsto 100 \cdot \frac{i + \color{blue}{\left(i \cdot i\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}{\frac{i}{n}} \]

   2. associate-*r/ 38.8%

      \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)}{\frac{i}{n}} \]

   3. metadata-eval 38.8%

      \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)}{\frac{i}{n}} \]

4. Simplified 38.8%

   \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)}}{\frac{i}{n}} \]

5. Taylor expanded in n around 0 2.6%

   \[\leadsto \color{blue}{-50 \cdot i} \]
6. Step-by-step derivation

   1. *-commutative 2.6%

      \[\leadsto \color{blue}{i \cdot -50} \]

7. Simplified 2.6%

   \[\leadsto \color{blue}{i \cdot -50} \]

8. Final simplification 2.6%

   \[\leadsto i \cdot -50 \]

Alternative 24: 50.3% 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 31.6%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

2. Taylor expanded in i around 0 47.3%

   \[\leadsto \color{blue}{100 \cdot n} \]
3. Step-by-step derivation

   1. *-commutative 47.3%

      \[\leadsto \color{blue}{n \cdot 100} \]

4. Simplified 47.3%

   \[\leadsto \color{blue}{n \cdot 100} \]

5. Final simplification 47.3%

   \[\leadsto n \cdot 100 \]

Developer target: 33.7% 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 2023271 
(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))))