Compound Interest

Percentage Accurate: 28.5% → 98.1%

Time: 21.4s

Alternatives: 18

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.5% 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: 98.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.2%

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

      1. sub-neg 24.2%

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

      2. metadata-eval 24.2%

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

   4. Applied egg-rr 24.2%

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

      1. metadata-eval 24.2%

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

      2. sub-neg 24.2%

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

      3. exp-to-pow 22.4%

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

      4. log1p-undefine 43.2%

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

      5. *-commutative 43.2%

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

      6. expm1-undefine 97.8%

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

   6. Simplified 97.8%

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

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

   1. Initial program 99.8%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-/r/ 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-undefine 99.7%

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

      9. clear-num 99.7%

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

      10. associate-*l/ 99.8%

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

      11. *-un-lft-identity 99.8%

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

      12. fma-undefine 99.8%

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

      13. *-commutative 99.8%

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

      14. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around 0 15.6%

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

      1. +-commutative 15.6%

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

   7. Simplified 15.6%

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

      1. expm1-log1p-u 15.6%

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

      2. expm1-undefine 15.6%

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

      3. associate-/r/ 17.3%

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

      4. *-commutative 17.3%

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

      5. *-commutative 17.3%

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

      6. fma-define 17.3%

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

   9. Applied egg-rr 17.3%

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

       1. expm1-define 2.8%

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

       2. associate-*r/ 2.8%

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

       3. associate-*l/ 15.6%

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

       4. fma-undefine 15.6%

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

       5. *-commutative 15.6%

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

       6. distribute-lft1-in 15.6%

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

       7. associate-+l+ 18.2%

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

       8. metadata-eval 18.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{i} \cdot \left(i \cdot 100 + \color{blue}{0}\right)\right)\right) \]

       9. +-rgt-identity 18.2%

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

       10. *-commutative 18.2%

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

       11. expm1-define 18.2%

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

       12. log1p-undefine 18.2%

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

       13. rem-exp-log 21.0%

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

       14. associate-+r- 21.0%

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

       15. *-commutative 21.0%

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

       16. associate-*r* 21.0%

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

       17. *-commutative 21.0%

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

       18. fmm-def 21.0%

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

   11. Simplified 100.0%

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

4. Final simplification 98.6%

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

Alternative 2: 83.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* 100.0 (* n (/ (expm1 i) i)))
     (if (<= t_0 INFINITY) (* t_0 100.0) (+ 1.0 (fma 100.0 n -1.0))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 1.0 + fma(100.0, n, -1.0);
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/l* 38.9%

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

      3. expm1-define 76.0%

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

   5. Simplified 76.0%

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

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around 0 15.6%

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

      1. +-commutative 15.6%

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

   7. Simplified 15.6%

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

      1. expm1-log1p-u 15.6%

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

      2. expm1-undefine 15.6%

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

      3. associate-/r/ 17.3%

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

      4. *-commutative 17.3%

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

      5. *-commutative 17.3%

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

      6. fma-define 17.3%

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

   9. Applied egg-rr 17.3%

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

       1. expm1-define 2.8%

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

       2. associate-*r/ 2.8%

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

       3. associate-*l/ 15.6%

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

       4. fma-undefine 15.6%

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

       5. *-commutative 15.6%

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

       6. distribute-lft1-in 15.6%

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

       7. associate-+l+ 18.2%

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

       8. metadata-eval 18.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{i} \cdot \left(i \cdot 100 + \color{blue}{0}\right)\right)\right) \]

       9. +-rgt-identity 18.2%

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

       10. *-commutative 18.2%

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

       11. expm1-define 18.2%

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

       12. log1p-undefine 18.2%

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

       13. rem-exp-log 21.0%

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

       14. associate-+r- 21.0%

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

       15. *-commutative 21.0%

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

       16. associate-*r* 21.0%

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

       17. *-commutative 21.0%

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

       18. fmm-def 21.0%

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

   11. Simplified 100.0%

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

4. Final simplification 84.3%

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

Alternative 3: 97.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* 100.0 (* n (/ (expm1 (* n (log1p (/ i n)))) i)))
     (if (<= t_0 INFINITY) (* t_0 100.0) (+ 1.0 (fma 100.0 n -1.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.2%

