Compound Interest

Percentage Accurate: 28.6% → 98.8%

Time: 21.5s

Alternatives: 16

Speedup: 8.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.6% 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: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left(t\_0, 100, -100\right)}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, t\_0, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

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 -4e-153)
     (/ (* n (fma t_0 100.0 -100.0)) i)
     (if (<= t_1 0.0)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (* n (/ (fma 100.0 t_0 -100.0) 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 <= -4e-153) {
		tmp = (n * fma(t_0, 100.0, -100.0)) / i;
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (fma(100.0, t_0, -100.0) / i);
	} else {
		tmp = n * (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 <= -4e-153)
		tmp = Float64(Float64(n * fma(t_0, 100.0, -100.0)) / i);
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(fma(100.0, t_0, -100.0) / i));
	else
		tmp = Float64(n * Float64(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, -4e-153], N[(N[(n * N[(t$95$0 * 100.0 + -100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(100.0 * t$95$0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(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 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -4.00000000000000016e-153

   1. Initial program 99.7%

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

      1. associate-*r/ 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-/r/ 99.8%

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

      7. *-commutative 99.8%

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

      8. fma-undefine 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*r/ 99.9%

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

      11. fma-undefine 99.9%

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

      12. *-commutative 99.9%

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

      13. fma-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -4.00000000000000016e-153 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 19.6%

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

      1. associate-/r/ 19.6%

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

      2. associate-*r* 19.6%

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

      3. *-commutative 19.6%

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

      4. associate-*r/ 19.6%

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

      5. sub-neg 19.6%

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

      6. distribute-lft-in 19.6%

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

      7. metadata-eval 19.6%

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

      8. metadata-eval 19.6%

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

      9. metadata-eval 19.6%

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

      10. fma-define 19.6%

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

      11. metadata-eval 19.6%

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

   3. Simplified 19.6%

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

      1. fma-undefine 19.6%

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

      2. metadata-eval 19.6%

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

      3. metadata-eval 19.6%

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

      4. distribute-lft-in 19.6%

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

      5. sub-neg 19.6%

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

      6. *-commutative 19.6%

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

      7. add-exp-log 19.6%

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

      8. expm1-define 19.6%

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

      9. log-pow 30.2%

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

      10. log1p-define 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.5%

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

      1. associate-/r/ 99.8%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      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 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. fma-define 99.9%

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

      11. 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. Add Preprocessing

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-undefine 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. sub-neg 1.9%

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

      6. *-commutative 1.9%

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

      7. add-exp-log 1.9%

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

      8. expm1-define 1.9%

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

      9. log-pow 1.9%

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

      10. log1p-define 1.9%

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

   6. Applied egg-rr 1.9%

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

      1. clear-num 1.9%

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

      2. inv-pow 1.9%

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

   8. Applied egg-rr 1.9%

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

      1. unpow-1 1.9%

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

      2. *-commutative 1.9%

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

   10. Simplified 1.9%

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

   11. Taylor expanded in i around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. *-commutative 100.0%

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

       3. sub-neg 100.0%

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

       4. associate-*r/ 100.0%

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

       5. metadata-eval 100.0%

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

       6. metadata-eval 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, t\_0, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

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 (- INFINITY))
     (*
      (/ 100.0 i)
      (*
       i
       (+
        n
        (*
         i
         (+
          (* n 0.5)
          (*
           i
           (+ (* 0.041666666666666664 (* i n)) (* n 0.16666666666666666))))))))
     (if (<= t_1 0.0)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (* n (/ (fma 100.0 t_0 -100.0) 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 <= -((double) INFINITY)) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (fma(100.0, t_0, -100.0) / i);
	} else {
		tmp = n * (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 <= Float64(-Inf))
		tmp = Float64(Float64(100.0 / i) * Float64(i * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666))))))));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(fma(100.0, t_0, -100.0) / i));
	else
		tmp = Float64(n * Float64(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, (-Infinity)], N[(N[(100.0 / i), $MachinePrecision] * N[(i * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(100.0 * t$95$0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(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 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.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. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in n around inf 44.9%

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

   6. Taylor expanded in i around inf 44.9%

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

      1. sub-neg 44.9%

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

      2. metadata-eval 44.9%

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

      3. metadata-eval 44.9%

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

      4. distribute-lft-in 44.9%

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

      5. metadata-eval 44.9%

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

      6. sub-neg 44.9%

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

      7. expm1-define 44.9%

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

   8. Simplified 44.9%

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

      1. div-inv 44.9%

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

      2. times-frac 44.9%

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

   10. Applied egg-rr 44.9%

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

   11. Taylor expanded in i around 0 100.0%

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(i \cdot \left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)\right)} \]

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 22.2%

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

      1. associate-/r/ 22.3%

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

      2. associate-*r* 22.3%

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

      3. *-commutative 22.3%

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

      4. associate-*r/ 22.3%

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

      5. sub-neg 22.3%

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

      6. distribute-lft-in 22.2%

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

      7. metadata-eval 22.2%

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

      8. metadata-eval 22.2%

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

      9. metadata-eval 22.2%

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

      10. fma-define 22.3%

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

      11. metadata-eval 22.3%

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

   3. Simplified 22.3%

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

      1. fma-undefine 22.2%

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

      2. metadata-eval 22.2%

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

      3. metadata-eval 22.2%

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

      4. distribute-lft-in 22.3%

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

      5. sub-neg 22.3%

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

      6. *-commutative 22.3%

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

      7. add-exp-log 22.3%

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

      8. expm1-define 22.3%

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

      9. log-pow 32.5%

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

      10. log1p-define 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.5%

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

      1. associate-/r/ 99.8%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      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 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. fma-define 99.9%

