Compound Interest

Percentage Accurate: 28.4% → 98.5%

Time: 21.9s

Alternatives: 16

Speedup: 16.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.4% 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: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ t_2 := \frac{t_0}{i}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-205}:\\ \;\;\;\;100 \cdot \left(\left(\frac{n}{i} - \frac{n}{i}\right) + \left(n \cdot t_2 - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(t_2 + \frac{-1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (/ t_0 i)))
   (if (<= t_1 -2e-205)
     (* 100.0 (+ (- (/ n i) (/ n i)) (- (* n t_2) (/ n i))))
     (if (<= t_1 0.0)
       (* (/ (expm1 (* n (log1p (/ i n)))) i) (* n 100.0))
       (if (<= t_1 INFINITY)
         (* (* n 100.0) (+ t_2 (/ -1.0 i)))
         (*
          100.0
          (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = t_0 / i;
	double tmp;
	if (t_1 <= -2e-205) {
		tmp = 100.0 * (((n / i) - (n / i)) + ((n * t_2) - (n / i)));
	} else if (t_1 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (n * 100.0) * (t_2 + (-1.0 / i));
	} else {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(t_0 / i)
	tmp = 0.0
	if (t_1 <= -2e-205)
		tmp = Float64(100.0 * Float64(Float64(Float64(n / i) - Float64(n / i)) + Float64(Float64(n * t_2) - Float64(n / i))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i) * Float64(n * 100.0));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(n * 100.0) * Float64(t_2 + Float64(-1.0 / i)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-205], N[(100.0 * N[(N[(N[(n / i), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$2), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$2 + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2e-205

   1. Initial program 99.6%

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

      1. div-sub 99.6%

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

      2. associate-/r/ 99.6%

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

      3. clear-num 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. prod-diff 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-udef 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. fma-udef 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -2e-205 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/r/ 23.2%

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

      3. sub-neg 23.2%

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

      4. metadata-eval 23.2%

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

      5. associate-*r* 23.2%

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

      6. metadata-eval 23.2%

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

      7. sub-neg 23.2%

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

      8. associate-*l/ 23.2%

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

      9. associate-/l* 23.2%

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

      10. pow-to-exp 23.2%

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

      11. expm1-def 33.6%

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

      12. *-commutative 33.6%

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

      13. log1p-udef 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-/r/ 99.6%

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

   5. Simplified 99.6%

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

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

   1. Initial program 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-/r/ 97.0%

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

      3. associate-*l* 97.1%

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

      4. sub-neg 97.1%

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

      5. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

      1. metadata-eval 97.1%

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

      2. sub-neg 97.1%

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

      3. div-sub 97.3%

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

   5. Applied egg-rr 97.3%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 85.1%

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

   4. Simplified 85.1%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      2. +-commutative 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      3. *-commutative 100.0%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. fma-def 100.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. *-commutative 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      6. unpow2 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      7. associate-*l* 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 100.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-205}:\\ \;\;\;\;100 \cdot \left(\left(\frac{n}{i} - \frac{n}{i}\right) + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ t_2 := \frac{t_0}{i}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-137}:\\ \;\;\;\;100 \cdot \left(\left(\frac{n}{i} - \frac{n}{i}\right) + \left(n \cdot t_2 - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(t_2 + \frac{-1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (/ t_0 i)))
   (if (<= t_1 -2e-137)
     (* 100.0 (+ (- (/ n i) (/ n i)) (- (* n t_2) (/ n i))))
     (if (<= t_1 0.0)
       (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 (/ i n)))
       (if (<= t_1 INFINITY)
         (* (* n 100.0) (+ t_2 (/ -1.0 i)))
         (*
          100.0
          (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = t_0 / i;
	double tmp;
	if (t_1 <= -2e-137) {
		tmp = 100.0 * (((n / i) - (n / i)) + ((n * t_2) - (n / i)));
	} else if (t_1 <= 0.0) {
		tmp = expm1((n * log1p((i / n)))) * (100.0 / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (n * 100.0) * (t_2 + (-1.0 / i));
	} else {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(t_0 / i)
	tmp = 0.0
	if (t_1 <= -2e-137)
		tmp = Float64(100.0 * Float64(Float64(Float64(n / i) - Float64(n / i)) + Float64(Float64(n * t_2) - Float64(n / i))));
	elseif (t_1 <= 0.0)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(n * 100.0) * Float64(t_2 + Float64(-1.0 / i)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / i), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-137], N[(100.0 * N[(N[(N[(n / i), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$2), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$2 + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -1.99999999999999996e-137

   1. Initial program 99.5%

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

      1. div-sub 99.5%

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

      2. associate-/r/ 99.5%

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

      3. clear-num 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. prod-diff 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-udef 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. fma-udef 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -1.99999999999999996e-137 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   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. clear-num 25.1%

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

      2. un-div-inv 25.1%

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

      3. pow-to-exp 25.1%

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

      4. expm1-def 33.3%

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

      5. *-commutative 33.3%

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

      6. log1p-udef 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. associate-/r/ 97.9%