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

      1. associate-/r/ 23.4%

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

      2. add-exp-log 23.4%

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

      3. expm1-define 23.4%

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

      4. log-pow 36.0%

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

      5. log1p-define 95.3%

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

   4. Applied egg-rr 95.3%

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

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around 0 15.6%

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

      1. +-commutative 15.6%

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

   7. Simplified 15.6%

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

      1. expm1-log1p-u 15.6%

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

      2. expm1-undefine 15.6%

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

      3. associate-/r/ 17.3%

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

      4. *-commutative 17.3%

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

      5. *-commutative 17.3%

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

      6. fma-define 17.3%

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

   9. Applied egg-rr 17.3%

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

       1. expm1-define 2.8%

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

       2. associate-*r/ 2.8%

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

       3. associate-*l/ 15.6%

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

       4. fma-undefine 15.6%

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

       5. *-commutative 15.6%

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

       6. distribute-lft1-in 15.6%

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

       7. associate-+l+ 18.2%

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

       8. metadata-eval 18.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{i} \cdot \left(i \cdot 100 + \color{blue}{0}\right)\right)\right) \]

       9. +-rgt-identity 18.2%

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

       10. *-commutative 18.2%

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

       11. expm1-define 18.2%

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

       12. log1p-undefine 18.2%

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

       13. rem-exp-log 21.0%

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

       14. associate-+r- 21.0%

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

       15. *-commutative 21.0%

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

       16. associate-*r* 21.0%

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

       17. *-commutative 21.0%

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

       18. fmm-def 21.0%

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

   11. Simplified 100.0%

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

4. Final simplification 96.9%

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

Alternative 4: 98.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_0 INFINITY) (* t_0 100.0) (+ 1.0 (fma 100.0 n -1.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.2%

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

      1. sub-neg 24.2%

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

      2. metadata-eval 24.2%

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

   4. Applied egg-rr 24.2%

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

      1. metadata-eval 24.2%

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

      2. sub-neg 24.2%

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

      3. exp-to-pow 22.4%

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

      4. log1p-undefine 43.2%

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

      5. *-commutative 43.2%

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

      6. expm1-undefine 97.8%

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

   6. Simplified 97.8%

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

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-rgt-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around 0 15.6%

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

      1. +-commutative 15.6%

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

   7. Simplified 15.6%

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

      1. expm1-log1p-u 15.6%

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

      2. expm1-undefine 15.6%

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

      3. associate-/r/ 17.3%

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

      4. *-commutative 17.3%

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

      5. *-commutative 17.3%

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

      6. fma-define 17.3%

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

   9. Applied egg-rr 17.3%

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

       1. expm1-define 2.8%

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

       2. associate-*r/ 2.8%

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

       3. associate-*l/ 15.6%

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

       4. fma-undefine 15.6%

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

       5. *-commutative 15.6%

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

       6. distribute-lft1-in 15.6%

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

       7. associate-+l+ 18.2%

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

       8. metadata-eval 18.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{i} \cdot \left(i \cdot 100 + \color{blue}{0}\right)\right)\right) \]

       9. +-rgt-identity 18.2%

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

       10. *-commutative 18.2%

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

       11. expm1-define 18.2%

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

       12. log1p-undefine 18.2%

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

       13. rem-exp-log 21.0%

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

       14. associate-+r- 21.0%

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

       15. *-commutative 21.0%

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

       16. associate-*r* 21.0%

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

       17. *-commutative 21.0%

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

       18. fmm-def 21.0%

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

   11. Simplified 100.0%

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

4. Final simplification 98.6%

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

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{-223} \lor \neg \left(n \leq 1.2 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.8e-223) (not (<= n 1.2e-123)))
   (* n (* (expm1 i) (/ 100.0 i)))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.8e-223) || !(n <= 1.2e-123)) {
		tmp = n * (expm1(i) * (100.0 / i));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.8e-223) || !(n <= 1.2e-123)) {
		tmp = n * (Math.expm1(i) * (100.0 / i));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.8e-223) or not (n <= 1.2e-123):
		tmp = n * (math.expm1(i) * (100.0 / i))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.8e-223) || !(n <= 1.2e-123))
		tmp = Float64(n * Float64(expm1(i) * Float64(100.0 / i)));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.8e-223], N[Not[LessEqual[n, 1.2e-123]], $MachinePrecision]], N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.8000000000000002e-223 or 1.2e-123 < n