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

      11. 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. Add Preprocessing

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-undefine 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. sub-neg 1.9%

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

      6. *-commutative 1.9%

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

      7. add-exp-log 1.9%

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

      8. expm1-define 1.9%

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

      9. log-pow 1.9%

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

      10. log1p-define 1.9%

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

   6. Applied egg-rr 1.9%

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

      1. clear-num 1.9%

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

      2. inv-pow 1.9%

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

   8. Applied egg-rr 1.9%

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

      1. unpow-1 1.9%

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

      2. *-commutative 1.9%

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

   10. Simplified 1.9%

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

   11. Taylor expanded in i around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. *-commutative 100.0%

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

       3. sub-neg 100.0%

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

       4. associate-*r/ 100.0%

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

       5. metadata-eval 100.0%

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

       6. metadata-eval 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{t\_0}{i} + \frac{-1}{i}\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

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 (- INFINITY))
     (*
      (/ 100.0 i)
      (*
       i
       (+
        n
        (*
         i
         (+
          (* n 0.5)
          (*
           i
           (+ (* 0.041666666666666664 (* i n)) (* n 0.16666666666666666))))))))
     (if (<= t_1 0.0)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (* (+ (/ t_0 i) (/ -1.0 i)) (* n 100.0))
         (* 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 <= -((double) INFINITY)) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 / i) + (-1.0 / i)) * (n * 100.0);
	} 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 <= -Double.POSITIVE_INFINITY) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * Math.expm1((n * Math.log1p((i / n))))) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 / i) + (-1.0 / i)) * (n * 100.0);
	} 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 <= -math.inf:
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))))
	elif t_1 <= 0.0:
		tmp = n * ((100.0 * math.expm1((n * math.log1p((i / n))))) / i)
	elif t_1 <= math.inf:
		tmp = ((t_0 / i) + (-1.0 / i)) * (n * 100.0)
	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 <= Float64(-Inf))
		tmp = Float64(Float64(100.0 / i) * Float64(i * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666))))))));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 / i) + Float64(-1.0 / i)) * Float64(n * 100.0));
	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, (-Infinity)], N[(N[(100.0 / i), $MachinePrecision] * N[(i * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 / i), $MachinePrecision] + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision] * N[(n * 100.0), $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 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.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. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in n around inf 44.9%

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

   6. Taylor expanded in i around inf 44.9%

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

      1. sub-neg 44.9%

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

      2. metadata-eval 44.9%

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

      3. metadata-eval 44.9%

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

      4. distribute-lft-in 44.9%

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

      5. metadata-eval 44.9%

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

      6. sub-neg 44.9%

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

      7. expm1-define 44.9%

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

   8. Simplified 44.9%

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

      1. div-inv 44.9%

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

      2. times-frac 44.9%

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

   10. Applied egg-rr 44.9%

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

   11. Taylor expanded in i around 0 100.0%

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(i \cdot \left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)\right)} \]

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 22.2%

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

      1. associate-/r/ 22.3%

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

      2. associate-*r* 22.3%

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

      3. *-commutative 22.3%

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

      4. associate-*r/ 22.3%

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

      5. sub-neg 22.3%

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

      6. distribute-lft-in 22.2%

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

      7. metadata-eval 22.2%

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

      8. metadata-eval 22.2%

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

      9. metadata-eval 22.2%

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

      10. fma-define 22.3%

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

      11. metadata-eval 22.3%

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

   3. Simplified 22.3%

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

      1. fma-undefine 22.2%

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

      2. metadata-eval 22.2%

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

      3. metadata-eval 22.2%

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

      4. distribute-lft-in 22.3%

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

      5. sub-neg 22.3%

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

      6. *-commutative 22.3%

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

      7. add-exp-log 22.3%

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

      8. expm1-define 22.3%

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

      9. log-pow 32.5%

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

      10. log1p-define 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.5%

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

      1. associate-*r/ 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-rgt-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. metadata-eval 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-rgt-in 99.6%

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

      4. sub-neg 99.6%

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

      5. associate-*r/ 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-/r/ 99.8%

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

      8. associate-*l* 99.8%

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

      9. add-exp-log 99.8%

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

      10. expm1-define 99.8%

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

      11. log-pow 60.0%

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

      12. log1p-define 60.0%

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

   6. Applied egg-rr 60.0%

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

      1. expm1-undefine 59.9%

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

      2. div-sub 59.9%

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

      3. *-commutative 59.9%

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

      4. log1p-undefine 59.9%

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

      5. exp-to-pow 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. fma-undefine 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. sub-neg 1.9%