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

   5. Simplified 97.9%

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

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

   1. Initial program 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-/r/ 97.0%

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

      3. associate-*l* 97.1%

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

      4. sub-neg 97.1%

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

      5. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

      1. metadata-eval 97.1%

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

      2. sub-neg 97.1%

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

      3. div-sub 97.3%

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

   5. Applied egg-rr 97.3%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 85.1%

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

   4. Simplified 85.1%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      2. +-commutative 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      3. *-commutative 100.0%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. fma-def 100.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. *-commutative 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      6. unpow2 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      7. associate-*l* 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 100.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-137}:\\ \;\;\;\;100 \cdot \left(\left(\frac{n}{i} - \frac{n}{i}\right) + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 3: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -1.02 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-231}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.2:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -1.02e-176)
     t_0
     (if (<= n 2.9e-231)
       0.0
       (if (<= n 1.2)
         (* 100.0 (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333)))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -1.02e-176) {
		tmp = t_0;
	} else if (n <= 2.9e-231) {
		tmp = 0.0;
	} else if (n <= 1.2) {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -1.02e-176)
		tmp = t_0;
	elseif (n <= 2.9e-231)
		tmp = 0.0;
	elseif (n <= 1.2)
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.02e-176], t$95$0, If[LessEqual[n, 2.9e-231], 0.0, If[LessEqual[n, 1.2], N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.02000000000000002e-176 or 1.19999999999999996 < n

   1. Initial program 25.5%

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

   2. Taylor expanded in n around inf 42.3%

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

      1. *-commutative 42.3%

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

      2. associate-/l* 42.3%

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

      3. expm1-def 90.1%

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

   4. Simplified 90.1%

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

   if -1.02000000000000002e-176 < n < 2.9000000000000001e-231

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-/r/ 76.4%

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

      3. associate-*l* 76.4%

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

      4. sub-neg 76.4%

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

      5. metadata-eval 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in i around 0 87.0%

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

   5. Taylor expanded in i around 0 87.0%

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

   if 2.9000000000000001e-231 < n < 1.19999999999999996

   1. Initial program 21.0%

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

   2. Taylor expanded in n around inf 6.3%

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

      1. *-commutative 6.3%

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

      2. associate-/l* 6.3%

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

      3. expm1-def 40.1%

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

   4. Simplified 40.1%

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

   5. Taylor expanded in i around 0 63.2%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. associate-+r+ 63.2%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      2. +-commutative 63.2%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      3. *-commutative 63.2%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. fma-def 63.2%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. *-commutative 63.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      6. unpow2 63.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      7. associate-*l* 63.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 63.2%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.02 \cdot 10^{-176}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-231}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.2:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ t_1 := i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\\ \mathbf{if}\;n \leq -1.2 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-170}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 190000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{+94}:\\ \;\;\;\;100 \cdot \frac{n \cdot n - t_1 \cdot t_1}{n - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* (expm1 i) (/ 100.0 i))))
        (t_1 (* i (* n (+ 0.5 (/ -0.5 n))))))
   (if (<= n -1.2e-180)
     t_0
     (if (<= n 1.05e-170)
       0.0
       (if (<= n 190000000.0)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 2.05e+94)
           (* 100.0 (/ (- (* n n) (* t_1 t_1)) (- n t_1)))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (expm1(i) * (100.0 / i));
	double t_1 = i * (n * (0.5 + (-0.5 / n)));
	double tmp;
	if (n <= -1.2e-180) {
		tmp = t_0;
	} else if (n <= 1.05e-170) {
		tmp = 0.0;
	} else if (n <= 190000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.05e+94) {
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (Math.expm1(i) * (100.0 / i));
	double t_1 = i * (n * (0.5 + (-0.5 / n)));
	double tmp;
	if (n <= -1.2e-180) {
		tmp = t_0;
	} else if (n <= 1.05e-170) {
		tmp = 0.0;
	} else if (n <= 190000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.05e+94) {
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (math.expm1(i) * (100.0 / i))
	t_1 = i * (n * (0.5 + (-0.5 / n)))
	tmp = 0
	if n <= -1.2e-180:
		tmp = t_0
	elif n <= 1.05e-170:
		tmp = 0.0
	elif n <= 190000000.0:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.05e+94:
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(expm1(i) * Float64(100.0 / i)))
	t_1 = Float64(i * Float64(n * Float64(0.5 + Float64(-0.5 / n))))
	tmp = 0.0
	if (n <= -1.2e-180)
		tmp = t_0;
	elseif (n <= 1.05e-170)
		tmp = 0.0;
	elseif (n <= 190000000.0)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.05e+94)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * n) - Float64(t_1 * t_1)) / Float64(n - t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(n * N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.2e-180], t$95$0, If[LessEqual[n, 1.05e-170], 0.0, If[LessEqual[n, 190000000.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.05e+94], N[(100.0 * N[(N[(N[(n * n), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(n - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.1999999999999999e-180 or 2.05000000000000015e94 < n