   1. Initial program 21.8%

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

   3. Taylor expanded in n around inf 30.7%

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

      1. *-commutative 30.7%

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

      2. associate-/l* 30.7%

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

      3. expm1-define 84.5%

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

   5. Simplified 84.5%

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

   6. Taylor expanded in n around 0 30.7%

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

      1. associate-*r/ 30.3%

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

      2. expm1-define 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-*r* 79.5%

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

      5. associate-/l* 84.5%

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

      6. associate-*r/ 83.6%

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

   8. Simplified 83.6%

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

   if -1.8000000000000002e-223 < n < 1.2e-123

   1. Initial program 46.8%

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

      1. associate-*r/ 46.8%

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

      2. sub-neg 46.8%

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

      3. distribute-rgt-in 46.8%

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

      4. metadata-eval 46.8%

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

      5. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in i around 0 60.6%

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

      1. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{-223} \lor \neg \left(n \leq 1.2 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-222} \lor \neg \left(n \leq 1.2 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{i \cdot 0.01}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.6e-222) (not (<= n 1.2e-123)))
   (* n (/ (expm1 i) (* i 0.01)))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.6e-222) || !(n <= 1.2e-123)) {
		tmp = n * (expm1(i) / (i * 0.01));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.6e-222) || !(n <= 1.2e-123)) {
		tmp = n * (Math.expm1(i) / (i * 0.01));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.6e-222) or not (n <= 1.2e-123):
		tmp = n * (math.expm1(i) / (i * 0.01))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.6e-222) || !(n <= 1.2e-123))
		tmp = Float64(n * Float64(expm1(i) / Float64(i * 0.01)));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.6e-222], N[Not[LessEqual[n, 1.2e-123]], $MachinePrecision]], N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i * 0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.5999999999999998e-222 or 1.2e-123 < n

   1. Initial program 21.8%

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

   3. Taylor expanded in n around inf 30.7%

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

      1. *-commutative 30.7%

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

      2. associate-/l* 30.7%

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

      3. expm1-define 84.5%

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

   5. Simplified 84.5%

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

   6. Taylor expanded in n around 0 30.7%

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

      1. associate-*r/ 30.3%

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

      2. expm1-define 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-*r* 79.5%

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

      5. associate-/l* 84.5%

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

      6. associate-*r/ 83.6%

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

   8. Simplified 83.6%

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

      1. clear-num 83.4%

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

      2. un-div-inv 84.4%

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

      3. div-inv 84.4%

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

      4. metadata-eval 84.4%

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

   10. Applied egg-rr 84.4%

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

   if -2.5999999999999998e-222 < n < 1.2e-123

   1. Initial program 46.8%

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

      1. associate-*r/ 46.8%

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

      2. sub-neg 46.8%

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

      3. distribute-rgt-in 46.8%

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

      4. metadata-eval 46.8%

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

      5. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in i around 0 60.6%

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

      1. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-222} \lor \neg \left(n \leq 1.2 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{i \cdot 0.01}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-222} \lor \neg \left(n \leq 1.2 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-222) (not (<= n 1.2e-123)))
   (* n (/ (* 100.0 (expm1 i)) i))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-222) || !(n <= 1.2e-123)) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-222) || !(n <= 1.2e-123)) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-222) or not (n <= 1.2e-123):
		tmp = n * ((100.0 * math.expm1(i)) / i)
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-222) || !(n <= 1.2e-123))
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-222], N[Not[LessEqual[n, 1.2e-123]], $MachinePrecision]], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.00000000000000008e-222 or 1.2e-123 < n

   1. Initial program 21.8%

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

      1. associate-/r/ 22.0%

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

      2. associate-*r* 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-*r/ 22.1%

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

      5. sub-neg 22.1%

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

      6. distribute-lft-in 22.1%

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

      7. metadata-eval 22.1%

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

      8. metadata-eval 22.1%

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

      9. metadata-eval 22.1%

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

      10. fma-define 22.1%

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

      11. metadata-eval 22.1%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in n around inf 30.8%