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

      6. *-commutative 1.9%

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

      7. add-exp-log 1.9%

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

      8. expm1-define 1.9%

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

      9. log-pow 1.9%

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

      10. log1p-define 1.9%

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

   6. Applied egg-rr 1.9%

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

      1. clear-num 1.9%

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

      2. inv-pow 1.9%

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

   8. Applied egg-rr 1.9%

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

      1. unpow-1 1.9%

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

      2. *-commutative 1.9%

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

   10. Simplified 1.9%

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

   11. Taylor expanded in i around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. *-commutative 100.0%

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

       3. sub-neg 100.0%

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

       4. associate-*r/ 100.0%

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

       5. metadata-eval 100.0%

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

       6. metadata-eval 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -600000000 \lor \neg \left(n \leq 1\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -600000000.0) (not (<= n 1.0)))
   (* n (* 100.0 (/ (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 <= -600000000.0) || !(n <= 1.0)) {
		tmp = n * (100.0 * (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 <= -600000000.0) || !(n <= 1.0)) {
		tmp = n * (100.0 * (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 <= -600000000.0) or not (n <= 1.0):
		tmp = n * (100.0 * (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 <= -600000000.0) || !(n <= 1.0))
		tmp = Float64(n * Float64(100.0 * Float64(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, -600000000.0], N[Not[LessEqual[n, 1.0]], $MachinePrecision]], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(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 < -6e8 or 1 < n

   1. Initial program 25.1%

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

      1. associate-/r/ 25.7%

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

      2. associate-*r* 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-*r/ 25.7%

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

      5. sub-neg 25.7%

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

      6. distribute-lft-in 25.7%

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

      7. metadata-eval 25.7%

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

      8. metadata-eval 25.7%

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

      9. metadata-eval 25.7%

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

      10. fma-define 25.7%

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

      11. metadata-eval 25.7%

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

   3. Simplified 25.7%

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

   5. Taylor expanded in n around inf 35.7%

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

      1. div-sub 36.1%

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

      2. associate-*r/ 36.2%

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

      3. metadata-eval 36.2%

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

      4. associate-*r/ 36.1%

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

      5. distribute-lft-out-- 36.1%

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

      6. div-sub 35.7%

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

      7. *-commutative 35.7%

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

      8. expm1-define 91.2%

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

   7. Simplified 91.2%

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

   if -6e8 < n < 1

   1. Initial program 29.4%

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

      1. associate-/r/ 29.5%

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

      2. associate-*r* 29.5%

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

      3. *-commutative 29.5%

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

      4. associate-*r/ 29.5%

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

      5. sub-neg 29.5%

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

      6. distribute-lft-in 29.4%

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

      7. metadata-eval 29.4%

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

      8. metadata-eval 29.4%

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

      9. metadata-eval 29.4%

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

      10. fma-define 29.5%

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

      11. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

      1. fma-undefine 29.4%

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

      2. metadata-eval 29.4%

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

      3. metadata-eval 29.4%

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

      4. distribute-lft-in 29.5%

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

      5. sub-neg 29.5%

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

      6. *-commutative 29.5%

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

      7. add-exp-log 29.5%

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

      8. expm1-define 29.5%

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

      9. log-pow 46.8%

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

      10. log1p-define 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. clear-num 94.2%

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

      2. inv-pow 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. unpow-1 94.2%

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

      2. *-commutative 94.2%

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

   10. Simplified 94.2%

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

   11. Taylor expanded in i around 0 84.3%

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

       1. associate-*r* 84.3%

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

       2. *-commutative 84.3%

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

       3. sub-neg 84.3%

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

       4. associate-*r/ 84.3%

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

       5. metadata-eval 84.3%

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

       6. metadata-eval 84.3%

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

   13. Simplified 84.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -600000000 \lor \neg \left(n \leq 1\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -600000000:\\ \;\;\;\;n \cdot \left(100 \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -600000000.0)
     (* n (* 100.0 t_0))
     (if (<= n 1.0)
       (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))
       (* 100.0 (* n t_0))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -600000000.0) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 1.0) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = 100.0 * (n * t_0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -600000000.0) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 1.0) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = 100.0 * (n * t_0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -600000000.0:
		tmp = n * (100.0 * t_0)
	elif n <= 1.0:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	else:
		tmp = 100.0 * (n * t_0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -600000000.0)
		tmp = Float64(n * Float64(100.0 * t_0));
	elseif (n <= 1.0)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(100.0 * Float64(n * t_0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -600000000.0], N[(n * N[(100.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.0], 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[(100.0 * N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6e8

   1. Initial program 30.7%

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

      1. associate-/r/ 31.2%

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

      2. associate-*r* 31.2%

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

      3. *-commutative 31.2%

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

      4. associate-*r/ 31.2%

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

      5. sub-neg 31.2%

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

      6. distribute-lft-in 31.2%

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

      7. metadata-eval 31.2%

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

      8. metadata-eval 31.2%

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

      9. metadata-eval 31.2%

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

      10. fma-define 31.2%

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

      11. metadata-eval 31.2%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in n around inf 34.5%

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

      1. div-sub 35.4%

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

      2. associate-*r/ 35.7%

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

      3. metadata-eval 35.7%

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

      4. associate-*r/ 35.4%

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

      5. distribute-lft-out-- 35.4%

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

      6. div-sub 34.5%

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

      7. *-commutative 34.5%

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

      8. expm1-define 88.1%

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

   7. Simplified 88.1%

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

   if -6e8 < n < 1

   1. Initial program 29.4%

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

      1. associate-/r/ 29.5%

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

      2. associate-*r* 29.5%

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

      3. *-commutative 29.5%

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

      4. associate-*r/ 29.5%

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

      5. sub-neg 29.5%

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

      6. distribute-lft-in 29.4%

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

      7. metadata-eval 29.4%

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

      8. metadata-eval 29.4%

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

      9. metadata-eval 29.4%

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

      10. fma-define 29.5%

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

      11. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

      1. fma-undefine 29.4%

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

      2. metadata-eval 29.4%

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

      3. metadata-eval 29.4%

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

      4. distribute-lft-in 29.5%

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

      5. sub-neg 29.5%

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

      6. *-commutative 29.5%

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

      7. add-exp-log 29.5%

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

      8. expm1-define 29.5%

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

      9. log-pow 46.8%

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

      10. log1p-define 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. clear-num 94.2%