   1. Initial program 23.5%

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

   2. Taylor expanded in n around inf 43.3%

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

      1. *-commutative 43.3%

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

      2. associate-/l* 43.3%

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

      3. expm1-def 90.7%

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

   4. Simplified 90.7%

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

      1. associate-*l/ 90.6%

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

   6. Applied egg-rr 90.6%

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

   7. Taylor expanded in n around 0 43.3%

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

      1. expm1-def 89.5%

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

      2. associate-/l* 90.7%

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

      3. associate-*r/ 90.6%

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

      4. associate-*l/ 90.6%

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

      5. *-commutative 90.6%

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

      6. associate-/r/ 90.1%

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

   9. Simplified 90.1%

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

   if -1.1999999999999999e-180 < n < 1.05e-170

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

      2. associate-/r/ 71.3%

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

      3. associate-*l* 71.3%

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

      4. sub-neg 71.3%

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

      5. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in i around 0 79.5%

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

   5. Taylor expanded in i around 0 79.5%

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

   if 1.05e-170 < n < 1.9e8

   1. Initial program 16.2%

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

   2. Taylor expanded in i around 0 59.5%

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

   if 1.9e8 < n < 2.05000000000000015e94

   1. Initial program 44.2%

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

   2. Taylor expanded in i around 0 53.0%

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

      1. sub-neg 53.0%

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

      2. associate-*r/ 53.0%

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

      3. metadata-eval 53.0%

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

      4. distribute-neg-frac 53.0%

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

      5. metadata-eval 53.0%

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

   4. Simplified 53.0%

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

      1. flip-+ 93.1%

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

   6. Applied egg-rr 93.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-180}:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-170}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 190000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{+94}:\\ \;\;\;\;100 \cdot \frac{n \cdot n - \left(i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right) \cdot \left(i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)}{n - i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ \end{array} \]

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ t_1 := i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-170}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 190000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot n - t_1 \cdot t_1}{n - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i)))))
        (t_1 (* i (* n (+ 0.5 (/ -0.5 n))))))
   (if (<= n -1.35e-178)
     t_0
     (if (<= n 3.5e-170)
       0.0
       (if (<= n 190000000.0)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 3e+82)
           (* 100.0 (/ (- (* n n) (* t_1 t_1)) (- n t_1)))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double t_1 = i * (n * (0.5 + (-0.5 / n)));
	double tmp;
	if (n <= -1.35e-178) {
		tmp = t_0;
	} else if (n <= 3.5e-170) {
		tmp = 0.0;
	} else if (n <= 190000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3e+82) {
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double t_1 = i * (n * (0.5 + (-0.5 / n)));
	double tmp;
	if (n <= -1.35e-178) {
		tmp = t_0;
	} else if (n <= 3.5e-170) {
		tmp = 0.0;
	} else if (n <= 190000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3e+82) {
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	t_1 = i * (n * (0.5 + (-0.5 / n)))
	tmp = 0
	if n <= -1.35e-178:
		tmp = t_0
	elif n <= 3.5e-170:
		tmp = 0.0
	elif n <= 190000000.0:
		tmp = 100.0 * (i / (i / n))
	elif n <= 3e+82:
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	t_1 = Float64(i * Float64(n * Float64(0.5 + Float64(-0.5 / n))))
	tmp = 0.0
	if (n <= -1.35e-178)
		tmp = t_0;
	elseif (n <= 3.5e-170)
		tmp = 0.0;
	elseif (n <= 190000000.0)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 3e+82)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * n) - Float64(t_1 * t_1)) / Float64(n - t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(n * N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.35e-178], t$95$0, If[LessEqual[n, 3.5e-170], 0.0, If[LessEqual[n, 190000000.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3e+82], N[(100.0 * N[(N[(N[(n * n), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(n - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.35000000000000004e-178 or 2.99999999999999989e82 < n

