Compound Interest

Percentage Accurate: 28.0% → 98.6%

Time: 22.5s

Alternatives: 20

Speedup: 8.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-241}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, t\_0, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1e-158)
     (/ (+ (* t_0 100.0) -100.0) (/ i n))
     (if (<= t_1 2e-241)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (* n (/ (fma 100.0 t_0 -100.0) i))
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

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-158) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else if (t_1 <= 2e-241) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (fma(100.0, t_0, -100.0) / i);
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-158)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	elseif (t_1 <= 2e-241)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(fma(100.0, t_0, -100.0) / i));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	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-158], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-241], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(100.0 * t$95$0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -1.00000000000000006e-158

   1. Initial program 99.6%

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

      1. associate-*r/ 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. metadata-eval 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   if -1.00000000000000006e-158 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1.9999999999999999e-241

   1. Initial program 18.6%

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

      1. associate-/r/ 18.3%

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

      2. associate-*r* 18.3%

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

      3. *-commutative 18.3%

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

      4. associate-*r/ 18.3%

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

      5. sub-neg 18.3%

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

      6. distribute-lft-in 18.3%

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

      7. metadata-eval 18.3%

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

      8. metadata-eval 18.3%

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

      9. metadata-eval 18.3%

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

      10. fma-define 18.3%

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

      11. metadata-eval 18.3%

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

   3. Simplified 18.3%

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

      1. fma-undefine 18.3%

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

      2. metadata-eval 18.3%

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

      3. metadata-eval 18.3%

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

      4. distribute-lft-in 18.3%

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

      5. sub-neg 18.3%

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

      6. *-commutative 18.3%

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

      7. add-exp-log 18.3%

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

      8. expm1-define 18.3%

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

      9. log-pow 27.1%

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

      10. log1p-define 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 1.9999999999999999e-241 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 96.6%

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

      1. associate-/r/ 96.5%

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

      2. associate-*r* 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-*r/ 96.8%

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

      5. sub-neg 96.8%

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

      6. distribute-lft-in 96.6%

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

      7. metadata-eval 96.6%

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

      8. metadata-eval 96.6%

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

      9. metadata-eval 96.6%

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

      10. fma-define 96.8%

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

      11. metadata-eval 96.8%

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

   3. Simplified 96.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-define 77.6%

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

   5. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-commutative 77.6%

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

      6. clear-num 77.5%

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

      7. un-div-inv 77.3%

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

      8. *-un-lft-identity 77.3%

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

      9. times-frac 77.5%

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

      10. metadata-eval 77.5%

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

   7. Applied egg-rr 77.5%

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

   8. Taylor expanded in i around 0 99.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-158}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-241}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := t\_0 \cdot 100\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-294}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))) (t_1 (* t_0 100.0)))
   (if (<= t_0 -4e-294)
     t_1
     (if (<= t_0 0.0)
       (* 100.0 (* n (/ (expm1 i) i)))
       (if (<= t_0 INFINITY)
         t_1
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -4e-294) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -4e-294) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	t_1 = t_0 * 100.0
	tmp = 0
	if t_0 <= -4e-294:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	t_1 = Float64(t_0 * 100.0)
	tmp = 0.0
	if (t_0 <= -4e-294)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-294], t$95$1, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.00000000000000007e-294 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 96.7%

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

   if -4.00000000000000007e-294 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 17.5%

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

   3. Taylor expanded in n around inf 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/l* 35.3%

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

      3. expm1-define 84.0%

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

   5. Simplified 84.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-define 77.6%

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

   5. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-commutative 77.6%

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

      6. clear-num 77.5%

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

      7. un-div-inv 77.3%

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

      8. *-un-lft-identity 77.3%

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

      9. times-frac 77.5%

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

      10. metadata-eval 77.5%

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

   7. Applied egg-rr 77.5%

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

   8. Taylor expanded in i around 0 99.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-294}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-294}:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -4e-294)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* 100.0 (* n (/ (expm1 i) i)))
       (if (<= t_1 INFINITY)
         (* n (/ (+ (* t_0 100.0) -100.0) i))
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

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 <= -4e-294) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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 <= -4e-294) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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 <= -4e-294:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	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 <= -4e-294)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	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, -4e-294], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.00000000000000007e-294

   1. Initial program 97.6%

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

   if -4.00000000000000007e-294 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 17.5%

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

   3. Taylor expanded in n around inf 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/l* 35.3%

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

      3. expm1-define 84.0%

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

   5. Simplified 84.0%

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

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

   1. Initial program 96.0%

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

      1. associate-/r/ 95.9%

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

      2. associate-*r* 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-*r/ 96.2%

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

      5. sub-neg 96.2%

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

      6. distribute-lft-in 96.1%

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

      7. metadata-eval 96.1%

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

      8. metadata-eval 96.1%

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

      9. metadata-eval 96.1%

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

      10. fma-define 96.2%

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

      11. metadata-eval 96.2%

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

   3. Simplified 96.2%

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

      1. fma-undefine 96.1%

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

      2. *-commutative 96.1%

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

   6. Applied egg-rr 96.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-define 77.6%

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

   5. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-commutative 77.6%

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

      6. clear-num 77.5%

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

      7. un-div-inv 77.3%

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

      8. *-un-lft-identity 77.3%

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

      9. times-frac 77.5%

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

      10. metadata-eval 77.5%

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

   7. Applied egg-rr 77.5%

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

   8. Taylor expanded in i around 0 99.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-294}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := t\_0 \cdot 100 + -100\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-294}:\\ \;\;\;\;\frac{t\_2}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t\_2}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (+ (* t_0 100.0) -100.0)))
   (if (<= t_1 -4e-294)
     (/ t_2 (/ i n))
     (if (<= t_1 0.0)
       (* 100.0 (* n (/ (expm1 i) i)))
       (if (<= t_1 INFINITY)
         (* n (/ t_2 i))
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = (t_0 * 100.0) + -100.0;
	double tmp;
	if (t_1 <= -4e-294) {
		tmp = t_2 / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (t_2 / i);
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = (t_0 * 100.0) + -100.0;
	double tmp;
	if (t_1 <= -4e-294) {
		tmp = t_2 / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (t_2 / i);
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	t_2 = (t_0 * 100.0) + -100.0
	tmp = 0
	if t_1 <= -4e-294:
		tmp = t_2 / (i / n)
	elif t_1 <= 0.0:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	elif t_1 <= math.inf:
		tmp = n * (t_2 / i)
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(Float64(t_0 * 100.0) + -100.0)
	tmp = 0.0
	if (t_1 <= -4e-294)
		tmp = Float64(t_2 / Float64(i / n));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(t_2 / i));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-294], N[(t$95$2 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(t$95$2 / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.00000000000000007e-294

   1. Initial program 97.6%

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

      1. associate-*r/ 97.9%

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

      2. sub-neg 97.9%

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

      3. distribute-rgt-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. metadata-eval 97.8%

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

   3. Simplified 97.8%

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

   if -4.00000000000000007e-294 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 17.5%

