Compound Interest

Percentage Accurate: 28.9% → 98.4%

Time: 19.4s

Alternatives: 11

Speedup: 16.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-191}:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left(t_0, 100, -100\right)}{i}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-191}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\frac{t_0}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -2e-191)
     (/ (* n (fma t_0 100.0 -100.0)) i)
     (if (<= t_1 5e-191)
       (* n (/ 100.0 (/ i (expm1 (* n (log1p (/ i n)))))))
       (if (<= t_1 INFINITY)
         (* n (* 100.0 (+ (/ t_0 i) (/ -1.0 i))))
         (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-191) {
		tmp = (n * fma(t_0, 100.0, -100.0)) / i;
	} else if (t_1 <= 5e-191) {
		tmp = n * (100.0 / (i / expm1((n * log1p((i / n))))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (100.0 * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-191)
		tmp = Float64(Float64(n * fma(t_0, 100.0, -100.0)) / i);
	elseif (t_1 <= 5e-191)
		tmp = Float64(n * Float64(100.0 / Float64(i / expm1(Float64(n * log1p(Float64(i / n)))))));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(100.0 * Float64(Float64(t_0 / i) + Float64(-1.0 / i))));
	else
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-191], N[(N[(n * N[(t$95$0 * 100.0 + -100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$1, 5e-191], N[(n * N[(100.0 / N[(i / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(100.0 * N[(N[(t$95$0 / i), $MachinePrecision] + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(1.0 / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2e-191

   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/ 99.6%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. fma-def 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -2e-191 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 5.0000000000000001e-191

   1. Initial program 15.8%

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

      1. associate-/r/ 15.7%

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

      2. associate-*r* 15.7%

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

      3. *-commutative 15.7%

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

      4. associate-*r/ 15.7%

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

      5. sub-neg 15.7%

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

      6. distribute-lft-in 15.7%

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

      7. fma-def 15.7%

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

      8. metadata-eval 15.7%

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

      9. metadata-eval 15.7%

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

   3. Simplified 15.7%

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

      1. clear-num 15.7%

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

      2. inv-pow 15.7%

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

      3. fma-udef 15.7%

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

      4. metadata-eval 15.7%

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

      5. metadata-eval 15.7%

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

      6. distribute-lft-in 15.7%

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

      7. sub-neg 15.7%

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

      8. *-commutative 15.7%

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

      9. pow-to-exp 15.7%

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

      10. expm1-def 32.1%

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

      11. add-log-exp 15.7%

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

      12. pow-to-exp 15.7%

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

      13. log-pow 32.1%

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

      14. log1p-udef 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. unpow-1 98.1%

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

      2. *-commutative 98.1%

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

   7. Simplified 98.1%

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

      1. clear-num 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. times-frac 98.2%

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

      4. metadata-eval 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. clear-num 98.1%

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

       2. un-div-inv 98.2%

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

   11. Applied egg-rr 98.2%

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

   if 5.0000000000000001e-191 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.3%

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

      1. associate-/r/ 98.2%

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

      2. associate-*r* 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*r/ 98.3%

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

      5. sub-neg 98.3%

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

      6. distribute-lft-in 98.2%

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

      7. fma-def 98.3%

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

      8. metadata-eval 98.3%

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

      9. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. fma-udef 98.2%

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

      4. metadata-eval 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-lft-in 98.3%

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

      7. sub-neg 98.3%

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

      8. *-commutative 98.3%

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

      9. pow-to-exp 66.8%

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

      10. expm1-def 68.4%

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

      11. add-log-exp 66.8%

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

      12. pow-to-exp 98.3%

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

      13. log-pow 68.4%

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

      14. log1p-udef 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. unpow-1 68.4%

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

      2. *-commutative 68.4%

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

   7. Simplified 68.4%

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

      1. clear-num 68.3%

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

      2. *-un-lft-identity 68.3%

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

      3. times-frac 68.2%

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

      4. metadata-eval 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. expm1-udef 66.8%