   1. Initial program 23.7%

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

   2. Taylor expanded in n around inf 43.1%

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

      1. *-commutative 43.1%

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

      2. associate-/l* 43.1%

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

      3. expm1-def 90.9%

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

   4. Simplified 90.9%

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

   if -1.35000000000000004e-178 < n < 3.49999999999999985e-170

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

      2. associate-/r/ 71.3%

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

      3. associate-*l* 71.3%

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

      4. sub-neg 71.3%

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

      5. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in i around 0 79.5%

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

   5. Taylor expanded in i around 0 79.5%

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

   if 3.49999999999999985e-170 < n < 1.9e8

   1. Initial program 16.2%

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

   2. Taylor expanded in i around 0 59.5%

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

   if 1.9e8 < n < 2.99999999999999989e82

   1. Initial program 46.9%

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

   2. Taylor expanded in i around 0 48.6%

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

      1. sub-neg 48.6%

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

      2. associate-*r/ 48.6%

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

      3. metadata-eval 48.6%

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

      4. distribute-neg-frac 48.6%

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

      5. metadata-eval 48.6%

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

   4. Simplified 48.6%

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

      1. flip-+ 91.3%

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

   6. Applied egg-rr 91.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-178}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-170}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 190000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot n - \left(i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right) \cdot \left(i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)}{n - i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 6: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -8.8 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.65 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -8.8e-66)
     t_0
     (if (<= i 2.25e-237)
       (* 100.0 (+ n (* i (* n (+ 0.5 (/ -0.5 n))))))
       (if (<= i 1.65e-53) (* 100.0 (/ (* i n) i)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -8.8e-66) {
		tmp = t_0;
	} else if (i <= 2.25e-237) {
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))));
	} else if (i <= 1.65e-53) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -8.8e-66) {
		tmp = t_0;
	} else if (i <= 2.25e-237) {
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))));
	} else if (i <= 1.65e-53) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -8.8e-66:
		tmp = t_0
	elif i <= 2.25e-237:
		tmp = 100.0 * (n + (i * (n * (0.5 + (-0.5 / n)))))
	elif i <= 1.65e-53:
		tmp = 100.0 * ((i * n) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -8.8e-66)
		tmp = t_0;
	elseif (i <= 2.25e-237)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(0.5 + Float64(-0.5 / n))))));
	elseif (i <= 1.65e-53)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -8.8e-66], t$95$0, If[LessEqual[i, 2.25e-237], N[(100.0 * N[(n + N[(i * N[(n * N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.65e-53], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -8.8000000000000004e-66 or 1.65000000000000002e-53 < i

   1. Initial program 46.7%

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

   2. Taylor expanded in n around inf 63.2%

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

      1. expm1-def 69.4%

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

   4. Simplified 69.4%

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

   if -8.8000000000000004e-66 < i < 2.25000000000000005e-237

   1. Initial program 3.9%

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

   2. Taylor expanded in i around 0 92.6%

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

      1. sub-neg 92.6%

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

      2. associate-*r/ 92.6%

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

      3. metadata-eval 92.6%

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

      4. distribute-neg-frac 92.6%

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

      5. metadata-eval 92.6%

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

   4. Simplified 92.6%

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

   if 2.25000000000000005e-237 < i < 1.65000000000000002e-53

   1. Initial program 15.6%

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

      1. *-commutative 15.6%

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

      2. associate-/r/ 16.1%

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

      3. associate-*l* 16.1%

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

      4. sub-neg 16.1%

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

      5. metadata-eval 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in i around 0 7.0%

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

      1. unpow2 7.0%

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

      2. associate-*r/ 7.0%

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

      3. metadata-eval 7.0%

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

   6. Simplified 7.0%

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

   7. Taylor expanded in n around inf 90.1%

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

   8. Taylor expanded in i around 0 90.1%

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

      1. *-commutative 90.1%

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

   10. Simplified 90.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.8 \cdot 10^{-66}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.65 \cdot 10^{-53}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \end{array} \]

Alternative 7: 67.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ t_1 := i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.52 \cdot 10^{-180}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-169}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 190000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot n - t_1 \cdot t_1}{n - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (* 100.0 (+ n (* n (+ (* i 0.5) (* i (* i 0.16666666666666666)))))))
        (t_1 (* i (* n (+ 0.5 (/ -0.5 n))))))
   (if (<= n -1.25e+194)
     t_0
     (if (<= n -1.52e-180)
       (/ (* n 100.0) (+ 1.0 (* i -0.5)))
       (if (<= n 1.5e-169)
         0.0
         (if (<= n 190000000.0)
           (* 100.0 (/ i (/ i n)))
           (if (<= n 7.5e+82)
             (* 100.0 (/ (- (* n n) (* t_1 t_1)) (- n t_1)))
             t_0)))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.16666666666666666)))));
	double t_1 = i * (n * (0.5 + (-0.5 / n)));
	double tmp;
	if (n <= -1.25e+194) {
		tmp = t_0;
	} else if (n <= -1.52e-180) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 1.5e-169) {
		tmp = 0.0;
	} else if (n <= 190000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 7.5e+82) {
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 100.0d0 * (n + (n * ((i * 0.5d0) + (i * (i * 0.16666666666666666d0)))))
    t_1 = i * (n * (0.5d0 + ((-0.5d0) / n)))
    if (n <= (-1.25d+194)) then
        tmp = t_0
    else if (n <= (-1.52d-180)) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    else if (n <= 1.5d-169) then
        tmp = 0.0d0
    else if (n <= 190000000.0d0) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 7.5d+82) then
        tmp = 100.0d0 * (((n * n) - (t_1 * t_1)) / (n - t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.16666666666666666)))));
	double t_1 = i * (n * (0.5 + (-0.5 / n)));
	double tmp;
	if (n <= -1.25e+194) {
		tmp = t_0;
	} else if (n <= -1.52e-180) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 1.5e-169) {
		tmp = 0.0;
	} else if (n <= 190000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 7.5e+82) {
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.16666666666666666)))))
	t_1 = i * (n * (0.5 + (-0.5 / n)))
	tmp = 0
	if n <= -1.25e+194:
		tmp = t_0
	elif n <= -1.52e-180:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	elif n <= 1.5e-169:
		tmp = 0.0
	elif n <= 190000000.0:
		tmp = 100.0 * (i / (i / n))
	elif n <= 7.5e+82:
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(i * 0.5) + Float64(i * Float64(i * 0.16666666666666666))))))
	t_1 = Float64(i * Float64(n * Float64(0.5 + Float64(-0.5 / n))))
	tmp = 0.0
	if (n <= -1.25e+194)
		tmp = t_0;
	elseif (n <= -1.52e-180)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	elseif (n <= 1.5e-169)
		tmp = 0.0;
	elseif (n <= 190000000.0)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 7.5e+82)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * n) - Float64(t_1 * t_1)) / Float64(n - t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.16666666666666666)))));
	t_1 = i * (n * (0.5 + (-0.5 / n)));
	tmp = 0.0;
	if (n <= -1.25e+194)
		tmp = t_0;
	elseif (n <= -1.52e-180)
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	elseif (n <= 1.5e-169)
		tmp = 0.0;
	elseif (n <= 190000000.0)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 7.5e+82)
		tmp = 100.0 * (((n * n) - (t_1 * t_1)) / (n - t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n + N[(n * N[(N[(i * 0.5), $MachinePrecision] + N[(i * N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(n * N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.25e+194], t$95$0, If[LessEqual[n, -1.52e-180], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5e-169], 0.0, If[LessEqual[n, 190000000.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.5e+82], N[(100.0 * N[(N[(N[(n * n), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(n - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.24999999999999997e194 or 7.4999999999999999e82 < n