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

   3. Taylor expanded in n around inf 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/l* 35.3%

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

      3. expm1-define 84.0%

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

   5. Simplified 84.0%

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

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

   1. Initial program 96.0%

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

      1. associate-/r/ 95.9%

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

      2. associate-*r* 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-*r/ 96.2%

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

      5. sub-neg 96.2%

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

      6. distribute-lft-in 96.1%

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

      7. metadata-eval 96.1%

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

      8. metadata-eval 96.1%

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

      9. metadata-eval 96.1%

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

      10. fma-define 96.2%

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

      11. metadata-eval 96.2%

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

   3. Simplified 96.2%

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

      1. fma-undefine 96.1%

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

      2. *-commutative 96.1%

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

   6. Applied egg-rr 96.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-define 77.6%

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

   5. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-commutative 77.6%

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

      6. clear-num 77.5%

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

      7. un-div-inv 77.3%

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

      8. *-un-lft-identity 77.3%

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

      9. times-frac 77.5%

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

      10. metadata-eval 77.5%

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

   7. Applied egg-rr 77.5%

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

   8. Taylor expanded in i around 0 99.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-294}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-241}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1e-158)
     (/ (+ (* t_0 100.0) -100.0) (/ i n))
     (if (<= t_1 2e-241)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (* t_1 100.0)
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

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-158) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else if (t_1 <= 2e-241) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1 * 100.0;
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-158) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else if (t_1 <= 2e-241) {
		tmp = n * ((100.0 * Math.expm1((n * Math.log1p((i / n))))) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * 100.0;
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-158:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	elif t_1 <= 2e-241:
		tmp = n * ((100.0 * math.expm1((n * math.log1p((i / n))))) / i)
	elif t_1 <= math.inf:
		tmp = t_1 * 100.0
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	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-158)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	elseif (t_1 <= 2e-241)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(t_1 * 100.0);
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	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-158], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-241], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$1 * 100.0), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -1.00000000000000006e-158

   1. Initial program 99.6%

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

      1. associate-*r/ 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. metadata-eval 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   if -1.00000000000000006e-158 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1.9999999999999999e-241

   1. Initial program 18.6%

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

      1. associate-/r/ 18.3%

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

      2. associate-*r* 18.3%

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

      3. *-commutative 18.3%

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

      4. associate-*r/ 18.3%

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

      5. sub-neg 18.3%

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

      6. distribute-lft-in 18.3%

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

      7. metadata-eval 18.3%

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

      8. metadata-eval 18.3%

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

      9. metadata-eval 18.3%

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

      10. fma-define 18.3%

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

      11. metadata-eval 18.3%

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

   3. Simplified 18.3%

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

      1. fma-undefine 18.3%

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

      2. metadata-eval 18.3%

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

      3. metadata-eval 18.3%

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

      4. distribute-lft-in 18.3%

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

      5. sub-neg 18.3%

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

      6. *-commutative 18.3%

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

      7. add-exp-log 18.3%

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

      8. expm1-define 18.3%

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

      9. log-pow 27.1%

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

      10. log1p-define 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 1.9999999999999999e-241 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 96.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-define 77.6%

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

   5. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-commutative 77.6%

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

      6. clear-num 77.5%

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

      7. un-div-inv 77.3%

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

      8. *-un-lft-identity 77.3%

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

      9. times-frac 77.5%

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

      10. metadata-eval 77.5%

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

   7. Applied egg-rr 77.5%

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

   8. Taylor expanded in i around 0 99.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-158}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-241}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{-34}:\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;i \leq 2.35 \cdot 10^{+247}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{+295}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -2.5e-32)
     t_0
     (if (<= i 3.8e-34)
       (+ (* i -50.0) (* n (+ 100.0 (* i 50.0))))
       (if (<= i 2.35e+247)
         t_0
         (if (<= i 2.4e+295) (/ n (+ 0.01 (* i -0.005))) (* 50.0 (* i n))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -2.5e-32) {
		tmp = t_0;
	} else if (i <= 3.8e-34) {
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)));
	} else if (i <= 2.35e+247) {
		tmp = t_0;
	} else if (i <= 2.4e+295) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 50.0 * (i * n);
	}
	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 <= -2.5e-32) {
		tmp = t_0;
	} else if (i <= 3.8e-34) {
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)));
	} else if (i <= 2.35e+247) {
		tmp = t_0;
	} else if (i <= 2.4e+295) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -2.5e-32:
		tmp = t_0
	elif i <= 3.8e-34:
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)))
	elif i <= 2.35e+247:
		tmp = t_0
	elif i <= 2.4e+295:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -2.5e-32)
		tmp = t_0;
	elseif (i <= 3.8e-34)
		tmp = Float64(Float64(i * -50.0) + Float64(n * Float64(100.0 + Float64(i * 50.0))));
	elseif (i <= 2.35e+247)
		tmp = t_0;
	elseif (i <= 2.4e+295)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(50.0 * Float64(i * n));
	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, -2.5e-32], t$95$0, If[LessEqual[i, 3.8e-34], N[(N[(i * -50.0), $MachinePrecision] + N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.35e+247], t$95$0, If[LessEqual[i, 2.4e+295], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -2.5e-32 or 3.8000000000000001e-34 < i < 2.3500000000000001e247

   1. Initial program 52.2%

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

   3. Taylor expanded in n around inf 67.3%

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

      1. expm1-define 69.9%

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

   5. Simplified 69.9%

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

   if -2.5e-32 < i < 3.8000000000000001e-34

   1. Initial program 6.8%

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

      1. associate-/r/ 7.3%

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

      2. associate-*r* 7.3%

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

      3. *-commutative 7.3%

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

      4. associate-*r/ 7.3%

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

      5. sub-neg 7.3%

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

      6. distribute-lft-in 7.3%

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

      7. metadata-eval 7.3%

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

      8. metadata-eval 7.3%

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

      9. metadata-eval 7.3%

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

      10. fma-define 7.3%

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

      11. metadata-eval 7.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in i around 0 90.3%

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

      1. *-commutative 90.3%

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

      2. associate-*r/ 90.3%

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

      3. metadata-eval 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in n around 0 90.4%

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

   if 2.3500000000000001e247 < i < 2.40000000000000005e295

   1. Initial program 30.0%

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

   3. Taylor expanded in n around inf 21.2%

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

      1. *-commutative 21.2%

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

      2. associate-/l* 21.2%

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

      3. expm1-define 21.2%

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

   5. Simplified 21.2%

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

      1. *-commutative 21.2%

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

      2. *-commutative 21.2%

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

      3. associate-*r* 21.2%

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

      4. associate-/l* 21.2%

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

      5. *-commutative 21.2%

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

      6. clear-num 21.2%

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

      7. un-div-inv 21.2%

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

      8. *-un-lft-identity 21.2%

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

      9. times-frac 21.2%

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

      10. metadata-eval 21.2%

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

   7. Applied egg-rr 21.2%

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

   8. Taylor expanded in i around 0 62.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   10. Simplified 62.8%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if 2.40000000000000005e295 < i