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

       2. div-sub 66.9%

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

       3. *-commutative 66.9%

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

       4. log1p-udef 66.9%

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

       5. exp-to-pow 98.4%

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

   11. Applied egg-rr 98.4%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. associate-*r* 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r/ 1.8%

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

      5. sub-neg 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. fma-def 1.8%

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

      8. metadata-eval 1.8%

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

      9. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. inv-pow 1.8%

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

      3. fma-udef 1.8%

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

      4. metadata-eval 1.8%

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

      5. metadata-eval 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. sub-neg 1.8%

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

      8. *-commutative 1.8%

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

      9. pow-to-exp 1.8%

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

      10. expm1-def 1.8%

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

      11. add-log-exp 1.8%

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

      12. pow-to-exp 1.8%

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

      13. log-pow 1.8%

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

      14. log1p-udef 1.8%

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

   5. Applied egg-rr 1.8%

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

      1. unpow-1 1.8%

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

      2. *-commutative 1.8%

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

   7. Simplified 1.8%

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

   8. Taylor expanded in i around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

   10. Simplified 99.9%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-191}:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-191}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-42}:\\ \;\;\;\;n \cdot \frac{-100 + t_0 \cdot 100}{i}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-191}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\frac{t_0}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-42)
     (* n (/ (+ -100.0 (* t_0 100.0)) i))
     (if (<= t_1 5e-191)
       (* n (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) i)))
       (if (<= t_1 INFINITY)
         (* n (* 100.0 (+ (/ t_0 i) (/ -1.0 i))))
         (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-42) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else if (t_1 <= 5e-191) {
		tmp = n * (100.0 * (expm1((n * log1p((i / n)))) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (100.0 * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-42) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else if (t_1 <= 5e-191) {
		tmp = n * (100.0 * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (100.0 * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -5e-42:
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i)
	elif t_1 <= 5e-191:
		tmp = n * (100.0 * (math.expm1((n * math.log1p((i / n)))) / i))
	elif t_1 <= math.inf:
		tmp = n * (100.0 * ((t_0 / i) + (-1.0 / i)))
	else:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-42)
		tmp = Float64(n * Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / i));
	elseif (t_1 <= 5e-191)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(100.0 * Float64(Float64(t_0 / i) + Float64(-1.0 / i))));
	else
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-42], N[(n * N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-191], N[(n * N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(100.0 * N[(N[(t$95$0 / i), $MachinePrecision] + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(1.0 / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -5.00000000000000003e-42

   1. Initial program 99.7%

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

      1. associate-/r/ 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. fma-def 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -5.00000000000000003e-42 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 5.0000000000000001e-191

   1. Initial program 16.7%

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

      1. associate-/r/ 16.6%

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

      2. associate-*r* 16.6%

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

      3. *-commutative 16.6%

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

      4. associate-*r/ 16.6%

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

      5. sub-neg 16.6%

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

      6. distribute-lft-in 16.6%

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

      7. fma-def 16.6%

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

      8. metadata-eval 16.6%

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

      9. metadata-eval 16.6%

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

   3. Simplified 16.6%

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

      1. clear-num 16.6%

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

      2. inv-pow 16.6%

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

      3. fma-udef 16.6%

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

      4. metadata-eval 16.6%

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

      5. metadata-eval 16.6%

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

      6. distribute-lft-in 16.6%

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

      7. sub-neg 16.6%

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

      8. *-commutative 16.6%

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

      9. pow-to-exp 16.6%

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

      10. expm1-def 32.8%

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

      11. add-log-exp 16.6%

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

      12. pow-to-exp 16.6%

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

      13. log-pow 32.8%

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

      14. log1p-udef 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. unpow-1 98.1%

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

      2. *-commutative 98.1%

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

   7. Simplified 98.1%

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

      1. clear-num 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. times-frac 98.2%

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

      4. metadata-eval 98.2%

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

   9. Applied egg-rr 98.2%

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

   if 5.0000000000000001e-191 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.3%