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

      2. inv-pow 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. unpow-1 94.2%

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

      2. *-commutative 94.2%

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

   10. Simplified 94.2%

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

   11. Taylor expanded in i around 0 84.3%

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

       1. associate-*r* 84.3%

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

       2. *-commutative 84.3%

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

       3. sub-neg 84.3%

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

       4. associate-*r/ 84.3%

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

       5. metadata-eval 84.3%

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

       6. metadata-eval 84.3%

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

   13. Simplified 84.3%

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

   if 1 < n

   1. Initial program 20.6%

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

   3. Taylor expanded in n around inf 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-/l* 36.6%

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

      3. expm1-define 93.9%

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

   5. Simplified 93.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -600000000:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{+76} \lor \neg \left(n \leq 3\right):\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.4e+76) (not (<= n 3.0)))
   (*
    (/ 100.0 i)
    (*
     i
     (+
      n
      (*
       i
       (+
        (* n 0.5)
        (*
         i
         (+ (* 0.041666666666666664 (* i n)) (* n 0.16666666666666666))))))))
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.4e+76) || !(n <= 3.0)) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-5.4d+76)) .or. (.not. (n <= 3.0d0))) then
        tmp = (100.0d0 / i) * (i * (n + (i * ((n * 0.5d0) + (i * ((0.041666666666666664d0 * (i * n)) + (n * 0.16666666666666666d0)))))))
    else
        tmp = n * (1.0d0 / (0.01d0 + ((i * 0.01d0) * ((0.5d0 / n) + (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5.4e+76) || !(n <= 3.0)) {
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	} 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 <= -5.4e+76) or not (n <= 3.0):
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))))
	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 <= -5.4e+76) || !(n <= 3.0))
		tmp = Float64(Float64(100.0 / i) * Float64(i * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666))))))));
	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

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -5.4e+76) || ~((n <= 3.0)))
		tmp = (100.0 / i) * (i * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666)))))));
	else
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.4e+76], N[Not[LessEqual[n, 3.0]], $MachinePrecision]], N[(N[(100.0 / i), $MachinePrecision] * N[(i * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 2 regimes

2. if n < -5.3999999999999998e76 or 3 < n

   1. Initial program 23.4%

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

      1. associate-*r/ 23.4%

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

      2. sub-neg 23.4%

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

      3. distribute-rgt-in 23.4%

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

      4. metadata-eval 23.4%

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

      5. metadata-eval 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in n around inf 34.9%

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

   6. Taylor expanded in i around inf 34.9%

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

      1. sub-neg 34.9%

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

      2. metadata-eval 34.9%

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

      3. metadata-eval 34.9%

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

      4. distribute-lft-in 34.9%

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

      5. metadata-eval 34.9%

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

      6. sub-neg 34.9%

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

      7. expm1-define 61.2%

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

   8. Simplified 61.2%

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

      1. div-inv 61.1%

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

      2. times-frac 91.0%

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

   10. Applied egg-rr 91.0%

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

   11. Taylor expanded in i around 0 79.9%

       \[\leadsto \frac{100}{i} \cdot \color{blue}{\left(i \cdot \left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)\right)} \]