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. expm1-define 100.0%

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

   5. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*r/ 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in i around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in i around inf 100.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{-34}:\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;i \leq 2.35 \cdot 10^{+247}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{+295}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -4.4 \cdot 10^{-220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.95:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (* 100.0 (expm1 i)) i))))
   (if (<= n -4.4e-220)
     t_0
     (if (<= n 1e-227)
       0.0
       (if (<= n 1.95)
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 * expm1(i)) / i);
	double tmp;
	if (n <= -4.4e-220) {
		tmp = t_0;
	} else if (n <= 1e-227) {
		tmp = 0.0;
	} else if (n <= 1.95) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * ((100.0 * Math.expm1(i)) / i);
	double tmp;
	if (n <= -4.4e-220) {
		tmp = t_0;
	} else if (n <= 1e-227) {
		tmp = 0.0;
	} else if (n <= 1.95) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * ((100.0 * math.expm1(i)) / i)
	tmp = 0
	if n <= -4.4e-220:
		tmp = t_0
	elif n <= 1e-227:
		tmp = 0.0
	elif n <= 1.95:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 * expm1(i)) / i))
	tmp = 0.0
	if (n <= -4.4e-220)
		tmp = t_0;
	elseif (n <= 1e-227)
		tmp = 0.0;
	elseif (n <= 1.95)
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.4e-220], t$95$0, If[LessEqual[n, 1e-227], 0.0, If[LessEqual[n, 1.95], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.39999999999999973e-220 or 1.94999999999999996 < n

   1. Initial program 25.3%

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

      1. associate-/r/ 25.3%

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

      2. associate-*r* 25.4%

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

      3. *-commutative 25.4%

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

      4. associate-*r/ 25.4%

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

      5. sub-neg 25.4%

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

      6. distribute-lft-in 25.4%

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

      7. metadata-eval 25.4%

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

      8. metadata-eval 25.4%

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

      9. metadata-eval 25.4%

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

      10. fma-define 25.4%

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

      11. metadata-eval 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in n around inf 37.0%

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

      1. sub-neg 37.0%

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

      2. metadata-eval 37.0%

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

      3. metadata-eval 37.0%

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

      4. distribute-lft-in 37.0%

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

      5. metadata-eval 37.0%

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

      6. sub-neg 37.0%

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

      7. expm1-define 89.0%

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

   7. Simplified 89.0%

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

   if -4.39999999999999973e-220 < n < 9.99999999999999945e-228

   1. Initial program 73.3%

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

      1. associate-*r/ 73.3%

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

      2. sub-neg 73.3%

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

      3. distribute-rgt-in 73.3%

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

      4. metadata-eval 73.3%

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

      5. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in i around 0 74.1%

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

      1. +-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in i around 0 82.4%

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

   9. Taylor expanded in i around 0 82.4%

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

   if 9.99999999999999945e-228 < n < 1.94999999999999996

   1. Initial program 15.3%

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

   3. Taylor expanded in n around inf 3.4%

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

      1. *-commutative 3.4%

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

      2. associate-/l* 3.4%

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

      3. expm1-define 53.1%

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

   5. Simplified 53.1%

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

      1. *-commutative 53.1%

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

      2. *-commutative 53.1%

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

      3. associate-*r* 53.1%

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

      4. associate-/l* 53.0%

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

      5. *-commutative 53.0%

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

      6. clear-num 53.0%

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

      7. un-div-inv 52.9%

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

      8. *-un-lft-identity 52.9%

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

      9. times-frac 53.0%

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

      10. metadata-eval 53.0%

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

   7. Applied egg-rr 53.0%

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

   8. Taylor expanded in i around 0 77.7%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-220}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.95:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -6 \cdot 10^{-222}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* n (/ (expm1 i) i)))))
   (if (<= n -6e-222)
     t_0
     (if (<= n 5.8e-228)
       0.0
       (if (<= n 1.0)
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (expm1(i) / i));
	double tmp;
	if (n <= -6e-222) {
		tmp = t_0;
	} else if (n <= 5.8e-228) {
		tmp = 0.0;
	} else if (n <= 1.0) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n * (Math.expm1(i) / i));
	double tmp;
	if (n <= -6e-222) {
		tmp = t_0;
	} else if (n <= 5.8e-228) {
		tmp = 0.0;
	} else if (n <= 1.0) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (math.expm1(i) / i))
	tmp = 0
	if n <= -6e-222:
		tmp = t_0
	elif n <= 5.8e-228:
		tmp = 0.0
	elif n <= 1.0:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -6e-222)
		tmp = t_0;
	elseif (n <= 5.8e-228)
		tmp = 0.0;
	elseif (n <= 1.0)
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6e-222], t$95$0, If[LessEqual[n, 5.8e-228], 0.0, If[LessEqual[n, 1.0], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.00000000000000059e-222 or 1 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf 37.0%