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

      1. associate-/r/ 98.2%

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

      2. associate-*r* 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*r/ 98.3%

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

      5. sub-neg 98.3%

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

      6. distribute-lft-in 98.2%

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

      7. fma-def 98.3%

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

      8. metadata-eval 98.3%

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

      9. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. fma-udef 98.2%

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

      4. metadata-eval 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-lft-in 98.3%

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

      7. sub-neg 98.3%

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

      8. *-commutative 98.3%

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

      9. pow-to-exp 66.8%

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

      10. expm1-def 68.4%

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

      11. add-log-exp 66.8%

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

      12. pow-to-exp 98.3%

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

      13. log-pow 68.4%

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

      14. log1p-udef 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. unpow-1 68.4%

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

      2. *-commutative 68.4%

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

   7. Simplified 68.4%

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

      1. clear-num 68.3%

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

      2. *-un-lft-identity 68.3%

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

      3. times-frac 68.2%

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

      4. metadata-eval 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. expm1-udef 66.8%

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

       2. div-sub 66.9%

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

       3. *-commutative 66.9%

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

       4. log1p-udef 66.9%

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

       5. exp-to-pow 98.4%

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

   11. Applied egg-rr 98.4%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. associate-*r* 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r/ 1.8%

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

      5. sub-neg 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. fma-def 1.8%

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

      8. metadata-eval 1.8%

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

      9. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. inv-pow 1.8%

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

      3. fma-udef 1.8%

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

      4. metadata-eval 1.8%

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

      5. metadata-eval 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. sub-neg 1.8%

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

      8. *-commutative 1.8%

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

      9. pow-to-exp 1.8%

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

      10. expm1-def 1.8%

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

      11. add-log-exp 1.8%

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

      12. pow-to-exp 1.8%

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

      13. log-pow 1.8%

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

      14. log1p-udef 1.8%

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

   5. Applied egg-rr 1.8%

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

      1. unpow-1 1.8%

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

      2. *-commutative 1.8%

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

   7. Simplified 1.8%

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

   8. Taylor expanded in i around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

   10. Simplified 99.9%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-42}:\\ \;\;\;\;n \cdot \frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-191}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 3: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot t_1\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-191}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\frac{t_0}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 (- INFINITY))
     (* 100.0 (* (/ n i) t_1))
     (if (<= t_2 5e-191)
       (* n (/ 100.0 (/ i (expm1 (* n (log1p (/ i n)))))))
       (if (<= t_2 INFINITY)
         (* n (* 100.0 (+ (/ t_0 i) (/ -1.0 i))))
         (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 100.0 * ((n / i) * t_1);
	} else if (t_2 <= 5e-191) {
		tmp = n * (100.0 / (i / expm1((n * log1p((i / n))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = n * (100.0 * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((n / i) * t_1);
	} else if (t_2 <= 5e-191) {
		tmp = n * (100.0 / (i / Math.expm1((n * Math.log1p((i / n))))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = n * (100.0 * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = t_0 + -1.0
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = 100.0 * ((n / i) * t_1)
	elif t_2 <= 5e-191:
		tmp = n * (100.0 / (i / math.expm1((n * math.log1p((i / n))))))
	elif t_2 <= math.inf:
		tmp = n * (100.0 * ((t_0 / i) + (-1.0 / i)))
	else:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(100.0 * Float64(Float64(n / i) * t_1));
	elseif (t_2 <= 5e-191)
		tmp = Float64(n * Float64(100.0 / Float64(i / expm1(Float64(n * log1p(Float64(i / n)))))));
	elseif (t_2 <= Inf)
		tmp = Float64(n * Float64(100.0 * Float64(Float64(t_0 / i) + Float64(-1.0 / i))));
	else
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-191], N[(n * N[(100.0 / N[(i / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(n * N[(100.0 * N[(N[(t$95$0 / i), $MachinePrecision] + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(1.0 / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. clear-num 100.0%

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

      3. sub-neg 100.0%

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

      4. div-inv 100.0%

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

      5. clear-num 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-rgt-out 100.0%