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

      1. sub-neg 30.8%

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

      2. metadata-eval 30.8%

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

      3. metadata-eval 30.8%

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

      4. distribute-lft-in 30.8%

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

      5. metadata-eval 30.8%

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

      6. sub-neg 30.8%

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

      7. expm1-define 84.5%

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

   7. Simplified 84.5%

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

   if -5.00000000000000008e-222 < n < 1.2e-123

   1. Initial program 46.8%

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

      1. associate-*r/ 46.8%

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

      2. sub-neg 46.8%

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

      3. distribute-rgt-in 46.8%

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

      4. metadata-eval 46.8%

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

      5. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in i around 0 60.6%

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

      1. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in i around 0 70.5%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-222} \lor \neg \left(n \leq 1.2 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-220}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.4e-220)
   (* n (/ (* 100.0 (expm1 i)) i))
   (if (<= n 1.2e-123) (/ 0.0 (/ i n)) (* 100.0 (* n (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.4e-220) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else if (n <= 1.2e-123) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n * (expm1(i) / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.4e-220) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else if (n <= 1.2e-123) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.4e-220:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	elif n <= 1.2e-123:
		tmp = 0.0 / (i / n)
	else:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.4e-220)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	elseif (n <= 1.2e-123)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.4e-220], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-123], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.4e-220

   1. Initial program 24.8%

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

      1. associate-/r/ 24.7%

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

      2. associate-*r* 24.8%

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

      3. *-commutative 24.8%

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

      4. associate-*r/ 24.8%

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

      5. sub-neg 24.8%

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

      6. distribute-lft-in 24.8%

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

      7. metadata-eval 24.8%

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

      8. metadata-eval 24.8%

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

      9. metadata-eval 24.8%

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

      10. fma-define 24.8%

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

      11. metadata-eval 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in n around inf 29.7%

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

      1. sub-neg 29.7%

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

      2. metadata-eval 29.7%

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

      3. metadata-eval 29.7%

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

      4. distribute-lft-in 29.7%

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

      5. metadata-eval 29.7%

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

      6. sub-neg 29.7%

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

      7. expm1-define 81.0%

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

   7. Simplified 81.0%

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

   if -1.4e-220 < n < 1.2e-123

   1. Initial program 46.8%

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

      1. associate-*r/ 46.8%

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

      2. sub-neg 46.8%

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

      3. distribute-rgt-in 46.8%

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

      4. metadata-eval 46.8%

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

      5. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in i around 0 60.6%

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

      1. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in i around 0 70.5%

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

   if 1.2e-123 < n

   1. Initial program 18.5%

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

   3. Taylor expanded in n around inf 32.1%

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

      1. *-commutative 32.1%

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

      2. associate-/l* 32.1%

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

      3. expm1-define 88.6%

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

   5. Simplified 88.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-220}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{-150}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(i \cdot 0.16666666666666666 + 0.5\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;1 + \mathsf{fma}\left(100, n, -1\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.3e-150)
   (* 100.0 (+ n (* i (* n (+ (* i 0.16666666666666666) 0.5)))))
   (if (<= n 1.2e-123)
     (+ 1.0 (fma 100.0 n -1.0))
     (*
      n
      (+
       100.0
       (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.3e-150) {
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))));
	} else if (n <= 1.2e-123) {
		tmp = 1.0 + fma(100.0, n, -1.0);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.3e-150)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(Float64(i * 0.16666666666666666) + 0.5)))));
	elseif (n <= 1.2e-123)
		tmp = Float64(1.0 + fma(100.0, n, -1.0));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.3e-150], N[(100.0 * N[(n + N[(i * N[(n * N[(N[(i * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-123], N[(1.0 + N[(100.0 * n + -1.0), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.30000000000000004e-150

   1. Initial program 21.9%

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

   3. Taylor expanded in n around inf 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-/l* 28.6%

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

      3. expm1-define 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in i around 0 62.2%

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

      1. associate-*r* 62.2%

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

      2. distribute-rgt-out 62.2%

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

   8. Simplified 62.2%

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

   if -4.30000000000000004e-150 < n < 1.2e-123

   1. Initial program 49.1%

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

      1. associate-*r/ 49.1%

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

      2. sub-neg 49.1%

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

      3. distribute-rgt-in 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in i around 0 58.3%