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

      1. *-commutative 37.0%

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

      2. associate-/l* 36.9%

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

      3. expm1-define 89.1%

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

   5. Simplified 89.1%

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

   if -6.00000000000000059e-222 < n < 5.8000000000000002e-228

   1. Initial program 73.3%

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

      1. associate-*r/ 73.3%

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

      2. sub-neg 73.3%

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

      3. distribute-rgt-in 73.3%

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

      4. metadata-eval 73.3%

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

      5. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in i around 0 74.1%

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

      1. +-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in i around 0 82.4%

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

   9. Taylor expanded in i around 0 82.4%

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

   if 5.8000000000000002e-228 < n < 1

   1. Initial program 15.3%

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

   3. Taylor expanded in n around inf 3.4%

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

      1. *-commutative 3.4%

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

      2. associate-/l* 3.4%

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

      3. expm1-define 53.1%

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

   5. Simplified 53.1%

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

      1. *-commutative 53.1%

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

      2. *-commutative 53.1%

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

      3. associate-*r* 53.1%

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

      4. associate-/l* 53.0%

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

      5. *-commutative 53.0%

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

      6. clear-num 53.0%

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

      7. un-div-inv 52.9%

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

      8. *-un-lft-identity 52.9%

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

      9. times-frac 53.0%

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

      10. metadata-eval 53.0%

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

   7. Applied egg-rr 53.0%

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

   8. Taylor expanded in i around 0 77.7%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-222}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.75 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + i \cdot \left(n \cdot 50 + \left(i \cdot n\right) \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.75e+194)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 1.0)
     (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))
     (/
      (*
       i
       (+
        (* n 100.0)
        (*
         i
         (+
          (* n 50.0)
          (* (* i n) (+ 16.666666666666668 (* i 4.166666666666667)))))))
      i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.75e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 1.0) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = (i * ((n * 100.0) + (i * ((n * 50.0) + ((i * n) * (16.666666666666668 + (i * 4.166666666666667))))))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.75d+194)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 1.0d0) then
        tmp = n / (0.01d0 + (i * ((i * 0.0008333333333333334d0) - 0.005d0)))
    else
        tmp = (i * ((n * 100.0d0) + (i * ((n * 50.0d0) + ((i * n) * (16.666666666666668d0 + (i * 4.166666666666667d0))))))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.75e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 1.0) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = (i * ((n * 100.0) + (i * ((n * 50.0) + ((i * n) * (16.666666666666668 + (i * 4.166666666666667))))))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.75e+194:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 1.0:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	else:
		tmp = (i * ((n * 100.0) + (i * ((n * 50.0) + ((i * n) * (16.666666666666668 + (i * 4.166666666666667))))))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.75e+194)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 1.0)
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	else
		tmp = Float64(Float64(i * Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(n * 50.0) + Float64(Float64(i * n) * Float64(16.666666666666668 + Float64(i * 4.166666666666667))))))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.75e+194)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 1.0)
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	else
		tmp = (i * ((n * 100.0) + (i * ((n * 50.0) + ((i * n) * (16.666666666666668 + (i * 4.166666666666667))))))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.75e+194], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.0], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(n * 50.0), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.75e194

   1. Initial program 8.9%

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

      1. associate-/r/ 9.5%

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

      2. associate-*r* 9.6%

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

      3. *-commutative 9.6%

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

      4. associate-*r/ 9.6%

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

      5. sub-neg 9.6%

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

      6. distribute-lft-in 9.6%

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

      7. metadata-eval 9.6%

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

      8. metadata-eval 9.6%

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

      9. metadata-eval 9.6%

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

      10. fma-define 9.6%

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

      11. metadata-eval 9.6%

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

   3. Simplified 9.6%

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

   5. Taylor expanded in n around inf 48.0%

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

      1. sub-neg 48.0%

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

      2. metadata-eval 48.0%

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

      3. metadata-eval 48.0%

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

      4. distribute-lft-in 48.0%

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

      5. metadata-eval 48.0%

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

      6. sub-neg 48.0%

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

      7. expm1-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in i around 0 68.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto n \cdot \left(100 + i \cdot \left(50 + \color{blue}{i \cdot 16.666666666666668}\right)\right) \]

   10. Simplified 68.7%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]

   if -2.75e194 < n < 1

   1. Initial program 34.1%

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

   3. Taylor expanded in n around inf 25.3%

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

      1. *-commutative 25.3%

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

      2. associate-/l* 25.3%

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

      3. expm1-define 66.0%

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

   5. Simplified 66.0%

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

      1. *-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*r* 66.0%

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

      4. associate-/l* 65.9%

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

      5. *-commutative 65.9%

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

      6. clear-num 65.9%

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

      7. un-div-inv 65.9%

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

      8. *-un-lft-identity 65.9%

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

      9. times-frac 66.0%

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

      10. metadata-eval 66.0%

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

   7. Applied egg-rr 66.0%

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

   8. Taylor expanded in i around 0 67.0%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]

   if 1 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-/l* 43.8%

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

      3. expm1-define 94.4%

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

   5. Simplified 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*r/ 94.3%

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

      3. associate-*r/ 94.2%

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

   7. Applied egg-rr 94.2%

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

   8. Taylor expanded in i around 0 82.4%

      \[\leadsto \frac{\color{blue}{i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(4.166666666666667 \cdot \left(i \cdot n\right) + 16.666666666666668 \cdot n\right)\right)\right)}}{i} \]

   9. Taylor expanded in n around 0 82.4%

      \[\leadsto \frac{i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n + \color{blue}{i \cdot \left(n \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)}\right)\right)}{i} \]
   10. Step-by-step derivation

       1. associate-*r* 82.4%

          \[\leadsto \frac{i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n + \color{blue}{\left(i \cdot n\right) \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)}\right)\right)}{i} \]

       2. *-commutative 82.4%

          \[\leadsto \frac{i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n + \color{blue}{\left(n \cdot i\right)} \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)\right)}{i} \]

       3. *-commutative 82.4%

          \[\leadsto \frac{i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n + \left(n \cdot i\right) \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right)}{i} \]

   11. Simplified 82.4%

       \[\leadsto \frac{i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n + \color{blue}{\left(n \cdot i\right) \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)}\right)\right)}{i} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.75 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + i \cdot \left(n \cdot 50 + \left(i \cdot n\right) \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.45 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(\left(i \cdot n\right) \cdot 4.166666666666667 + n \cdot 16.666666666666668\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.45e+194)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 1.0)
     (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))
     (+
      (* n 100.0)
      (*
       i
       (+
        (* n 50.0)
        (* i (+ (* (* i n) 4.166666666666667) (* n 16.666666666666668)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.45e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 1.0) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (((i * n) * 4.166666666666667) + (n * 16.666666666666668)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.45d+194)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 1.0d0) then
        tmp = n / (0.01d0 + (i * ((i * 0.0008333333333333334d0) - 0.005d0)))
    else
        tmp = (n * 100.0d0) + (i * ((n * 50.0d0) + (i * (((i * n) * 4.166666666666667d0) + (n * 16.666666666666668d0)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.45e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 1.0) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (((i * n) * 4.166666666666667) + (n * 16.666666666666668)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.45e+194:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 1.0:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	else:
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (((i * n) * 4.166666666666667) + (n * 16.666666666666668)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.45e+194)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 1.0)
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(n * 50.0) + Float64(i * Float64(Float64(Float64(i * n) * 4.166666666666667) + Float64(n * 16.666666666666668))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.45e+194)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 1.0)
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	else
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (((i * n) * 4.166666666666667) + (n * 16.666666666666668)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.45e+194], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.0], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(n * 50.0), $MachinePrecision] + N[(i * N[(N[(N[(i * n), $MachinePrecision] * 4.166666666666667), $MachinePrecision] + N[(n * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.45000000000000013e194

   1. Initial program 8.9%

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

      1. associate-/r/ 9.5%

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

      2. associate-*r* 9.6%

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

      3. *-commutative 9.6%

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

      4. associate-*r/ 9.6%

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

      5. sub-neg 9.6%

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

      6. distribute-lft-in 9.6%

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

      7. metadata-eval 9.6%

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

      8. metadata-eval 9.6%

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

      9. metadata-eval 9.6%

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

      10. fma-define 9.6%

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

      11. metadata-eval 9.6%

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

   3. Simplified 9.6%

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

   5. Taylor expanded in n around inf 48.0%

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

      1. sub-neg 48.0%

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

      2. metadata-eval 48.0%

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

      3. metadata-eval 48.0%

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

      4. distribute-lft-in 48.0%

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

      5. metadata-eval 48.0%

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

      6. sub-neg 48.0%

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

      7. expm1-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in i around 0 68.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto n \cdot \left(100 + i \cdot \left(50 + \color{blue}{i \cdot 16.666666666666668}\right)\right) \]