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

   5. Simplified 100.0%

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

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 5.0000000000000001e-191

   1. Initial program 18.1%

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

      1. associate-/r/ 17.9%

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

      2. associate-*r* 18.0%

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

      3. *-commutative 18.0%

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

      4. associate-*r/ 18.0%

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

      5. sub-neg 18.0%

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

      6. distribute-lft-in 18.0%

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

      7. fma-def 18.0%

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

      8. metadata-eval 18.0%

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

      9. metadata-eval 18.0%

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

   3. Simplified 18.0%

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

      1. clear-num 18.0%

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

      2. inv-pow 18.0%

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

      3. fma-udef 18.0%

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

      4. metadata-eval 18.0%

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

      5. metadata-eval 18.0%

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

      6. distribute-lft-in 18.0%

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

      7. sub-neg 18.0%

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

      8. *-commutative 18.0%

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

      9. pow-to-exp 18.0%

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

      10. expm1-def 33.9%

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

      11. add-log-exp 18.0%

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

      12. pow-to-exp 18.0%

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

      13. log-pow 33.9%

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

      14. log1p-udef 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. unpow-1 98.1%

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

      2. *-commutative 98.1%

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

   7. Simplified 98.1%

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

      1. clear-num 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. times-frac 98.2%

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

      4. metadata-eval 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. clear-num 98.1%

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

       2. un-div-inv 98.2%

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

   11. Applied egg-rr 98.2%

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

   if 5.0000000000000001e-191 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.3%

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

      1. associate-/r/ 98.2%

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

      2. associate-*r* 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*r/ 98.3%

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

      5. sub-neg 98.3%

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

      6. distribute-lft-in 98.2%

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

      7. fma-def 98.3%

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

      8. metadata-eval 98.3%

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

      9. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. fma-udef 98.2%

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

      4. metadata-eval 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-lft-in 98.3%

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

      7. sub-neg 98.3%

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

      8. *-commutative 98.3%

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

      9. pow-to-exp 66.8%

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

      10. expm1-def 68.4%

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

      11. add-log-exp 66.8%

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

      12. pow-to-exp 98.3%

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

      13. log-pow 68.4%

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

      14. log1p-udef 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. unpow-1 68.4%

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

      2. *-commutative 68.4%

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

   7. Simplified 68.4%

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

      1. clear-num 68.3%

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

      2. *-un-lft-identity 68.3%

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

      3. times-frac 68.2%

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

      4. metadata-eval 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. expm1-udef 66.8%

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

       2. div-sub 66.9%

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

       3. *-commutative 66.9%

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

       4. log1p-udef 66.9%

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

       5. exp-to-pow 98.4%

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

   11. Applied egg-rr 98.4%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. associate-*r* 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r/ 1.8%

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

      5. sub-neg 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. fma-def 1.8%

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

      8. metadata-eval 1.8%

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

      9. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. inv-pow 1.8%

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

      3. fma-udef 1.8%

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

      4. metadata-eval 1.8%

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

      5. metadata-eval 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. sub-neg 1.8%

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

      8. *-commutative 1.8%

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

      9. pow-to-exp 1.8%

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

      10. expm1-def 1.8%

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

      11. add-log-exp 1.8%

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

      12. pow-to-exp 1.8%

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

      13. log-pow 1.8%

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

      14. log1p-udef 1.8%

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

   5. Applied egg-rr 1.8%

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

      1. unpow-1 1.8%

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

      2. *-commutative 1.8%

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

   7. Simplified 1.8%

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

   8. Taylor expanded in i around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

   10. Simplified 99.9%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-191}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 4: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.75 \cdot 10^{+15} \lor \neg \left(i \leq 1.35 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.75e+15) (not (<= i 1.35e-5)))
   (* (expm1 i) (/ (* n 100.0) i))
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.75e+15) || !(i <= 1.35e-5)) {
		tmp = expm1(i) * ((n * 100.0) / i);
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.75e+15) || !(i <= 1.35e-5)) {
		tmp = Math.expm1(i) * ((n * 100.0) / i);
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.75e+15) or not (i <= 1.35e-5):
		tmp = math.expm1(i) * ((n * 100.0) / i)
	else:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.75e+15) || !(i <= 1.35e-5))
		tmp = Float64(expm1(i) * Float64(Float64(n * 100.0) / i));
	else
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.75e+15], N[Not[LessEqual[i, 1.35e-5]], $MachinePrecision]], N[(N[(Exp[i] - 1), $MachinePrecision] * N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(1.0 / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.75e15 or 1.3499999999999999e-5 < i