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

      1. +-commutative 58.3%

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

   7. Simplified 58.3%

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

      1. expm1-log1p-u 58.3%

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

      2. expm1-undefine 68.3%

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

      3. associate-/r/ 68.3%

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

      4. *-commutative 68.3%

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

      5. *-commutative 68.3%

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

      6. fma-define 68.3%

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

   9. Applied egg-rr 68.3%

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

       1. expm1-define 20.6%

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

       2. associate-*r/ 20.6%

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

       3. associate-*l/ 52.3%

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

       4. fma-undefine 52.3%

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

       5. *-commutative 52.3%

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

       6. distribute-lft1-in 52.3%

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

       7. associate-+l+ 51.2%

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

       8. metadata-eval 51.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{i} \cdot \left(i \cdot 100 + \color{blue}{0}\right)\right)\right) \]

       9. +-rgt-identity 51.2%

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

       10. *-commutative 51.2%

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

       11. expm1-define 68.3%

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

       12. log1p-undefine 68.3%

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

       13. rem-exp-log 68.3%

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

       14. associate-+r- 68.3%

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

       15. *-commutative 68.3%

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

       16. associate-*r* 68.3%

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

       17. *-commutative 68.3%

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

       18. fmm-def 68.3%

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

   11. Simplified 68.3%

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

   if 1.2e-123 < n

   1. Initial program 18.5%

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

      1. associate-/r/ 18.9%

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

      2. associate-*r* 18.9%

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

      3. *-commutative 18.9%

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

      4. associate-*r/ 18.9%

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

      5. sub-neg 18.9%

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

      6. distribute-lft-in 18.9%

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

      7. metadata-eval 18.9%

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

      8. metadata-eval 18.9%

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

      9. metadata-eval 18.9%

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

      10. fma-define 18.9%

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

      11. metadata-eval 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in n around inf 32.1%

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

      1. sub-neg 32.1%

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

      2. metadata-eval 32.1%

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

      3. metadata-eval 32.1%

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

      4. distribute-lft-in 32.1%

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

      5. metadata-eval 32.1%

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

      6. sub-neg 32.1%

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

      7. expm1-define 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in i around 0 76.3%

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

      1. *-commutative 76.3%

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

   10. Simplified 76.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{-150}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(i \cdot 0.16666666666666666 + 0.5\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;1 + \mathsf{fma}\left(100, n, -1\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-153}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(i \cdot 0.16666666666666666 + 0.5\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8e-153)
   (* 100.0 (+ n (* i (* n (+ (* i 0.16666666666666666) 0.5)))))
   (if (<= n 1.2e-123)
     (/ 0.0 (/ i n))
     (*
      n
      (+
       100.0
       (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8e-153) {
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))));
	} else if (n <= 1.2e-123) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-8d-153)) then
        tmp = 100.0d0 * (n + (i * (n * ((i * 0.16666666666666666d0) + 0.5d0))))
    else if (n <= 1.2d-123) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8e-153) {
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))));
	} else if (n <= 1.2e-123) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8e-153:
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))))
	elif n <= 1.2e-123:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8e-153)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(Float64(i * 0.16666666666666666) + 0.5)))));
	elseif (n <= 1.2e-123)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -8e-153)
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))));
	elseif (n <= 1.2e-123)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8e-153], N[(100.0 * N[(n + N[(i * N[(n * N[(N[(i * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-123], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.00000000000000031e-153

   1. Initial program 21.9%

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

   3. Taylor expanded in n around inf 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-/l* 28.6%

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

      3. expm1-define 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in i around 0 62.2%

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

      1. associate-*r* 62.2%

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

      2. distribute-rgt-out 62.2%

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

   8. Simplified 62.2%

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

   if -8.00000000000000031e-153 < n < 1.2e-123

   1. Initial program 49.1%

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

      1. associate-*r/ 49.1%

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

      2. sub-neg 49.1%

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

      3. distribute-rgt-in 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in i around 0 58.3%

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

      1. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in i around 0 68.3%

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

   if 1.2e-123 < n

   1. Initial program 18.5%

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

      1. associate-/r/ 18.9%

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

      2. associate-*r* 18.9%

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

      3. *-commutative 18.9%

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

      4. associate-*r/ 18.9%

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

      5. sub-neg 18.9%

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

      6. distribute-lft-in 18.9%

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

      7. metadata-eval 18.9%

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

      8. metadata-eval 18.9%

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

      9. metadata-eval 18.9%

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

      10. fma-define 18.9%

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

      11. metadata-eval 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in n around inf 32.1%