   10. Simplified 68.7%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]

   if -2.45000000000000013e194 < n < 1

   1. Initial program 34.1%

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

   3. Taylor expanded in n around inf 25.3%

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

      1. *-commutative 25.3%

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

      2. associate-/l* 25.3%

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

      3. expm1-define 66.0%

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

   5. Simplified 66.0%

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

      1. *-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*r* 66.0%

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

      4. associate-/l* 65.9%

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

      5. *-commutative 65.9%

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

      6. clear-num 65.9%

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

      7. un-div-inv 65.9%

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

      8. *-un-lft-identity 65.9%

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

      9. times-frac 66.0%

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

      10. metadata-eval 66.0%

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

   7. Applied egg-rr 66.0%

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

   8. Taylor expanded in i around 0 67.0%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]

   if 1 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-/l* 43.8%

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

      3. expm1-define 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in i around 0 81.2%

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(4.166666666666667 \cdot \left(i \cdot n\right) + 16.666666666666668 \cdot n\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.45 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(\left(i \cdot n\right) \cdot 4.166666666666667 + n \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{-220}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.7:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.3e-220)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 3.9e-228)
     0.0
     (if (<= n 1.7)
       (* 100.0 (/ i (/ i n)))
       (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.3e-220) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 3.9e-228) {
		tmp = 0.0;
	} else if (n <= 1.7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.3d-220)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 3.9d-228) then
        tmp = 0.0d0
    else if (n <= 1.7d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.3e-220) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 3.9e-228) {
		tmp = 0.0;
	} else if (n <= 1.7) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.3e-220:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 3.9e-228:
		tmp = 0.0
	elif n <= 1.7:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.3e-220)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 3.9e-228)
		tmp = 0.0;
	elseif (n <= 1.7)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.3e-220)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 3.9e-228)
		tmp = 0.0;
	elseif (n <= 1.7)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.3e-220], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.9e-228], 0.0, If[LessEqual[n, 1.7], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.29999999999999981e-220

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/l* 32.8%

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

      3. expm1-define 85.8%

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

   5. Simplified 85.8%

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

      1. *-commutative 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-*r* 85.9%

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

      4. associate-/l* 85.8%

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

      5. *-commutative 85.8%

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

      6. clear-num 85.7%

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

      7. un-div-inv 85.7%

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

      8. *-un-lft-identity 85.7%

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

      9. times-frac 85.8%

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

      10. metadata-eval 85.8%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in i around 0 63.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   10. Simplified 63.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -2.29999999999999981e-220 < n < 3.90000000000000029e-228

   1. Initial program 73.3%

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

      1. associate-*r/ 73.3%

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

      2. sub-neg 73.3%

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

      3. distribute-rgt-in 73.3%

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

      4. metadata-eval 73.3%

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

      5. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in i around 0 74.1%

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

      1. +-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in i around 0 82.4%

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

   9. Taylor expanded in i around 0 82.4%

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

   if 3.90000000000000029e-228 < n < 1.69999999999999996

   1. Initial program 15.3%

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

   3. Taylor expanded in i around 0 72.5%

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

   if 1.69999999999999996 < n

   1. Initial program 25.0%

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

      1. associate-/r/ 25.3%

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

      2. associate-*r* 25.3%

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

      3. *-commutative 25.3%

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

      4. associate-*r/ 25.3%

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

      5. sub-neg 25.3%

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

      6. distribute-lft-in 25.3%

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

      7. metadata-eval 25.3%

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

      8. metadata-eval 25.3%

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

      9. metadata-eval 25.3%

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

      10. fma-define 25.3%

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

      11. metadata-eval 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in n around inf 43.8%

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

      1. sub-neg 43.8%

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

      2. metadata-eval 43.8%

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

      3. metadata-eval 43.8%

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

      4. distribute-lft-in 43.8%

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

      5. metadata-eval 43.8%

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

      6. sub-neg 43.8%

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

      7. expm1-define 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in i around 0 73.5%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 73.5%

         \[\leadsto n \cdot \left(100 + i \cdot \left(50 + \color{blue}{i \cdot 16.666666666666668}\right)\right) \]

   10. Simplified 73.5%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{-220}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.7:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{-219}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.55:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot 50\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.6e-219)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 2e-230)
     0.0
     (if (<= n 1.55)
       (* 100.0 (/ i (/ i n)))
       (* n (/ (* i (+ 100.0 (* i 50.0))) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.6e-219) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 2e-230) {
		tmp = 0.0;
	} else if (n <= 1.55) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((i * (100.0 + (i * 50.0))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.6d-219)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 2d-230) then
        tmp = 0.0d0
    else if (n <= 1.55d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * ((i * (100.0d0 + (i * 50.0d0))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.6e-219) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 2e-230) {
		tmp = 0.0;
	} else if (n <= 1.55) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((i * (100.0 + (i * 50.0))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.6e-219:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 2e-230:
		tmp = 0.0
	elif n <= 1.55:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * ((i * (100.0 + (i * 50.0))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.6e-219)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 2e-230)
		tmp = 0.0;
	elseif (n <= 1.55)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(i * 50.0))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.6e-219)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 2e-230)
		tmp = 0.0;
	elseif (n <= 1.55)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * ((i * (100.0 + (i * 50.0))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.6e-219], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2e-230], 0.0, If[LessEqual[n, 1.55], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(i * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.60000000000000056e-219

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/l* 32.8%

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

      3. expm1-define 85.8%

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

   5. Simplified 85.8%

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

      1. *-commutative 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-*r* 85.9%

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

      4. associate-/l* 85.8%

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

      5. *-commutative 85.8%

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

      6. clear-num 85.7%

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

      7. un-div-inv 85.7%

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

      8. *-un-lft-identity 85.7%

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

      9. times-frac 85.8%

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

      10. metadata-eval 85.8%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in i around 0 63.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   10. Simplified 63.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -9.60000000000000056e-219 < n < 2.00000000000000009e-230

   1. Initial program 73.3%

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

      1. associate-*r/ 73.3%

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

      2. sub-neg 73.3%

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

      3. distribute-rgt-in 73.3%

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

      4. metadata-eval 73.3%

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

      5. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in i around 0 74.1%

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

      1. +-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in i around 0 82.4%

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

   9. Taylor expanded in i around 0 82.4%

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

   if 2.00000000000000009e-230 < n < 1.55000000000000004

   1. Initial program 15.3%

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

   3. Taylor expanded in i around 0 72.5%

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

   if 1.55000000000000004 < n

   1. Initial program 25.0%

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

      1. associate-/r/ 25.3%

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

      2. associate-*r* 25.3%

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

      3. *-commutative 25.3%

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

      4. associate-*r/ 25.3%

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

      5. sub-neg 25.3%

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

      6. distribute-lft-in 25.3%

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

      7. metadata-eval 25.3%

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

      8. metadata-eval 25.3%

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

      9. metadata-eval 25.3%

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

      10. fma-define 25.3%

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

      11. metadata-eval 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in i around 0 74.4%

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

      1. *-commutative 74.4%

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

      2. associate-*r/ 74.4%

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

      3. metadata-eval 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in n around inf 74.4%