   1. Initial program 47.6%

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

      1. associate-*r/ 47.7%

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

      2. sub-neg 47.7%

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

      3. distribute-lft-in 47.7%

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

      4. metadata-eval 47.7%

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

      5. metadata-eval 47.7%

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

      6. fma-udef 47.7%

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

      7. associate-/r/ 47.6%

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

      8. *-commutative 47.6%

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

      9. expm1-log1p-u 37.9%

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

      10. expm1-udef 27.8%

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

   3. Applied egg-rr 44.3%

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

      1. expm1-def 62.1%

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

      2. expm1-log1p 77.4%

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

      3. associate-*l/ 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in n around inf 57.3%

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

   if -1.75e15 < i < 1.3499999999999999e-5

   1. Initial program 7.6%

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

      1. associate-/r/ 8.0%

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

      2. associate-*r* 8.0%

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

      3. *-commutative 8.0%

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

      4. associate-*r/ 8.0%

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

      5. sub-neg 8.0%

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

      6. distribute-lft-in 8.0%

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

      7. fma-def 8.0%

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

      8. metadata-eval 8.0%

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

      9. metadata-eval 8.0%

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

   3. Simplified 8.0%

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

      1. clear-num 8.0%

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

      2. inv-pow 8.0%

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

      3. fma-udef 8.0%

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

      4. metadata-eval 8.0%

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

      5. metadata-eval 8.0%

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

      6. distribute-lft-in 8.0%

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

      7. sub-neg 8.0%

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

      8. *-commutative 8.0%

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

      9. pow-to-exp 8.0%

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

      10. expm1-def 18.5%

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

      11. add-log-exp 8.0%

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

      12. pow-to-exp 8.0%

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

      13. log-pow 18.5%

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

      14. log1p-udef 76.1%

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

   5. Applied egg-rr 76.1%

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

      1. unpow-1 76.1%

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

      2. *-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in i around 0 90.7%

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

      1. associate-*r* 90.7%

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

      2. *-commutative 90.7%

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

      3. sub-neg 90.7%

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

      4. associate-*r/ 90.7%

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

      5. metadata-eval 90.7%

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

      6. metadata-eval 90.7%

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

   10. Simplified 90.7%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.75 \cdot 10^{+15} \lor \neg \left(i \leq 1.35 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 5: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.16 \cdot 10^{+64} \lor \neg \left(n \leq 1.55\right):\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.16e+64) (not (<= n 1.55)))
   (* 100.0 (/ (* n (expm1 i)) i))
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.16e+64) || !(n <= 1.55)) {
		tmp = 100.0 * ((n * expm1(i)) / i);
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.16e+64) || !(n <= 1.55)) {
		tmp = 100.0 * ((n * Math.expm1(i)) / i);
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.16e+64) or not (n <= 1.55):
		tmp = 100.0 * ((n * math.expm1(i)) / i)
	else:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.16e+64) || !(n <= 1.55))
		tmp = Float64(100.0 * Float64(Float64(n * expm1(i)) / i));
	else
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.16e+64], N[Not[LessEqual[n, 1.55]], $MachinePrecision]], N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(1.0 / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.16e64 or 1.55000000000000004 < n