   if -5.3999999999999998e76 < n < 3

   1. Initial program 30.5%

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

      1. associate-/r/ 30.6%

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

      2. associate-*r* 30.6%

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

      3. *-commutative 30.6%

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

      4. associate-*r/ 30.6%

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

      5. sub-neg 30.6%

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

      6. distribute-lft-in 30.5%

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

      7. metadata-eval 30.5%

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

      8. metadata-eval 30.5%

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

      9. metadata-eval 30.5%

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

      10. fma-define 30.6%

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

      11. metadata-eval 30.6%

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

   3. Simplified 30.6%

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

      1. fma-undefine 30.5%

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

      2. metadata-eval 30.5%

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

      3. metadata-eval 30.5%

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

      4. distribute-lft-in 30.6%

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

      5. sub-neg 30.6%

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

      6. *-commutative 30.6%

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

      7. add-exp-log 30.6%

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

      8. expm1-define 30.6%

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

      9. log-pow 43.6%

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

      10. log1p-define 93.6%

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

   6. Applied egg-rr 93.6%

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

      1. clear-num 93.5%

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

      2. inv-pow 93.5%

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

   8. Applied egg-rr 93.5%

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

      1. unpow-1 93.5%

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

      2. *-commutative 93.5%

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

   10. Simplified 93.5%

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

   11. Taylor expanded in i around 0 80.7%

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

       1. associate-*r* 80.7%

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

       2. *-commutative 80.7%

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

       3. sub-neg 80.7%

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

       4. associate-*r/ 80.7%

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

       5. metadata-eval 80.7%

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

       6. metadata-eval 80.7%

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

   13. Simplified 80.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{+76} \lor \neg \left(n \leq 3\right):\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+109} \lor \neg \left(n \leq 1\right):\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(\left(i \cdot n\right) \cdot 4.166666666666667 + n \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.6e+109) (not (<= n 1.0)))
   (+
    (* n 100.0)
    (*
     i
     (+
      (* n 50.0)
      (* i (+ (* (* i n) 4.166666666666667) (* n 16.666666666666668))))))
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.6e+109) || !(n <= 1.0)) {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (((i * n) * 4.166666666666667) + (n * 16.666666666666668)))));
	} else {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-5.6d+109)) .or. (.not. (n <= 1.0d0))) then
        tmp = (n * 100.0d0) + (i * ((n * 50.0d0) + (i * (((i * n) * 4.166666666666667d0) + (n * 16.666666666666668d0)))))
    else
        tmp = n * (1.0d0 / (0.01d0 + ((i * 0.01d0) * ((0.5d0 / n) + (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5.6e+109) || !(n <= 1.0)) {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (((i * n) * 4.166666666666667) + (n * 16.666666666666668)))));
	} 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 <= -5.6e+109) or not (n <= 1.0):
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (((i * n) * 4.166666666666667) + (n * 16.666666666666668)))))
	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 <= -5.6e+109) || !(n <= 1.0))
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(n * 50.0) + Float64(i * Float64(Float64(Float64(i * n) * 4.166666666666667) + Float64(n * 16.666666666666668))))));
	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

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -5.6e+109) || ~((n <= 1.0)))
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (((i * n) * 4.166666666666667) + (n * 16.666666666666668)))));
	else
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.6e+109], N[Not[LessEqual[n, 1.0]], $MachinePrecision]], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(n * 50.0), $MachinePrecision] + N[(i * N[(N[(N[(i * n), $MachinePrecision] * 4.166666666666667), $MachinePrecision] + N[(n * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < -5.6000000000000004e109 or 1 < n

   1. Initial program 20.8%

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

   3. Taylor expanded in n around inf 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/l* 35.4%

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

      3. expm1-define 92.5%

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

   5. Simplified 92.5%

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

   6. Taylor expanded in i around 0 80.6%

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

   if -5.6000000000000004e109 < n < 1

   1. Initial program 32.4%

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

      1. associate-/r/ 32.5%

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

      2. associate-*r* 32.5%

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

      3. *-commutative 32.5%

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

      4. associate-*r/ 32.6%

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

      5. sub-neg 32.6%

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

      6. distribute-lft-in 32.5%

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

      7. metadata-eval 32.5%

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

      8. metadata-eval 32.5%

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

      9. metadata-eval 32.5%

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

      10. fma-define 32.6%

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

      11. metadata-eval 32.6%

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

   3. Simplified 32.6%

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

      1. fma-undefine 32.5%

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

      2. metadata-eval 32.5%

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

      3. metadata-eval 32.5%

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

      4. distribute-lft-in 32.6%

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

      5. sub-neg 32.6%

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

      6. *-commutative 32.6%

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

      7. add-exp-log 32.6%

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

      8. expm1-define 32.6%

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

      9. log-pow 43.3%

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

      10. log1p-define 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. clear-num 92.4%

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

      2. inv-pow 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. *-commutative 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in i around 0 78.7%

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

       1. associate-*r* 78.7%

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

       2. *-commutative 78.7%

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

       3. sub-neg 78.7%

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

       4. associate-*r/ 78.7%

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

       5. metadata-eval 78.7%

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

       6. metadata-eval 78.7%

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

   13. Simplified 78.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+109} \lor \neg \left(n \leq 1\right):\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(\left(i \cdot n\right) \cdot 4.166666666666667 + n \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 9 \cdot 10^{+47}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 9e+47)
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))
   (/
    (*
     i
     (+
      100.0
      (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
    (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 9e+47) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / (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 <= 9d+47) then
        tmp = n * (1.0d0 / (0.01d0 + ((i * 0.01d0) * ((0.5d0 / n) + (-0.5d0)))))
    else
        tmp = (i * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 9e+47) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 9e+47:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	else:
		tmp = (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 9e+47)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667))))))) / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 9e+47)
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	else
		tmp = (i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 9e+47], 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[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 8.99999999999999958e47

   1. Initial program 21.1%

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

      1. associate-/r/ 21.5%

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

      2. associate-*r* 21.5%

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

      3. *-commutative 21.5%

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

      4. associate-*r/ 21.5%

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

      5. sub-neg 21.5%

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

      6. distribute-lft-in 21.4%

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

      7. metadata-eval 21.4%

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

      8. metadata-eval 21.4%

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

      9. metadata-eval 21.4%

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

      10. fma-define 21.5%

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

      11. metadata-eval 21.5%

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

   3. Simplified 21.5%

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

      1. fma-undefine 21.4%

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

      2. metadata-eval 21.4%

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

      3. metadata-eval 21.4%

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

      4. distribute-lft-in 21.5%

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

      5. sub-neg 21.5%

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

      6. *-commutative 21.5%

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

      7. add-exp-log 21.5%

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

      8. expm1-define 21.5%

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

      9. log-pow 26.5%

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

      10. log1p-define 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. clear-num 78.1%