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

      1. *-commutative 74.4%

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

   10. Simplified 74.4%

       \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + i \cdot 50\right)}}{i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{-219}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.55:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot 50\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.2e+194)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 1.0)
     (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))
     (*
      n
      (+
       100.0
       (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.2e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 1.0) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.2d+194)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 1.0d0) then
        tmp = n / (0.01d0 + (i * ((i * 0.0008333333333333334d0) - 0.005d0)))
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.2e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 1.0) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.2e+194:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 1.0:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.2e+194)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 1.0)
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.2e+194)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 1.0)
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	else
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.2e+194], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.0], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.20000000000000021e194

   1. Initial program 8.9%

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

      1. associate-/r/ 9.5%

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

      2. associate-*r* 9.6%

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

      3. *-commutative 9.6%

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

      4. associate-*r/ 9.6%

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

      5. sub-neg 9.6%

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

      6. distribute-lft-in 9.6%

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

      7. metadata-eval 9.6%

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

      8. metadata-eval 9.6%

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

      9. metadata-eval 9.6%

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

      10. fma-define 9.6%

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

      11. metadata-eval 9.6%

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

   3. Simplified 9.6%

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

   5. Taylor expanded in n around inf 48.0%

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

      1. sub-neg 48.0%

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

      2. metadata-eval 48.0%

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

      3. metadata-eval 48.0%

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

      4. distribute-lft-in 48.0%

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

      5. metadata-eval 48.0%

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

      6. sub-neg 48.0%

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

      7. expm1-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in i around 0 68.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto n \cdot \left(100 + i \cdot \left(50 + \color{blue}{i \cdot 16.666666666666668}\right)\right) \]

   10. Simplified 68.7%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]

   if -3.20000000000000021e194 < n < 1

   1. Initial program 34.1%

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

   3. Taylor expanded in n around inf 25.3%

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

      1. *-commutative 25.3%

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

      2. associate-/l* 25.3%

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

      3. expm1-define 66.0%

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

   5. Simplified 66.0%

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

      1. *-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*r* 66.0%

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

      4. associate-/l* 65.9%

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

      5. *-commutative 65.9%

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

      6. clear-num 65.9%

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

      7. un-div-inv 65.9%

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

      8. *-un-lft-identity 65.9%

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

      9. times-frac 66.0%

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

      10. metadata-eval 66.0%

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

   7. Applied egg-rr 66.0%

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

   8. Taylor expanded in i around 0 67.0%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]

   if 1 < n

   1. Initial program 25.0%

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

      1. associate-/r/ 25.3%

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

      2. associate-*r* 25.3%

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

      3. *-commutative 25.3%

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

      4. associate-*r/ 25.3%

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

      5. sub-neg 25.3%

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

      6. distribute-lft-in 25.3%

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

      7. metadata-eval 25.3%

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

      8. metadata-eval 25.3%

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

      9. metadata-eval 25.3%

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

      10. fma-define 25.3%

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

      11. metadata-eval 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in n around inf 43.8%

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

      1. sub-neg 43.8%

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

      2. metadata-eval 43.8%

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

      3. metadata-eval 43.8%

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

      4. distribute-lft-in 43.8%

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

      5. metadata-eval 43.8%

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

      6. sub-neg 43.8%

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

      7. expm1-define 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in i around 0 81.2%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 81.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right) \]

   10. Simplified 81.2%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-217}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-229}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.7e-217)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 1.65e-229)
     0.0
     (if (<= n 1.5) (* 100.0 (/ i (/ i n))) (* 100.0 (+ n (* n (* i 0.5))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.7e-217) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.65e-229) {
		tmp = 0.0;
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n + (n * (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 <= (-2.7d-217)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 1.65d-229) then
        tmp = 0.0d0
    else if (n <= 1.5d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * (n + (n * (i * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.7e-217) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.65e-229) {
		tmp = 0.0;
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n + (n * (i * 0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.7e-217:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 1.65e-229:
		tmp = 0.0
	elif n <= 1.5:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * (n + (n * (i * 0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.7e-217)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 1.65e-229)
		tmp = 0.0;
	elseif (n <= 1.5)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(i * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.7e-217)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 1.65e-229)
		tmp = 0.0;
	elseif (n <= 1.5)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * (n + (n * (i * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.7e-217], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e-229], 0.0, If[LessEqual[n, 1.5], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(n * N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.70000000000000016e-217

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/l* 32.8%

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

      3. expm1-define 85.8%

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

   5. Simplified 85.8%

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

      1. *-commutative 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-*r* 85.9%

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

      4. associate-/l* 85.8%

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

      5. *-commutative 85.8%

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

      6. clear-num 85.7%

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

      7. un-div-inv 85.7%

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

      8. *-un-lft-identity 85.7%

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

      9. times-frac 85.8%

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

      10. metadata-eval 85.8%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in i around 0 63.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   10. Simplified 63.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -2.70000000000000016e-217 < n < 1.65000000000000011e-229

   1. Initial program 73.3%

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

      1. associate-*r/ 73.3%

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

      2. sub-neg 73.3%

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

      3. distribute-rgt-in 73.3%

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

      4. metadata-eval 73.3%

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

      5. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in i around 0 74.1%

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

      1. +-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in i around 0 82.4%

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

   9. Taylor expanded in i around 0 82.4%

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

   if 1.65000000000000011e-229 < n < 1.5

   1. Initial program 15.3%

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

   3. Taylor expanded in i around 0 72.5%

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

   if 1.5 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-/l* 43.8%

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

      3. expm1-define 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in i around 0 69.5%

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

      1. associate-*r* 69.5%

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

   8. Simplified 69.5%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-217}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-229}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-223}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8e-223)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 7.5e-228)
     0.0
     (if (<= n 1.5) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8e-223) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 7.5e-228) {
		tmp = 0.0;
	} else if (n <= 1.5) {
		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 <= (-8d-223)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 7.5d-228) then
        tmp = 0.0d0
    else if (n <= 1.5d0) 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 <= -8e-223) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 7.5e-228) {
		tmp = 0.0;
	} else if (n <= 1.5) {
		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 <= -8e-223:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 7.5e-228:
		tmp = 0.0
	elif n <= 1.5:
		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 <= -8e-223)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 7.5e-228)
		tmp = 0.0;
	elseif (n <= 1.5)
		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 <= -8e-223)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 7.5e-228)
		tmp = 0.0;
	elseif (n <= 1.5)
		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, -8e-223], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.5e-228], 0.0, If[LessEqual[n, 1.5], 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 < -7.9999999999999998e-223

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/l* 32.8%

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

      3. expm1-define 85.8%

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

   5. Simplified 85.8%

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

      1. *-commutative 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-*r* 85.9%

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

      4. associate-/l* 85.8%

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

      5. *-commutative 85.8%

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

      6. clear-num 85.7%

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

      7. un-div-inv 85.7%

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

      8. *-un-lft-identity 85.7%

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

      9. times-frac 85.8%

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

      10. metadata-eval 85.8%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in i around 0 63.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   10. Simplified 63.9%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -7.9999999999999998e-223 < n < 7.4999999999999999e-228