   1. Initial program 22.8%

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

   2. Taylor expanded in n around inf 36.7%

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

      1. *-commutative 36.7%

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

      2. expm1-def 90.4%

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

   4. Simplified 90.4%

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

   if -1.16e64 < n < 1.55000000000000004

   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/ 25.2%

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

      2. associate-*r* 25.2%

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

      3. *-commutative 25.2%

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

      4. associate-*r/ 25.2%

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

      5. sub-neg 25.2%

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

      6. distribute-lft-in 25.2%

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

      7. fma-def 25.2%

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

      8. metadata-eval 25.2%

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

      9. metadata-eval 25.2%

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

   3. Simplified 25.2%

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

      1. clear-num 25.2%

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

      2. inv-pow 25.2%

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

      3. fma-udef 25.2%

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

      4. metadata-eval 25.2%

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

      5. metadata-eval 25.2%

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

      6. distribute-lft-in 25.2%

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

      7. sub-neg 25.2%

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

      8. *-commutative 25.2%

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

      9. pow-to-exp 25.2%

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

      10. expm1-def 52.8%

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

      11. add-log-exp 25.2%

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

      12. pow-to-exp 25.2%

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

      13. log-pow 52.8%

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

      14. log1p-udef 91.2%

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

   5. Applied egg-rr 91.2%

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

      1. unpow-1 91.2%

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

      2. *-commutative 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in i around 0 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

      3. sub-neg 74.1%

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

      4. associate-*r/ 74.1%

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

      5. metadata-eval 74.1%

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

      6. metadata-eval 74.1%

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

   10. Simplified 74.1%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.16 \cdot 10^{+64} \lor \neg \left(n \leq 1.55\right):\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 6: 71.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.095:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 0.095)
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))
   (* n (/ (* 100.0 (+ i (* (* i i) (- 0.5 (/ 0.5 n))))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 0.095) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.095d0) then
        tmp = n * (1.0d0 / (0.01d0 + ((i * 0.01d0) * ((0.5d0 / n) + (-0.5d0)))))
    else
        tmp = n * ((100.0d0 * (i + ((i * i) * (0.5d0 - (0.5d0 / n))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 0.095) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 0.095:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	else:
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 0.095)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(n * Float64(Float64(100.0 * Float64(i + Float64(Float64(i * i) * Float64(0.5 - Float64(0.5 / n))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 0.095)
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	else
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 0.095], N[(n * N[(1.0 / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(i + N[(N[(i * i), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.095000000000000001

   1. Initial program 24.5%

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

      1. associate-/r/ 24.6%

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

      2. associate-*r* 24.6%

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

      3. *-commutative 24.6%

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

      4. associate-*r/ 24.6%

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

      5. sub-neg 24.6%

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

      6. distribute-lft-in 24.6%

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

      7. fma-def 24.6%

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

      8. metadata-eval 24.6%

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

      9. metadata-eval 24.6%

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

   3. Simplified 24.6%

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

      1. clear-num 24.6%

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

      2. inv-pow 24.6%

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

      3. fma-udef 24.6%

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

      4. metadata-eval 24.6%

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

      5. metadata-eval 24.6%

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

      6. distribute-lft-in 24.6%

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

      7. sub-neg 24.6%

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

      8. *-commutative 24.6%

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

      9. pow-to-exp 20.9%

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

      10. expm1-def 39.2%

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

      11. add-log-exp 20.9%

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

      12. pow-to-exp 24.6%

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

      13. log-pow 39.2%

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

      14. log1p-udef 80.7%

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

   5. Applied egg-rr 80.7%

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

      1. unpow-1 80.7%

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

      2. *-commutative 80.7%

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

   7. Simplified 80.7%

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

   8. Taylor expanded in i around 0 69.3%

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

      1. associate-*r* 69.3%

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

      2. *-commutative 69.3%

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

      3. sub-neg 69.3%

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

      4. associate-*r/ 69.3%

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

      5. metadata-eval 69.3%

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

      6. metadata-eval 69.3%

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

   10. Simplified 69.3%

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

   if 0.095000000000000001 < n

   1. Initial program 22.9%

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

      1. associate-/r/ 23.2%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.2%

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

      5. sub-neg 23.2%

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

      6. distribute-lft-in 23.2%

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

      7. fma-def 23.2%

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

      8. metadata-eval 23.2%

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

      9. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in i around 0 78.9%

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

      1. distribute-lft-out 78.9%

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

      2. unpow2 78.9%

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

      3. associate-*r/ 78.9%

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

      4. metadata-eval 78.9%

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

   6. Simplified 78.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 0.095:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \]