   1. Initial program 10.0%

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

   2. Taylor expanded in n around inf 45.9%

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

      1. *-commutative 45.9%

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

      2. associate-/l* 45.9%

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

      3. expm1-def 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in i around 0 78.1%

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

      1. +-commutative 78.1%

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

      2. associate-*r* 78.1%

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

      3. associate-*r* 78.1%

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

      4. distribute-rgt-out 78.7%

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

      5. *-commutative 78.7%

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

      6. unpow2 78.7%

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

      7. associate-*r* 78.7%

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

   7. Simplified 78.7%

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

   if -1.24999999999999997e194 < n < -1.5199999999999999e-180

   1. Initial program 42.3%

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

   2. Taylor expanded in n around inf 39.3%

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

      1. *-commutative 39.3%

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

      2. associate-/l* 39.3%

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

      3. expm1-def 79.9%

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

   4. Simplified 79.9%

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

      1. associate-*l/ 79.8%

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

   6. Applied egg-rr 79.8%

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

   7. Taylor expanded in i around 0 54.3%

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

      1. *-commutative 54.3%

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

   9. Simplified 54.3%

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

   if -1.5199999999999999e-180 < n < 1.5e-169

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

      2. associate-/r/ 71.3%

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

      3. associate-*l* 71.3%

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

      4. sub-neg 71.3%

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

      5. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in i around 0 79.5%

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

   5. Taylor expanded in i around 0 79.5%

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

   if 1.5e-169 < n < 1.9e8

   1. Initial program 16.2%

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

   2. Taylor expanded in i around 0 59.5%

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

   if 1.9e8 < n < 7.4999999999999999e82

   1. Initial program 46.9%

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

   2. Taylor expanded in i around 0 48.6%

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

      1. sub-neg 48.6%

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

      2. associate-*r/ 48.6%

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

      3. metadata-eval 48.6%

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

      4. distribute-neg-frac 48.6%

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

      5. metadata-eval 48.6%

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

   4. Simplified 48.6%

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

      1. flip-+ 91.3%

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

   6. Applied egg-rr 91.3%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot n - \left(i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right) \cdot \left(i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)}{n - i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{+194}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -1.52 \cdot 10^{-180}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-169}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 190000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{+82}:\\ \;\;\;\;100 \cdot \frac{n \cdot n - \left(i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right) \cdot \left(i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)\right)}{n - i \cdot \left(n \cdot \left(0.5 + \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]

Alternative 8: 65.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;n \leq -1.3 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (* 100.0 (+ n (* n (+ (* i 0.5) (* i (* i 0.16666666666666666))))))))
   (if (<= n -1.3e+194)
     t_0
     (if (<= n -5.5e-178)
       (/ (* n 100.0) (+ 1.0 (* i -0.5)))
       (if (<= n 9.5e-93) 0.0 t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.16666666666666666)))));
	double tmp;
	if (n <= -1.3e+194) {
		tmp = t_0;
	} else if (n <= -5.5e-178) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 9.5e-93) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n + (n * ((i * 0.5d0) + (i * (i * 0.16666666666666666d0)))))
    if (n <= (-1.3d+194)) then
        tmp = t_0
    else if (n <= (-5.5d-178)) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    else if (n <= 9.5d-93) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.16666666666666666)))));
	double tmp;
	if (n <= -1.3e+194) {
		tmp = t_0;
	} else if (n <= -5.5e-178) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 9.5e-93) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.16666666666666666)))))
	tmp = 0
	if n <= -1.3e+194:
		tmp = t_0
	elif n <= -5.5e-178:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	elif n <= 9.5e-93:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(i * 0.5) + Float64(i * Float64(i * 0.16666666666666666))))))
	tmp = 0.0
	if (n <= -1.3e+194)
		tmp = t_0;
	elseif (n <= -5.5e-178)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	elseif (n <= 9.5e-93)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.16666666666666666)))));
	tmp = 0.0;
	if (n <= -1.3e+194)
		tmp = t_0;
	elseif (n <= -5.5e-178)
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	elseif (n <= 9.5e-93)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n + N[(n * N[(N[(i * 0.5), $MachinePrecision] + N[(i * N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.3e+194], t$95$0, If[LessEqual[n, -5.5e-178], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e-93], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2999999999999999e194 or 9.5000000000000001e-93 < n