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

      1. sub-neg 32.1%

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

      2. metadata-eval 32.1%

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

      3. metadata-eval 32.1%

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

      4. distribute-lft-in 32.1%

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

      5. metadata-eval 32.1%

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

      6. sub-neg 32.1%

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

      7. expm1-define 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in i around 0 76.3%

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

      1. *-commutative 76.3%

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

   10. Simplified 76.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-153}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(i \cdot 0.16666666666666666 + 0.5\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{-159} \lor \neg \left(n \leq 1.2 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -6.6e-159) (not (<= n 1.2e-123)))
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -6.6e-159) || !(n <= 1.2e-123)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-6.6d-159)) .or. (.not. (n <= 1.2d-123))) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else
        tmp = 0.0d0 / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -6.6e-159) || !(n <= 1.2e-123)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -6.6e-159) or not (n <= 1.2e-123):
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -6.6e-159) || !(n <= 1.2e-123))
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -6.6e-159) || ~((n <= 1.2e-123)))
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	else
		tmp = 0.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -6.6e-159], N[Not[LessEqual[n, 1.2e-123]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.6000000000000003e-159 or 1.2e-123 < n

   1. Initial program 20.2%

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

      1. associate-/r/ 20.7%

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

      2. associate-*r* 20.8%

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

      3. *-commutative 20.8%

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

      4. associate-*r/ 20.8%

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

      5. sub-neg 20.8%

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

      6. distribute-lft-in 20.8%

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

      7. metadata-eval 20.8%

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

      8. metadata-eval 20.8%

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

      9. metadata-eval 20.8%

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

      10. fma-define 20.8%

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

      11. metadata-eval 20.8%

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

   3. Simplified 20.8%

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

   5. Taylor expanded in n around inf 30.3%

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

      1. sub-neg 30.3%

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

      2. metadata-eval 30.3%

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

      3. metadata-eval 30.3%

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

      4. distribute-lft-in 30.3%

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

      5. metadata-eval 30.3%

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

      6. sub-neg 30.3%

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

      7. expm1-define 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in i around 0 67.4%

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

      1. *-commutative 67.4%

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

   10. Simplified 67.4%

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

   if -6.6000000000000003e-159 < n < 1.2e-123

   1. Initial program 49.1%

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

      1. associate-*r/ 49.1%

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

      2. sub-neg 49.1%

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

      3. distribute-rgt-in 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in i around 0 58.3%

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

      1. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in i around 0 68.3%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{-159} \lor \neg \left(n \leq 1.2 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-159}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(i \cdot 0.16666666666666666 + 0.5\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e-159)
   (* 100.0 (+ n (* i (* n (+ (* i 0.16666666666666666) 0.5)))))
   (if (<= n 1.2e-123)
     (/ 0.0 (/ i n))
     (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e-159) {
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))));
	} else if (n <= 1.2e-123) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.8d-159)) then
        tmp = 100.0d0 * (n + (i * (n * ((i * 0.16666666666666666d0) + 0.5d0))))
    else if (n <= 1.2d-123) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.8e-159) {
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))));
	} else if (n <= 1.2e-123) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.8e-159:
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))))
	elif n <= 1.2e-123:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e-159)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(Float64(i * 0.16666666666666666) + 0.5)))));
	elseif (n <= 1.2e-123)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.8e-159)
		tmp = 100.0 * (n + (i * (n * ((i * 0.16666666666666666) + 0.5))));
	elseif (n <= 1.2e-123)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e-159], N[(100.0 * N[(n + N[(i * N[(n * N[(N[(i * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-123], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.8000000000000002e-159

   1. Initial program 21.9%

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

   3. Taylor expanded in n around inf 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-/l* 28.6%

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

      3. expm1-define 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in i around 0 62.2%

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

      1. associate-*r* 62.2%

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

      2. distribute-rgt-out 62.2%

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

   8. Simplified 62.2%

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

   if -2.8000000000000002e-159 < n < 1.2e-123

   1. Initial program 49.1%

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

      1. associate-*r/ 49.1%

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

      2. sub-neg 49.1%

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

      3. distribute-rgt-in 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in i around 0 58.3%

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

      1. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in i around 0 68.3%

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

   if 1.2e-123 < n

   1. Initial program 18.5%

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

      1. associate-/r/ 18.9%

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

      2. associate-*r* 18.9%

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

      3. *-commutative 18.9%

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

      4. associate-*r/ 18.9%

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

      5. sub-neg 18.9%

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

      6. distribute-lft-in 18.9%

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

      7. metadata-eval 18.9%

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

      8. metadata-eval 18.9%

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

      9. metadata-eval 18.9%

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

      10. fma-define 18.9%

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

      11. metadata-eval 18.9%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in n around inf 32.1%