Alternative 7: 61.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.64:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i + 0.5 \cdot \left(i \cdot i\right)}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.64)
   0.0
   (if (<= i 3.2e-21)
     (* 100.0 (+ n (* (- 0.5 (/ 0.5 n)) (* i n))))
     (* 100.0 (/ (+ i (* 0.5 (* i i))) (/ i n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.64) {
		tmp = 0.0;
	} else if (i <= 3.2e-21) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else {
		tmp = 100.0 * ((i + (0.5 * (i * 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 (i <= (-0.64d0)) then
        tmp = 0.0d0
    else if (i <= 3.2d-21) then
        tmp = 100.0d0 * (n + ((0.5d0 - (0.5d0 / n)) * (i * n)))
    else
        tmp = 100.0d0 * ((i + (0.5d0 * (i * i))) / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.64) {
		tmp = 0.0;
	} else if (i <= 3.2e-21) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else {
		tmp = 100.0 * ((i + (0.5 * (i * i))) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.64:
		tmp = 0.0
	elif i <= 3.2e-21:
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)))
	else:
		tmp = 100.0 * ((i + (0.5 * (i * i))) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.64)
		tmp = 0.0;
	elseif (i <= 3.2e-21)
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * n))));
	else
		tmp = Float64(100.0 * Float64(Float64(i + Float64(0.5 * Float64(i * i))) / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.64)
		tmp = 0.0;
	elseif (i <= 3.2e-21)
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	else
		tmp = 100.0 * ((i + (0.5 * (i * i))) / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.64], 0.0, If[LessEqual[i, 3.2e-21], N[(100.0 * N[(n + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(i + N[(0.5 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -0.640000000000000013

   1. Initial program 58.7%

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

   2. Taylor expanded in i around 0 31.0%

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

   3. Taylor expanded in i around 0 31.0%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if -0.640000000000000013 < i < 3.2000000000000002e-21

   1. Initial program 6.5%

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

   2. Taylor expanded in i around 0 87.2%

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

      1. associate-*r* 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*r/ 87.5%

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

      4. metadata-eval 87.5%

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

   4. Simplified 87.5%

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

   if 3.2000000000000002e-21 < i

   1. Initial program 37.0%

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

      1. associate-/r/ 37.2%

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

      2. associate-*r* 37.2%

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

      3. *-commutative 37.2%

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

      4. associate-*r/ 37.2%

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

      5. sub-neg 37.2%

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

      6. distribute-lft-in 37.2%

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

      7. fma-def 37.2%

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

      8. metadata-eval 37.2%

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

      9. metadata-eval 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in i around 0 35.5%

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

      1. distribute-lft-out 35.5%

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

      2. unpow2 35.5%

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

      3. associate-*r/ 35.5%

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

      4. metadata-eval 35.5%

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

   6. Simplified 35.5%

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

   7. Taylor expanded in n around inf 37.5%

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

      1. associate-/l* 40.5%

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

      2. *-commutative 40.5%

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

      3. unpow2 40.5%

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

   9. Simplified 40.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.64:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i + 0.5 \cdot \left(i \cdot i\right)}{\frac{i}{n}}\\ \end{array} \]

Alternative 8: 70.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.095:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 0.095)
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))
   (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 0.095) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.095d0) then
        tmp = n * (1.0d0 / (0.01d0 + ((i * 0.01d0) * ((0.5d0 / n) + (-0.5d0)))))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 0.095) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 0.095:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 0.095)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 0.095)
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 0.095], N[(n * N[(1.0 / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.095000000000000001