   1. Initial program 14.3%

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

   2. Taylor expanded in n around inf 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-/l* 40.1%

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

      3. expm1-def 91.7%

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

   4. Simplified 91.7%

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

   5. Taylor expanded in i around 0 74.1%

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

      1. +-commutative 74.1%

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

      2. associate-*r* 74.1%

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

      3. associate-*r* 74.1%

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

      4. distribute-rgt-out 74.5%

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

      5. *-commutative 74.5%

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

      6. unpow2 74.5%

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

      7. associate-*r* 74.5%

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

   7. Simplified 74.5%

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

   if -1.2999999999999999e194 < n < -5.50000000000000028e-178

   1. Initial program 42.3%

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

   2. Taylor expanded in n around inf 39.3%

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

      1. *-commutative 39.3%

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

      2. associate-/l* 39.3%

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

      3. expm1-def 79.9%

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

   4. Simplified 79.9%

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

      1. associate-*l/ 79.8%

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

   6. Applied egg-rr 79.8%

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

   7. Taylor expanded in i around 0 54.3%

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

      1. *-commutative 54.3%

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

   9. Simplified 54.3%

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

   if -5.50000000000000028e-178 < n < 9.5000000000000001e-93

   1. Initial program 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-/r/ 55.0%

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

      3. associate-*l* 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in i around 0 68.8%

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

   5. Taylor expanded in i around 0 68.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{+194}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]

Alternative 9: 64.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{i + 0.5 \cdot \left(i \cdot i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.5e-180)
   (/ (* n 100.0) (+ 1.0 (* i -0.5)))
   (if (<= n 1.15e-92) 0.0 (* 100.0 (/ n (/ i (+ i (* 0.5 (* i i)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.5e-180) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 1.15e-92) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n / (i / (i + (0.5 * (i * i)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.5d-180)) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    else if (n <= 1.15d-92) then
        tmp = 0.0d0
    else
        tmp = 100.0d0 * (n / (i / (i + (0.5d0 * (i * i)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.5e-180) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 1.15e-92) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n / (i / (i + (0.5 * (i * i)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.5e-180:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	elif n <= 1.15e-92:
		tmp = 0.0
	else:
		tmp = 100.0 * (n / (i / (i + (0.5 * (i * i)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.5e-180)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	elseif (n <= 1.15e-92)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n / Float64(i / Float64(i + Float64(0.5 * Float64(i * i))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.5e-180)
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	elseif (n <= 1.15e-92)
		tmp = 0.0;
	else
		tmp = 100.0 * (n / (i / (i + (0.5 * (i * i)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.5e-180], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-92], 0.0, N[(100.0 * N[(n / N[(i / N[(i + N[(0.5 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.5000000000000001e-180

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-/l* 44.4%

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

      3. expm1-def 86.4%

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

   4. Simplified 86.4%

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

      1. associate-*l/ 86.3%

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

   6. Applied egg-rr 86.3%

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

   7. Taylor expanded in i around 0 54.6%

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

      1. *-commutative 54.6%

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

   9. Simplified 54.6%

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

   if -2.5000000000000001e-180 < n < 1.15000000000000008e-92

   1. Initial program 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-/r/ 55.0%

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

      3. associate-*l* 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in i around 0 68.8%

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

   5. Taylor expanded in i around 0 68.8%

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

   if 1.15000000000000008e-92 < n

   1. Initial program 19.2%

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

   2. Taylor expanded in n around inf 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/l* 34.4%

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

      3. expm1-def 88.6%

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

   4. Simplified 88.6%

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

   5. Taylor expanded in i around 0 73.8%

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

      1. *-commutative 73.8%

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

      2. unpow2 73.8%

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

   7. Simplified 73.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{i + 0.5 \cdot \left(i \cdot i\right)}}\\ \end{array} \]