   1. Initial program 73.3%

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

      1. associate-*r/ 73.3%

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

      2. sub-neg 73.3%

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

      3. distribute-rgt-in 73.3%

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

      4. metadata-eval 73.3%

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

      5. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in i around 0 74.1%

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

      1. +-commutative 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in i around 0 82.4%

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

   9. Taylor expanded in i around 0 82.4%

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

   if 7.4999999999999999e-228 < n < 1.5

   1. Initial program 15.3%

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

   3. Taylor expanded in i around 0 72.5%

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

   if 1.5 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-/l* 43.8%

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

      3. expm1-define 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in i around 0 69.5%

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

      1. +-commutative 69.5%

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

      2. associate-*r* 69.5%

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

      3. distribute-rgt-in 69.4%

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

      4. *-commutative 69.4%

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

   8. Simplified 69.4%

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot 50\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-223}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.7:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot 50\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+194)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.7)
     (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))
     (* n (/ (* i (+ 100.0 (* i 50.0))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.7) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = n * ((i * (100.0 + (i * 50.0))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.8d+194)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.7d0) then
        tmp = n / (0.01d0 + (i * ((i * 0.0008333333333333334d0) - 0.005d0)))
    else
        tmp = n * ((i * (100.0d0 + (i * 50.0d0))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.7) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = n * ((i * (100.0 + (i * 50.0))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.8e+194:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.7:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	else:
		tmp = n * ((i * (100.0 + (i * 50.0))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+194)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.7)
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	else
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(i * 50.0))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.8e+194)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.7)
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	else
		tmp = n * ((i * (100.0 + (i * 50.0))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+194], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.7], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(i * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.8000000000000001e194

   1. Initial program 8.9%

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

      1. associate-/r/ 9.5%

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

      2. associate-*r* 9.6%

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

      3. *-commutative 9.6%

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

      4. associate-*r/ 9.6%

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

      5. sub-neg 9.6%

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

      6. distribute-lft-in 9.6%

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

      7. metadata-eval 9.6%

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

      8. metadata-eval 9.6%

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

      9. metadata-eval 9.6%

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

      10. fma-define 9.6%

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

      11. metadata-eval 9.6%

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

   3. Simplified 9.6%

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

   5. Taylor expanded in n around inf 48.0%

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

      1. sub-neg 48.0%

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

      2. metadata-eval 48.0%

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

      3. metadata-eval 48.0%

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

      4. distribute-lft-in 48.0%

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

      5. metadata-eval 48.0%

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

      6. sub-neg 48.0%

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

      7. expm1-define 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in i around 0 68.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto n \cdot \left(100 + i \cdot \left(50 + \color{blue}{i \cdot 16.666666666666668}\right)\right) \]

   10. Simplified 68.7%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]

   if -2.8000000000000001e194 < n < 0.69999999999999996

   1. Initial program 34.1%

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

   3. Taylor expanded in n around inf 25.3%

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

      1. *-commutative 25.3%

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

      2. associate-/l* 25.3%

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

      3. expm1-define 66.0%

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

   5. Simplified 66.0%

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

      1. *-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*r* 66.0%

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

      4. associate-/l* 65.9%

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

      5. *-commutative 65.9%

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

      6. clear-num 65.9%

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

      7. un-div-inv 65.9%

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

      8. *-un-lft-identity 65.9%

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

      9. times-frac 66.0%

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

      10. metadata-eval 66.0%

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

   7. Applied egg-rr 66.0%

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

   8. Taylor expanded in i around 0 67.0%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]

   if 0.69999999999999996 < n

   1. Initial program 25.0%

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

      1. associate-/r/ 25.3%

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

      2. associate-*r* 25.3%

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

      3. *-commutative 25.3%

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

      4. associate-*r/ 25.3%

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

      5. sub-neg 25.3%

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

      6. distribute-lft-in 25.3%

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

      7. metadata-eval 25.3%

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

      8. metadata-eval 25.3%

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

      9. metadata-eval 25.3%

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

      10. fma-define 25.3%

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

      11. metadata-eval 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in i around 0 74.4%

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

      1. *-commutative 74.4%

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

      2. associate-*r/ 74.4%

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

      3. metadata-eval 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in n around inf 74.4%

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

      1. *-commutative 74.4%

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

   10. Simplified 74.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.7:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot 50\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.9:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+194)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.9)
     (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))
     (/ (* i (* n (+ 100.0 (* i 50.0)))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.9) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.8d+194)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.9d0) then
        tmp = n / (0.01d0 + (i * ((i * 0.0008333333333333334d0) - 0.005d0)))
    else
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+194) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.9) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.8e+194:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.9:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	else:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+194)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.9)
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	else
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.8e+194)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.9)
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	else
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+194], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.9], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.8000000000000001e194

   1. Initial program 8.9%

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

      1. associate-/r/ 9.5%

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

      2. associate-*r* 9.6%

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

      3. *-commutative 9.6%

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

      4. associate-*r/ 9.6%

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

      5. sub-neg 9.6%

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

      6. distribute-lft-in 9.6%

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

      7. metadata-eval 9.6%

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

      8. metadata-eval 9.6%

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

      9. metadata-eval 9.6%

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

      10. fma-define 9.6%

          \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{i} \]

      11. metadata-eval 9.6%

          \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 9.6%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around inf 48.0%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} - 100}}{i} \]
   6. Step-by-step derivation

      1. sub-neg 48.0%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot e^{i} + \left(-100\right)}}{i} \]

      2. metadata-eval 48.0%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{-100}}{i} \]

      3. metadata-eval 48.0%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{100 \cdot -1}}{i} \]

      4. distribute-lft-in 48.0%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot \left(e^{i} + -1\right)}}{i} \]

      5. metadata-eval 48.0%

         \[\leadsto n \cdot \frac{100 \cdot \left(e^{i} + \color{blue}{\left(-1\right)}\right)}{i} \]

      6. sub-neg 48.0%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left(e^{i} - 1\right)}}{i} \]

      7. expm1-define 99.7%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   7. Simplified 99.7%

      \[\leadsto n \cdot \frac{\color{blue}{100 \cdot \mathsf{expm1}\left(i\right)}}{i} \]

   8. Taylor expanded in i around 0 68.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto n \cdot \left(100 + i \cdot \left(50 + \color{blue}{i \cdot 16.666666666666668}\right)\right) \]

   10. Simplified 68.7%

       \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)} \]

   if -2.8000000000000001e194 < n < 0.900000000000000022

   1. Initial program 34.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 25.3%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 25.3%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 25.3%

         \[\leadsto \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \cdot 100 \]

      3. expm1-define 66.0%

         \[\leadsto \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \cdot 100 \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot 100} \]
   6. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

      2. *-commutative 66.0%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)} \]

      3. associate-*r* 66.0%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n} \]

      4. associate-/l* 65.9%

         \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}} \cdot n \]