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

      2. inv-pow 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. unpow-1 78.1%

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

      2. *-commutative 78.1%

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

   10. Simplified 78.1%

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

   11. Taylor expanded in i around 0 80.6%

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

       1. associate-*r* 80.6%

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

       2. *-commutative 80.6%

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

       3. sub-neg 80.6%

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

       4. associate-*r/ 80.6%

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

       5. metadata-eval 80.6%

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

       6. metadata-eval 80.6%

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

   13. Simplified 80.6%

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

   if 8.99999999999999958e47 < i

   1. Initial program 55.0%

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

      1. associate-*r/ 55.0%

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

      2. sub-neg 55.0%

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

      3. distribute-rgt-in 55.0%

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

      4. metadata-eval 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in n around inf 57.2%

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

   6. Taylor expanded in i around 0 50.9%

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

      1. *-commutative 50.9%

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

   8. Simplified 50.9%

      \[\leadsto \frac{\color{blue}{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 9 \cdot 10^{+47}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-156} \lor \neg \left(n \leq 8 \cdot 10^{-247}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1e-156) (not (<= n 8e-247)))
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1e-156) || !(n <= 8e-247)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} 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 ((n <= (-1d-156)) .or. (.not. (n <= 8d-247))) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1e-156) || !(n <= 8e-247)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1e-156) or not (n <= 8e-247):
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1e-156) || !(n <= 8e-247))
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1e-156) || ~((n <= 8e-247)))
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1e-156], N[Not[LessEqual[n, 8e-247]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.00000000000000004e-156 or 8.0000000000000002e-247 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf 26.1%

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

      1. *-commutative 26.1%

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

      2. associate-/l* 26.1%

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

      3. expm1-define 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in i around 0 68.7%

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

   7. Taylor expanded in n around 0 68.7%

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

   if -1.00000000000000004e-156 < n < 8.0000000000000002e-247

   1. Initial program 69.1%

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

      1. associate-*r/ 69.1%

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

      2. sub-neg 69.1%

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

      3. distribute-rgt-in 69.1%

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

      4. metadata-eval 69.1%

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

      5. metadata-eval 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in i around 0 79.5%

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

   6. Taylor expanded in i around 0 79.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-156} \lor \neg \left(n \leq 8 \cdot 10^{-247}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-156}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-247}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(\left(i \cdot n\right) \cdot 16.666666666666668\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1e-156)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 8e-247)
     0.0
     (+ (* n 100.0) (* i (* (* i n) 16.666666666666668))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1e-156) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 8e-247) {
		tmp = 0.0;
	} else {
		tmp = (n * 100.0) + (i * ((i * n) * 16.666666666666668));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1d-156)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 8d-247) then
        tmp = 0.0d0
    else
        tmp = (n * 100.0d0) + (i * ((i * n) * 16.666666666666668d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1e-156) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 8e-247) {
		tmp = 0.0;
	} else {
		tmp = (n * 100.0) + (i * ((i * n) * 16.666666666666668));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1e-156:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 8e-247:
		tmp = 0.0
	else:
		tmp = (n * 100.0) + (i * ((i * n) * 16.666666666666668))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1e-156)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 8e-247)
		tmp = 0.0;
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(i * n) * 16.666666666666668)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1e-156)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 8e-247)
		tmp = 0.0;
	else
		tmp = (n * 100.0) + (i * ((i * n) * 16.666666666666668));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1e-156], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8e-247], 0.0, N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(i * n), $MachinePrecision] * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.00000000000000004e-156

   1. Initial program 25.2%

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

   3. Taylor expanded in n around inf 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-/l* 27.1%

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

      3. expm1-define 84.2%

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

   5. Simplified 84.2%

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

   6. Taylor expanded in i around 0 66.2%

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

   7. Taylor expanded in n around 0 66.2%

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

   if -1.00000000000000004e-156 < n < 8.0000000000000002e-247

   1. Initial program 69.1%

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

      1. associate-*r/ 69.1%

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

      2. sub-neg 69.1%

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

      3. distribute-rgt-in 69.1%

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

      4. metadata-eval 69.1%

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

      5. metadata-eval 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in i around 0 79.5%

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

   6. Taylor expanded in i around 0 79.5%

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

   if 8.0000000000000002e-247 < n

   1. Initial program 18.8%

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

   3. Taylor expanded in n around inf 25.4%

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

      1. *-commutative 25.4%

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

      2. associate-/l* 25.4%

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

      3. expm1-define 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in i around 0 70.7%

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

   7. Taylor expanded in i around inf 70.7%

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

      1. *-commutative 70.7%

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

      2. *-commutative 70.7%

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

   9. Simplified 70.7%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-156}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-247}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(\left(i \cdot n\right) \cdot 16.666666666666668\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.7 \cdot 10^{+48}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(\left(i \cdot n\right) \cdot 16.666666666666668\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.7e+48)
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))
   (+ (* n 100.0) (* i (* (* i n) 16.666666666666668)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.7e+48) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = (n * 100.0) + (i * ((i * n) * 16.666666666666668));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 2.7d+48) then
        tmp = n * (1.0d0 / (0.01d0 + ((i * 0.01d0) * ((0.5d0 / n) + (-0.5d0)))))
    else
        tmp = (n * 100.0d0) + (i * ((i * n) * 16.666666666666668d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.7e+48) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = (n * 100.0) + (i * ((i * n) * 16.666666666666668));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.7e+48:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	else:
		tmp = (n * 100.0) + (i * ((i * n) * 16.666666666666668))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.7e+48)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(i * n) * 16.666666666666668)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 2.7e+48)
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	else
		tmp = (n * 100.0) + (i * ((i * n) * 16.666666666666668));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.7e+48], 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), $MachinePrecision] + N[(i * N[(N[(i * n), $MachinePrecision] * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2.70000000000000004e48