Alternative 10: 61.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-169}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 30:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -2.8e-115)
     t_0
     (if (<= n 1.5e-169) 0.0 (if (<= n 30.0) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double tmp;
	if (n <= -2.8e-115) {
		tmp = t_0;
	} else if (n <= 1.5e-169) {
		tmp = 0.0;
	} else if (n <= 30.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * ((i * n) / i)
    if (n <= (-2.8d-115)) then
        tmp = t_0
    else if (n <= 1.5d-169) then
        tmp = 0.0d0
    else if (n <= 30.0d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double tmp;
	if (n <= -2.8e-115) {
		tmp = t_0;
	} else if (n <= 1.5e-169) {
		tmp = 0.0;
	} else if (n <= 30.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((i * n) / i)
	tmp = 0
	if n <= -2.8e-115:
		tmp = t_0
	elif n <= 1.5e-169:
		tmp = 0.0
	elif n <= 30.0:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(i * n) / i))
	tmp = 0.0
	if (n <= -2.8e-115)
		tmp = t_0;
	elseif (n <= 1.5e-169)
		tmp = 0.0;
	elseif (n <= 30.0)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * ((i * n) / i);
	tmp = 0.0;
	if (n <= -2.8e-115)
		tmp = t_0;
	elseif (n <= 1.5e-169)
		tmp = 0.0;
	elseif (n <= 30.0)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.8e-115], t$95$0, If[LessEqual[n, 1.5e-169], 0.0, If[LessEqual[n, 30.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.79999999999999987e-115 or 30 < n

   1. Initial program 24.7%

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

      1. *-commutative 24.7%

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

      2. associate-/r/ 25.2%

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

      3. associate-*l* 25.2%

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

      4. sub-neg 25.2%

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

      5. metadata-eval 25.2%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in i around 0 15.3%

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

      1. unpow2 15.3%

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

      2. associate-*r/ 15.3%

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

      3. metadata-eval 15.3%

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

   6. Simplified 15.3%

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

   7. Taylor expanded in n around inf 65.5%

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

   8. Taylor expanded in i around 0 61.7%

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

      1. *-commutative 61.7%

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

   10. Simplified 61.7%

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

   if -2.79999999999999987e-115 < n < 1.5e-169

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-/r/ 66.1%

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

      3. associate-*l* 66.0%

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

      4. sub-neg 66.0%

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

      5. metadata-eval 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in i around 0 72.6%

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

   5. Taylor expanded in i around 0 72.6%

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

   if 1.5e-169 < n < 30

   1. Initial program 10.9%

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

   2. Taylor expanded in i around 0 62.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-115}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-169}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 30:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \]

Alternative 11: 61.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{-118} \lor \neg \left(n \leq 9.5 \cdot 10^{-93}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.6e-118) (not (<= n 9.5e-93)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.6e-118) || !(n <= 9.5e-93)) {
		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 <= (-1.6d-118)) .or. (.not. (n <= 9.5d-93))) 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 <= -1.6e-118) || !(n <= 9.5e-93)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.6e-118) or not (n <= 9.5e-93):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.6e-118) || !(n <= 9.5e-93))
		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 <= -1.6e-118) || ~((n <= 9.5e-93)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.6e-118], N[Not[LessEqual[n, 9.5e-93]], $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 < -1.60000000000000002e-118 or 9.5000000000000001e-93 < n

   1. Initial program 23.6%

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

   2. Taylor expanded in n around inf 39.9%

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

      1. *-commutative 39.9%

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

      2. associate-/l* 39.9%

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

      3. expm1-def 89.2%

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

   4. Simplified 89.2%

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

   5. Taylor expanded in i around 0 62.0%

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

      1. associate-*r* 62.0%

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

      2. distribute-rgt-out 62.0%

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

   7. Simplified 62.0%

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

   if -1.60000000000000002e-118 < n < 9.5000000000000001e-93

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

      2. associate-/r/ 53.5%

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

      3. associate-*l* 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in i around 0 65.2%

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

   5. Taylor expanded in i around 0 65.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{-118} \lor \neg \left(n \leq 9.5 \cdot 10^{-93}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 62.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{-179}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.7e-179)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 9.5e-93) 0.0 (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.7e-179) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 9.5e-93) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.7d-179)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 9.5d-93) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.7e-179) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 9.5e-93) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.7e-179:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 9.5e-93:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.7e-179)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 9.5e-93)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.7e-179)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 9.5e-93)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.7e-179], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e-93], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.7000000000000003e-179

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-/l* 44.4%

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

      3. expm1-def 86.4%

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

   4. Simplified 86.4%

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

   5. Taylor expanded in i around 0 54.6%

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

      1. *-commutative 54.6%

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

   7. Simplified 54.6%

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

   if -4.7000000000000003e-179 < n < 9.5000000000000001e-93

   1. Initial program 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-/r/ 55.0%

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

      3. associate-*l* 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in i around 0 68.8%

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

   5. Taylor expanded in i around 0 68.8%

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

   if 9.5000000000000001e-93 < n

   1. Initial program 19.2%

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

   2. Taylor expanded in n around inf 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/l* 34.4%

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

      3. expm1-def 88.6%

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

   4. Simplified 88.6%

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

   5. Taylor expanded in i around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. distribute-rgt-out 71.0%

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

   7. Simplified 71.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{-179}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13: 62.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.8e-180)
   (/ (* n 100.0) (+ 1.0 (* i -0.5)))
   (if (<= n 9.5e-93) 0.0 (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.8e-180) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 9.5e-93) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.8d-180)) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    else if (n <= 9.5d-93) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.8e-180) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 9.5e-93) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.8e-180:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	elif n <= 9.5e-93:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.8e-180)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	elseif (n <= 9.5e-93)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -5.8e-180)
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	elseif (n <= 9.5e-93)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.8e-180], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e-93], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.79999999999999961e-180