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

      1. sub-neg 32.1%

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

      2. metadata-eval 32.1%

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

      3. metadata-eval 32.1%

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

      4. distribute-lft-in 32.1%

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

      5. metadata-eval 32.1%

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

      6. sub-neg 32.1%

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

      7. expm1-define 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in i around 0 72.8%

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

      1. *-commutative 72.8%

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

   10. Simplified 72.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-159}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(i \cdot 0.16666666666666666 + 0.5\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{+31} \lor \neg \left(n \leq 9.5 \cdot 10^{-26}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.9e+31) (not (<= n 9.5e-26)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.9e+31) || !(n <= 9.5e-26)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-5.9d+31)) .or. (.not. (n <= 9.5d-26))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5.9e+31) || !(n <= 9.5e-26)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5.9e+31) or not (n <= 9.5e-26):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.9e+31) || !(n <= 9.5e-26))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -5.9e+31) || ~((n <= 9.5e-26)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.9e+31], N[Not[LessEqual[n, 9.5e-26]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.9000000000000004e31 or 9.4999999999999995e-26 < n

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf 36.0%

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

      1. *-commutative 36.0%

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

      2. associate-/l* 36.0%

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

      3. expm1-define 93.3%

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

   5. Simplified 93.3%

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

   6. Taylor expanded in i around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. associate-*r* 68.2%

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

      3. distribute-rgt-out 68.2%

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

      4. *-commutative 68.2%

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

   8. Simplified 68.2%

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

   if -5.9000000000000004e31 < n < 9.4999999999999995e-26

   1. Initial program 34.0%

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

   3. Taylor expanded in i around 0 55.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{+31} \lor \neg \left(n \leq 9.5 \cdot 10^{-26}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-153} \lor \neg \left(n \leq 1.45 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.9e-153) (not (<= n 1.45e-123)))
   (* n (+ 100.0 (* i 50.0)))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e-153) || !(n <= 1.45e-123)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.9d-153)) .or. (.not. (n <= 1.45d-123))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0 / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e-153) || !(n <= 1.45e-123)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.9e-153) or not (n <= 1.45e-123):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.9e-153) || !(n <= 1.45e-123))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.9e-153) || ~((n <= 1.45e-123)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.9e-153], N[Not[LessEqual[n, 1.45e-123]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.90000000000000011e-153 or 1.45000000000000002e-123 < n

   1. Initial program 20.2%

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

   3. Taylor expanded in n around inf 30.3%

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

      1. *-commutative 30.3%

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

      2. associate-/l* 30.3%

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

      3. expm1-define 85.5%

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

   5. Simplified 85.5%

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

   6. Taylor expanded in i around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. associate-*r* 64.6%

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

      3. distribute-rgt-out 64.6%

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

      4. *-commutative 64.6%

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

   8. Simplified 64.6%

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

   if -1.90000000000000011e-153 < n < 1.45000000000000002e-123

   1. Initial program 49.1%

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

      1. associate-*r/ 49.1%

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

      2. sub-neg 49.1%

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

      3. distribute-rgt-in 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in i around 0 58.3%

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

      1. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in i around 0 68.3%

      \[\leadsto \frac{\color{blue}{100} + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-153} \lor \neg \left(n \leq 1.45 \cdot 10^{-123}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.42 \cdot 10^{-158}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.42e-158)
   (* 100.0 (+ n (* 0.5 (* i n))))
   (if (<= n 1.2e-123) (/ 0.0 (/ i n)) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.42e-158) {
		tmp = 100.0 * (n + (0.5 * (i * n)));
	} else if (n <= 1.2e-123) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.42d-158)) then
        tmp = 100.0d0 * (n + (0.5d0 * (i * n)))
    else if (n <= 1.2d-123) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.42e-158) {
		tmp = 100.0 * (n + (0.5 * (i * n)));
	} else if (n <= 1.2e-123) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.42e-158:
		tmp = 100.0 * (n + (0.5 * (i * n)))
	elif n <= 1.2e-123:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.42e-158)
		tmp = Float64(100.0 * Float64(n + Float64(0.5 * Float64(i * n))));
	elseif (n <= 1.2e-123)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.42e-158)
		tmp = 100.0 * (n + (0.5 * (i * n)));
	elseif (n <= 1.2e-123)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.42e-158], N[(100.0 * N[(n + N[(0.5 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-123], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.42000000000000005e-158