   1. Initial program 21.1%

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

      1. associate-/r/ 21.5%

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

      2. associate-*r* 21.5%

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

      3. *-commutative 21.5%

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

      4. associate-*r/ 21.5%

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

      5. sub-neg 21.5%

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

      6. distribute-lft-in 21.4%

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

      7. metadata-eval 21.4%

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

      8. metadata-eval 21.4%

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

      9. metadata-eval 21.4%

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

      10. fma-define 21.5%

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

      11. metadata-eval 21.5%

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

   3. Simplified 21.5%

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

      1. fma-undefine 21.4%

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

      2. metadata-eval 21.4%

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

      3. metadata-eval 21.4%

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

      4. distribute-lft-in 21.5%

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

      5. sub-neg 21.5%

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

      6. *-commutative 21.5%

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

      7. add-exp-log 21.5%

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

      8. expm1-define 21.5%

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

      9. log-pow 26.5%

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

      10. log1p-define 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. clear-num 78.1%

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

      2. inv-pow 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. unpow-1 78.1%

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

      2. *-commutative 78.1%

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

   10. Simplified 78.1%

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

   11. Taylor expanded in i around 0 80.6%

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

       1. associate-*r* 80.6%

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

       2. *-commutative 80.6%

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

       3. sub-neg 80.6%

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

       4. associate-*r/ 80.6%

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

       5. metadata-eval 80.6%

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

       6. metadata-eval 80.6%

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

   13. Simplified 80.6%

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

   if 2.70000000000000004e48 < i

   1. Initial program 55.0%

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

   3. Taylor expanded in n around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. associate-/l* 57.4%

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

      3. expm1-define 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in i around 0 43.1%

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

   7. Taylor expanded in i around inf 43.1%

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

      1. *-commutative 43.1%

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

      2. *-commutative 43.1%

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

   9. Simplified 43.1%

      \[\leadsto 100 \cdot n + i \cdot \color{blue}{\left(\left(n \cdot i\right) \cdot 16.666666666666668\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.7 \cdot 10^{+48}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(\left(i \cdot n\right) \cdot 16.666666666666668\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 9 \cdot 10^{+47}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 9e+47)
   (* n (/ 1.0 (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5)))))
   (/ (* i (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668))))) (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 9e+47) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = (i * (100.0 + (i * (50.0 + (i * 16.666666666666668))))) / (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 <= 9d+47) then
        tmp = n * (1.0d0 / (0.01d0 + ((i * 0.01d0) * ((0.5d0 / n) + (-0.5d0)))))
    else
        tmp = (i * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 9e+47) {
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	} else {
		tmp = (i * (100.0 + (i * (50.0 + (i * 16.666666666666668))))) / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 9e+47:
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))))
	else:
		tmp = (i * (100.0 + (i * (50.0 + (i * 16.666666666666668))))) / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 9e+47)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668))))) / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 9e+47)
		tmp = n * (1.0 / (0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))));
	else
		tmp = (i * (100.0 + (i * (50.0 + (i * 16.666666666666668))))) / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 9e+47], 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[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 8.99999999999999958e47

   1. Initial program 21.1%

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

      1. associate-/r/ 21.5%

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

      2. associate-*r* 21.5%

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

      3. *-commutative 21.5%

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

      4. associate-*r/ 21.5%

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

      5. sub-neg 21.5%

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

      6. distribute-lft-in 21.4%

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

      7. metadata-eval 21.4%

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

      8. metadata-eval 21.4%

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

      9. metadata-eval 21.4%

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

      10. fma-define 21.5%

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

      11. metadata-eval 21.5%

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

   3. Simplified 21.5%

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

      1. fma-undefine 21.4%

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

      2. metadata-eval 21.4%

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

      3. metadata-eval 21.4%

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

      4. distribute-lft-in 21.5%

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

      5. sub-neg 21.5%

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

      6. *-commutative 21.5%

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

      7. add-exp-log 21.5%

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

      8. expm1-define 21.5%

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

      9. log-pow 26.5%

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

      10. log1p-define 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. clear-num 78.1%

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

      2. inv-pow 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. unpow-1 78.1%

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

      2. *-commutative 78.1%

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

   10. Simplified 78.1%

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

   11. Taylor expanded in i around 0 80.6%

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

       1. associate-*r* 80.6%

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

       2. *-commutative 80.6%

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

       3. sub-neg 80.6%

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

       4. associate-*r/ 80.6%

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

       5. metadata-eval 80.6%

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

       6. metadata-eval 80.6%

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

   13. Simplified 80.6%

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

   if 8.99999999999999958e47 < i

   1. Initial program 55.0%

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

      1. associate-*r/ 55.0%

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

      2. sub-neg 55.0%

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

      3. distribute-rgt-in 55.0%

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

      4. metadata-eval 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in n around inf 57.2%