   1. Initial program 24.5%

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

      1. associate-/r/ 24.6%

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

      2. associate-*r* 24.6%

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

      3. *-commutative 24.6%

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

      4. associate-*r/ 24.6%

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

      5. sub-neg 24.6%

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

      6. distribute-lft-in 24.6%

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

      7. fma-def 24.6%

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

      8. metadata-eval 24.6%

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

      9. metadata-eval 24.6%

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

   3. Simplified 24.6%

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

      1. clear-num 24.6%

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

      2. inv-pow 24.6%

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

      3. fma-udef 24.6%

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

      4. metadata-eval 24.6%

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

      5. metadata-eval 24.6%

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

      6. distribute-lft-in 24.6%

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

      7. sub-neg 24.6%

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

      8. *-commutative 24.6%

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

      9. pow-to-exp 20.9%

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

      10. expm1-def 39.2%

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

      11. add-log-exp 20.9%

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

      12. pow-to-exp 24.6%

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

      13. log-pow 39.2%

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

      14. log1p-udef 80.7%

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

   5. Applied egg-rr 80.7%

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

      1. unpow-1 80.7%

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

      2. *-commutative 80.7%

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

   7. Simplified 80.7%

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

   8. Taylor expanded in i around 0 69.3%

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

      1. associate-*r* 69.3%

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

      2. *-commutative 69.3%

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

      3. sub-neg 69.3%

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

      4. associate-*r/ 69.3%

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

      5. metadata-eval 69.3%

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

      6. metadata-eval 69.3%

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

   10. Simplified 69.3%

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

   if 0.095000000000000001 < n

   1. Initial program 22.9%

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

      1. associate-/r/ 23.2%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.2%

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

      5. sub-neg 23.2%

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

      6. distribute-lft-in 23.2%

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

      7. fma-def 23.2%

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

      8. metadata-eval 23.2%

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

      9. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in i around 0 74.0%

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

      1. associate-*r* 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-*r/ 74.0%

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

      4. metadata-eval 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in n around inf 74.0%

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

      1. *-commutative 74.0%

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

   9. Simplified 74.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 0.095:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 9: 59.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.82:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.82) 0.0 (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.82) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.82d0)) then
        tmp = 0.0d0
    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 (i <= -1.82) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.82:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.82)
		tmp = 0.0;
	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 (i <= -1.82)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.82], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.82000000000000006

   1. Initial program 58.7%

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

   2. Taylor expanded in i around 0 31.0%

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

   3. Taylor expanded in i around 0 31.0%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if -1.82000000000000006 < i

   1. Initial program 15.6%

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

      1. associate-/r/ 16.0%

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

      2. associate-*r* 16.0%

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

      3. *-commutative 16.0%

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

      4. associate-*r/ 16.0%

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

      5. sub-neg 16.0%

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

      6. distribute-lft-in 16.0%

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

      7. fma-def 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in i around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-*r/ 70.8%

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

      4. metadata-eval 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in n around inf 71.2%

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

      1. *-commutative 71.2%

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

   9. Simplified 71.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.82:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 10: 58.5% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -750000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1700000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -750000.0) 0.0 (if (<= i 1700000.0) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -750000.0) {
		tmp = 0.0;
	} else if (i <= 1700000.0) {
		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 <= (-750000.0d0)) then
        tmp = 0.0d0
    else if (i <= 1700000.0d0) 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 <= -750000.0) {
		tmp = 0.0;
	} else if (i <= 1700000.0) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -750000.0:
		tmp = 0.0
	elif i <= 1700000.0:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -750000.0)
		tmp = 0.0;
	elseif (i <= 1700000.0)
		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 <= -750000.0)
		tmp = 0.0;
	elseif (i <= 1700000.0)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -750000.0], 0.0, If[LessEqual[i, 1700000.0], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -7.5e5 or 1.7e6 < i

   1. Initial program 49.1%

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

   2. Taylor expanded in i around 0 26.1%

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

   3. Taylor expanded in i around 0 26.1%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if -7.5e5 < i < 1.7e6

   1. Initial program 6.3%

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

   2. Taylor expanded in i around 0 85.0%

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

      1. *-commutative 85.0%

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

   4. Simplified 85.0%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -750000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1700000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 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 23.9%

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

2. Taylor expanded in i around 0 14.5%

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

3. Taylor expanded in i around 0 14.7%

   \[\leadsto 100 \cdot \color{blue}{0} \]

4. Final simplification 14.7%

   \[\leadsto 0 \]

Developer target: 35.2% 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 2023192 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))