   1. Initial program 21.9%

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

   3. Taylor expanded in n around inf 28.6%

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

      1. *-commutative 28.6%

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

      2. associate-/l* 28.6%

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

      3. expm1-define 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in i around 0 60.3%

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

      1. *-commutative 60.3%

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

      2. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   if -1.42000000000000005e-158 < n < 1.2e-123

   1. Initial program 49.1%

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

      1. associate-*r/ 49.1%

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

      2. sub-neg 49.1%

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

      3. distribute-rgt-in 49.1%

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

      4. metadata-eval 49.1%

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

      5. metadata-eval 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in i around 0 58.3%

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

      1. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in i around 0 68.3%

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

   if 1.2e-123 < n

   1. Initial program 18.5%

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

   3. Taylor expanded in n around inf 32.1%

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

      1. *-commutative 32.1%

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

      2. associate-/l* 32.1%

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

      3. expm1-define 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in i around 0 69.0%

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

      1. +-commutative 69.0%

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

      2. associate-*r* 69.0%

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

      3. distribute-rgt-out 69.0%

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

      4. *-commutative 69.0%

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

   8. Simplified 69.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.42 \cdot 10^{-158}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+59}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{+123}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -5e+59)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 5.2e+123) (* n 100.0) (* 50.0 (* i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -5e+59) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 5.2e+123) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-5d+59)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 5.2d+123) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -5e+59) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 5.2e+123) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -5e+59:
		tmp = 100.0 * (i / (i / n))
	elif i <= 5.2e+123:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -5e+59)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 5.2e+123)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -5e+59)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 5.2e+123)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -5e+59], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.2e+123], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.9999999999999997e59

   1. Initial program 58.0%

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

   3. Taylor expanded in i around 0 27.1%

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

   if -4.9999999999999997e59 < i < 5.19999999999999971e123

   1. Initial program 11.8%

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

   3. Taylor expanded in i around 0 73.3%

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

      1. *-commutative 73.3%

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

   5. Simplified 73.3%

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

   if 5.19999999999999971e123 < i

   1. Initial program 53.4%

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

   3. Taylor expanded in n around inf 42.9%

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

      1. *-commutative 42.9%

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

      2. associate-/l* 42.9%

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

      3. expm1-define 42.9%

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

   5. Simplified 42.9%

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

   6. Taylor expanded in i around 0 33.5%

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

      1. +-commutative 33.5%

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

      2. associate-*r* 33.5%

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

      3. distribute-rgt-out 33.5%

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

      4. *-commutative 33.5%

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

   8. Simplified 33.5%

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

   9. Taylor expanded in i around inf 33.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+59}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{+123}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5.2 \cdot 10^{+123}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 5.2e+123) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 5.2e+123) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 5.2d+123) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 5.2e+123) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 5.2e+123:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 5.2e+123)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 5.2e+123)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 5.2e+123], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.19999999999999971e123

   1. Initial program 20.7%

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

   3. Taylor expanded in i around 0 60.1%

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

      1. *-commutative 60.1%

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

   5. Simplified 60.1%

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

   if 5.19999999999999971e123 < i

   1. Initial program 53.4%

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

   3. Taylor expanded in n around inf 42.9%

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

      1. *-commutative 42.9%

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

      2. associate-/l* 42.9%

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

      3. expm1-define 42.9%

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

   5. Simplified 42.9%

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

   6. Taylor expanded in i around 0 33.5%

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

      1. +-commutative 33.5%

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

      2. associate-*r* 33.5%

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

      3. distribute-rgt-out 33.5%

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

      4. *-commutative 33.5%

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

   8. Simplified 33.5%

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

   9. Taylor expanded in i around inf 33.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5.2 \cdot 10^{+123}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.0% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

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

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 25.5%

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

3. Taylor expanded in i around 0 51.9%

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

   1. *-commutative 51.9%

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

5. Simplified 51.9%

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

6. Final simplification 51.9%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer target: 34.9% 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 2024089 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

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

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