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-/l* 44.4%

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

      3. expm1-def 86.4%

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

   4. Simplified 86.4%

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

      1. associate-*l/ 86.3%

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

   6. Applied egg-rr 86.3%

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

   7. Taylor expanded in i around 0 54.6%

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

      1. *-commutative 54.6%

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

   9. Simplified 54.6%

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

   if -5.79999999999999961e-180 < n < 9.5000000000000001e-93

   1. Initial program 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-/r/ 55.0%

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

      3. associate-*l* 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in i around 0 68.8%

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

   5. Taylor expanded in i around 0 68.8%

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

   if 9.5000000000000001e-93 < n

   1. Initial program 19.2%

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

   2. Taylor expanded in n around inf 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-/l* 34.4%

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

      3. expm1-def 88.6%

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

   4. Simplified 88.6%

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

   5. Taylor expanded in i around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. distribute-rgt-out 71.0%

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

   7. Simplified 71.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 14: 60.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -6.3:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 170:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -6.3) 0.0 (if (<= i 170.0) (* n 100.0) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -6.3) {
		tmp = 0.0;
	} else if (i <= 170.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-6.3d0)) then
        tmp = 0.0d0
    else if (i <= 170.0d0) then
        tmp = n * 100.0d0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -6.3) {
		tmp = 0.0;
	} else if (i <= 170.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -6.3:
		tmp = 0.0
	elif i <= 170.0:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -6.3)
		tmp = 0.0;
	elseif (i <= 170.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -6.3)
		tmp = 0.0;
	elseif (i <= 170.0)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -6.3], 0.0, If[LessEqual[i, 170.0], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -6.29999999999999982

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-/r/ 61.9%

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

      3. associate-*l* 61.8%

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

      4. sub-neg 61.8%

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

      5. metadata-eval 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in i around 0 25.0%

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

   5. Taylor expanded in i around 0 25.0%

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

   if -6.29999999999999982 < i < 170

   1. Initial program 12.0%

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

   2. Taylor expanded in i around 0 80.4%

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

      1. *-commutative 80.4%

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

   4. Simplified 80.4%

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

   if 170 < i

   1. Initial program 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-/r/ 44.5%

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

      3. associate-*l* 44.5%

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

      4. sub-neg 44.5%

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

      5. metadata-eval 44.5%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in i around 0 34.7%

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

      1. unpow2 34.7%

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

      2. associate-*r/ 34.7%

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

      3. metadata-eval 34.7%

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

   6. Simplified 34.7%

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

   7. Taylor expanded in n around inf 41.5%

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

   8. Taylor expanded in i around inf 32.0%

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

      1. *-commutative 32.0%

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

   10. Simplified 32.0%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.3:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 170:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 15: 55.6% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-123}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.2e-123) (* n 100.0) (if (<= n 9.5e-93) 0.0 (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.2e-123) {
		tmp = n * 100.0;
	} else if (n <= 9.5e-93) {
		tmp = 0.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-6.2d-123)) then
        tmp = n * 100.0d0
    else if (n <= 9.5d-93) then
        tmp = 0.0d0
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -6.2e-123) {
		tmp = n * 100.0;
	} else if (n <= 9.5e-93) {
		tmp = 0.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -6.2e-123:
		tmp = n * 100.0
	elif n <= 9.5e-93:
		tmp = 0.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.2e-123)
		tmp = Float64(n * 100.0);
	elseif (n <= 9.5e-93)
		tmp = 0.0;
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -6.2e-123)
		tmp = n * 100.0;
	elseif (n <= 9.5e-93)
		tmp = 0.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.2e-123], N[(n * 100.0), $MachinePrecision], If[LessEqual[n, 9.5e-93], 0.0, N[(n * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.19999999999999996e-123 or 9.5000000000000001e-93 < n

   1. Initial program 23.6%

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

   2. Taylor expanded in i around 0 54.3%

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

      1. *-commutative 54.3%

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

   4. Simplified 54.3%

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

   if -6.19999999999999996e-123 < n < 9.5000000000000001e-93

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

      2. associate-/r/ 53.5%

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

      3. associate-*l* 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in i around 0 65.2%

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

   5. Taylor expanded in i around 0 65.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-123}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 16: 17.7% 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 30.4%

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

   1. *-commutative 30.4%

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

   2. associate-/r/ 30.8%

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

   3. associate-*l* 30.8%

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

   4. sub-neg 30.8%

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

   5. metadata-eval 30.8%

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

3. Simplified 30.8%

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

4. Taylor expanded in i around 0 17.9%

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

5. Taylor expanded in i around 0 18.0%

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

6. Final simplification 18.0%

   \[\leadsto 0 \]

Developer target: 34.1% 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 2023272 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

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

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