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

   6. Taylor expanded in i around 0 46.5%

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

      1. *-commutative 46.5%

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

   8. Simplified 46.5%

      \[\leadsto \frac{\color{blue}{i \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 9 \cdot 10^{+47}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.02 \cdot 10^{-156} \lor \neg \left(n \leq 8 \cdot 10^{-247}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.02e-156) (not (<= n 8e-247)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.02e-156) || !(n <= 8e-247)) {
		tmp = n * (100.0 + (i * 50.0));
	} 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 ((n <= (-1.02d-156)) .or. (.not. (n <= 8d-247))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.02e-156) || !(n <= 8e-247)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.02e-156) or not (n <= 8e-247):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.02e-156) || !(n <= 8e-247))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.02e-156) || ~((n <= 8e-247)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.02e-156], N[Not[LessEqual[n, 8e-247]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.02e-156 or 8.0000000000000002e-247 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf 26.1%

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

      1. *-commutative 26.1%

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

      2. associate-/l* 26.1%

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

      3. expm1-define 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in i around 0 67.3%

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

      1. +-commutative 67.3%

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

      2. associate-*r* 67.3%

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

      3. distribute-rgt-in 67.3%

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

      4. *-commutative 67.3%

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

   8. Simplified 67.3%

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

   if -1.02e-156 < n < 8.0000000000000002e-247

   1. Initial program 69.1%

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

      1. associate-*r/ 69.1%

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

      2. sub-neg 69.1%

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

      3. distribute-rgt-in 69.1%

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

      4. metadata-eval 69.1%

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

      5. metadata-eval 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in i around 0 79.5%

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

   6. Taylor expanded in i around 0 79.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.02 \cdot 10^{-156} \lor \neg \left(n \leq 8 \cdot 10^{-247}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.05 \cdot 10^{+26}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 9 \cdot 10^{+47}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.05e+26) 0.0 (if (<= i 9e+47) (* n 100.0) (* n (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.05e+26) {
		tmp = 0.0;
	} else if (i <= 9e+47) {
		tmp = n * 100.0;
	} else {
		tmp = n * (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 <= (-2.05d+26)) then
        tmp = 0.0d0
    else if (i <= 9d+47) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.05e+26) {
		tmp = 0.0;
	} else if (i <= 9e+47) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.05e+26:
		tmp = 0.0
	elif i <= 9e+47:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.05e+26)
		tmp = 0.0;
	elseif (i <= 9e+47)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.05e+26)
		tmp = 0.0;
	elseif (i <= 9e+47)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.05e+26], 0.0, If[LessEqual[i, 9e+47], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.04999999999999992e26

   1. Initial program 62.8%

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

      1. associate-*r/ 62.8%

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

      2. sub-neg 62.8%

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

      3. distribute-rgt-in 62.7%

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

      4. metadata-eval 62.7%

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

      5. metadata-eval 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in i around 0 26.1%

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

   6. Taylor expanded in i around 0 26.1%

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

   if -2.04999999999999992e26 < i < 8.99999999999999958e47

   1. Initial program 11.7%

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

   3. Taylor expanded in i around 0 81.7%

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

      1. *-commutative 81.7%

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

   5. Simplified 81.7%

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

   if 8.99999999999999958e47 < i

   1. Initial program 55.0%

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

   3. Taylor expanded in n around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. associate-/l* 57.4%

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

      3. expm1-define 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in i around 0 32.2%

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

   7. Taylor expanded in i around inf 32.2%

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

      1. *-commutative 32.2%

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

      2. *-commutative 32.2%

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

   9. Simplified 32.2%

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

   10. Taylor expanded in n around 0 32.2%

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

       1. associate-*r* 32.2%

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

   12. Simplified 32.2%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.05 \cdot 10^{+26}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 9 \cdot 10^{+47}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-156} \lor \neg \left(n \leq 8 \cdot 10^{-247}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1e-156) (not (<= n 8e-247))) (* n 100.0) 0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1e-156) || !(n <= 8e-247)) {
		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 ((n <= (-1d-156)) .or. (.not. (n <= 8d-247))) 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 ((n <= -1e-156) || !(n <= 8e-247)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1e-156) or not (n <= 8e-247):
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1e-156) || !(n <= 8e-247))
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1e-156) || ~((n <= 8e-247)))
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1e-156], N[Not[LessEqual[n, 8e-247]], $MachinePrecision]], N[(n * 100.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.00000000000000004e-156 or 8.0000000000000002e-247 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in i around 0 61.5%

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

      1. *-commutative 61.5%

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

   5. Simplified 61.5%

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

   if -1.00000000000000004e-156 < n < 8.0000000000000002e-247

   1. Initial program 69.1%

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

      1. associate-*r/ 69.1%

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

      2. sub-neg 69.1%

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

      3. distribute-rgt-in 69.1%

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

      4. metadata-eval 69.1%

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

      5. metadata-eval 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in i around 0 79.5%

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

   6. Taylor expanded in i around 0 79.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-156} \lor \neg \left(n \leq 8 \cdot 10^{-247}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 17.5% 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 26.9%

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

   1. associate-*r/ 26.9%

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

   2. sub-neg 26.9%

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

   3. distribute-rgt-in 26.9%

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

   4. metadata-eval 26.9%

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

   5. metadata-eval 26.9%

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

3. Simplified 26.9%

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

5. Taylor expanded in i around 0 15.0%

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

6. Taylor expanded in i around 0 15.3%

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

7. Final simplification 15.3%

   \[\leadsto 0 \]
8. Add Preprocessing

Developer target: 35.3% 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 2024075 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

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

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