      5. *-commutative 65.9%

         \[\leadsto \color{blue}{n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}} \]

      6. clear-num 65.9%

         \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}} \]

      7. un-div-inv 65.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}} \]

      8. *-un-lft-identity 65.9%

         \[\leadsto \frac{n}{\frac{\color{blue}{1 \cdot i}}{100 \cdot \mathsf{expm1}\left(i\right)}} \]

      9. times-frac 66.0%

         \[\leadsto \frac{n}{\color{blue}{\frac{1}{100} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

      10. metadata-eval 66.0%

          \[\leadsto \frac{n}{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}} \]

   7. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   8. Taylor expanded in i around 0 67.0%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]

   if 0.900000000000000022 < n

   1. Initial program 25.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 43.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 43.8%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 43.8%

         \[\leadsto \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \cdot 100 \]

      3. expm1-define 94.4%

         \[\leadsto \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \cdot 100 \]

   5. Simplified 94.4%

      \[\leadsto \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot 100} \]
   6. Step-by-step derivation

      1. *-commutative 94.4%

         \[\leadsto \color{blue}{100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

      2. associate-*r/ 94.3%

         \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

      3. associate-*r/ 94.2%

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}} \]

   7. Applied egg-rr 94.2%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}} \]

   8. Taylor expanded in i around 0 77.0%

      \[\leadsto \frac{\color{blue}{i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)}}{i} \]
   9. Step-by-step derivation

      1. +-commutative 77.0%

         \[\leadsto \frac{i \cdot \color{blue}{\left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}}{i} \]

      2. associate-*r* 77.0%

         \[\leadsto \frac{i \cdot \left(100 \cdot n + \color{blue}{\left(50 \cdot i\right) \cdot n}\right)}{i} \]

      3. *-commutative 77.0%

         \[\leadsto \frac{i \cdot \left(100 \cdot n + \color{blue}{\left(i \cdot 50\right)} \cdot n\right)}{i} \]

      4. distribute-rgt-out 77.0%

         \[\leadsto \frac{i \cdot \color{blue}{\left(n \cdot \left(100 + i \cdot 50\right)\right)}}{i} \]

   10. Simplified 77.0%

       \[\leadsto \frac{\color{blue}{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}}{i} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.9:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -12000 \lor \neg \left(n \leq 1.45\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -12000.0) (not (<= n 1.45)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -12000.0) || !(n <= 1.45)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-12000.0d0)) .or. (.not. (n <= 1.45d0))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -12000.0) || !(n <= 1.45)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -12000.0) or not (n <= 1.45):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -12000.0) || !(n <= 1.45))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -12000.0) || ~((n <= 1.45)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -12000.0], N[Not[LessEqual[n, 1.45]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -12000 or 1.44999999999999996 < n

   1. Initial program 25.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 41.0%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. *-commutative 41.0%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 40.9%

         \[\leadsto \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \cdot 100 \]

      3. expm1-define 91.8%

         \[\leadsto \left(n \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right) \cdot 100 \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot 100} \]

   6. Taylor expanded in i around 0 63.4%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. +-commutative 63.4%

         \[\leadsto \color{blue}{100 \cdot n + 50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* 63.4%

         \[\leadsto 100 \cdot n + \color{blue}{\left(50 \cdot i\right) \cdot n} \]

      3. distribute-rgt-in 63.3%

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

      4. *-commutative 63.3%

         \[\leadsto n \cdot \left(100 + \color{blue}{i \cdot 50}\right) \]

   8. Simplified 63.3%

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot 50\right)} \]

   if -12000 < n < 1.44999999999999996

   1. Initial program 31.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 67.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -12000 \lor \neg \left(n \leq 1.45\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 59.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.45 \cdot 10^{+36}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.45e+36) 0.0 (if (<= i 3.5e-18) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.45e+36) {
		tmp = 0.0;
	} else if (i <= 3.5e-18) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.45d+36)) then
        tmp = 0.0d0
    else if (i <= 3.5d-18) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.45e+36) {
		tmp = 0.0;
	} else if (i <= 3.5e-18) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.45e+36:
		tmp = 0.0
	elif i <= 3.5e-18:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.45e+36)
		tmp = 0.0;
	elseif (i <= 3.5e-18)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.45e+36)
		tmp = 0.0;
	elseif (i <= 3.5e-18)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.45e+36], 0.0, If[LessEqual[i, 3.5e-18], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.45e36 or 3.4999999999999999e-18 < i

   1. Initial program 57.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 57.2%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 57.2%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-rgt-in 57.2%

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \left(-1\right) \cdot 100}}{\frac{i}{n}} \]

      4. metadata-eval 57.2%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \color{blue}{-1} \cdot 100}{\frac{i}{n}} \]

      5. metadata-eval 57.2%

         \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \color{blue}{-100}}{\frac{i}{n}} \]

   3. Simplified 57.2%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 26.0%

      \[\leadsto \frac{\color{blue}{\left(1 + i\right)} \cdot 100 + -100}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. +-commutative 26.0%

         \[\leadsto \frac{\color{blue}{\left(i + 1\right)} \cdot 100 + -100}{\frac{i}{n}} \]

   7. Simplified 26.0%

      \[\leadsto \frac{\color{blue}{\left(i + 1\right)} \cdot 100 + -100}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0 29.1%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

   9. Taylor expanded in i around 0 29.1%

      \[\leadsto \color{blue}{0} \]

   if -1.45e36 < i < 3.4999999999999999e-18

   1. Initial program 7.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0 84.2%

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-commutative 84.2%

         \[\leadsto \color{blue}{n \cdot 100} \]

   5. Simplified 84.2%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.45 \cdot 10^{+36}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 17.8% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 27.8%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Step-by-step derivation

   1. associate-*r/ 27.8%

      \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

   2. sub-neg 27.8%

      \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

   3. distribute-rgt-in 27.8%

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \left(-1\right) \cdot 100}}{\frac{i}{n}} \]

   4. metadata-eval 27.8%

      \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \color{blue}{-1} \cdot 100}{\frac{i}{n}} \]

   5. metadata-eval 27.8%

      \[\leadsto \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \color{blue}{-100}}{\frac{i}{n}} \]

3. Simplified 27.8%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}} \]
4. Add Preprocessing

5. Taylor expanded in i around 0 15.0%

   \[\leadsto \frac{\color{blue}{\left(1 + i\right)} \cdot 100 + -100}{\frac{i}{n}} \]
6. Step-by-step derivation

   1. +-commutative 15.0%

      \[\leadsto \frac{\color{blue}{\left(i + 1\right)} \cdot 100 + -100}{\frac{i}{n}} \]

7. Simplified 15.0%

   \[\leadsto \frac{\color{blue}{\left(i + 1\right)} \cdot 100 + -100}{\frac{i}{n}} \]

8. Taylor expanded in i around 0 15.6%

   \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]

9. Taylor expanded in i around 0 15.9%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 15.9%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer target: 33.4% 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 2024076 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (* 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))))