Compound Interest

Percentage Accurate: 28.1% → 98.2%

Time: 17.3s

Alternatives: 19

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: 28.1% 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: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

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)
     (/ (* (expm1 (* n (log1p (/ i n)))) 100.0) (/ i n))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (/
        100.0
        (+
         (/ 1.0 n)
         (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n))))))))))

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 = (expm1((n * log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) * 100.0) / (i / n)
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	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(Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))));
	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[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 24.0%

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

      1. associate-/r/ 23.8%

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

      2. associate-*r* 23.8%

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

      3. *-commutative 23.8%

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

      4. associate-*r/ 23.8%

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

      5. sub-neg 23.8%

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

      6. distribute-lft-in 23.8%

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

      7. metadata-eval 23.8%

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

      8. metadata-eval 23.8%

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

      9. metadata-eval 23.8%

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

      10. fma-define 23.8%

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

      11. metadata-eval 23.8%

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

   3. Simplified 23.8%

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

      1. *-commutative 23.8%

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

      2. fma-undefine 23.8%

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

      3. *-commutative 23.8%

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

      4. associate-/r/ 24.0%

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

      5. metadata-eval 24.0%

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

      6. metadata-eval 24.0%

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

      7. distribute-rgt-in 24.1%

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

      8. sub-neg 24.1%

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

      9. associate-*r/ 24.0%

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

      10. *-commutative 24.0%

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

      11. frac-2neg 24.0%

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

      12. associate-*l/ 24.1%

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

   6. Applied egg-rr 98.5%

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

   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 96.1%

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

      1. associate-*r/ 96.1%

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

      2. sub-neg 96.1%

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

      3. distribute-rgt-in 96.3%

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

      4. metadata-eval 96.3%

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

      5. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. associate-*r* 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r/ 1.8%

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

      5. sub-neg 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. metadata-eval 1.8%

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

      8. metadata-eval 1.8%

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

      9. metadata-eval 1.8%

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

      10. fma-define 1.8%

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

      11. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

      1. *-commutative 1.8%

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

      2. fma-undefine 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-/r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. sub-neg 0.0%

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

      9. associate-*r/ 0.0%

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

      10. clear-num 0.0%

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

      11. un-div-inv 0.0%

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

      12. add-exp-log 0.0%

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

      13. expm1-define 0.0%

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

      14. log-pow 0.0%

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

      15. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in i around 0 99.5%

      \[\leadsto \frac{100}{\color{blue}{i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

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

Alternative 2: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

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 (/ (/ i n) (expm1 (* n (log1p (/ i n))))))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (/
        100.0
        (+
         (/ 1.0 n)
         (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n))))))))))

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 / ((i / n) / expm1((n * log1p((i / n)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 / ((i / n) / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 / ((i / n) / math.expm1((n * math.log1p((i / n)))))
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	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(Float64(i / n) / expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))));
	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[(i / n), $MachinePrecision] / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 24.0%

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

      1. associate-/r/ 23.8%

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

      2. associate-*r* 23.8%

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

      3. *-commutative 23.8%

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

      4. associate-*r/ 23.8%

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

      5. sub-neg 23.8%

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

      6. distribute-lft-in 23.8%

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

      7. metadata-eval 23.8%

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

      8. metadata-eval 23.8%

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

      9. metadata-eval 23.8%

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

      10. fma-define 23.8%

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

      11. metadata-eval 23.8%

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

   3. Simplified 23.8%

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

      1. *-commutative 23.8%

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

      2. fma-undefine 23.8%

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

      3. *-commutative 23.8%

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

      4. associate-/r/ 24.0%

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

      5. metadata-eval 24.0%

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

      6. metadata-eval 24.0%

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

      7. distribute-rgt-in 24.1%

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

      8. sub-neg 24.1%

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

      9. associate-*r/ 24.0%

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

      10. clear-num 24.0%

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

      11. un-div-inv 24.1%

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

      12. add-exp-log 24.1%

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

      13. expm1-define 24.1%

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

      14. log-pow 38.1%

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

      15. log1p-define 98.4%

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

   6. Applied egg-rr 98.4%

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

   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 96.1%

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

      1. associate-*r/ 96.1%

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

      2. sub-neg 96.1%

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

      3. distribute-rgt-in 96.3%

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

      4. metadata-eval 96.3%

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

      5. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. associate-*r* 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r/ 1.8%

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

      5. sub-neg 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. metadata-eval 1.8%

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

      8. metadata-eval 1.8%

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

      9. metadata-eval 1.8%

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

      10. fma-define 1.8%

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

      11. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

      1. *-commutative 1.8%

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

      2. fma-undefine 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-/r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. sub-neg 0.0%

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

      9. associate-*r/ 0.0%

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

      10. clear-num 0.0%

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

      11. un-div-inv 0.0%

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

      12. add-exp-log 0.0%

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

      13. expm1-define 0.0%

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

      14. log-pow 0.0%

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

      15. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in i around 0 99.5%

      \[\leadsto \frac{100}{\color{blue}{i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

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

Alternative 3: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

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)
     (* n (/ (* (expm1 (* n (log1p (/ i n)))) 100.0) i))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (/
        100.0
        (+
         (/ 1.0 n)
         (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n))))))))))

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 = n * ((expm1((n * log1p((i / n)))) * 100.0) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * ((Math.expm1((n * Math.log1p((i / n)))) * 100.0) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = n * ((math.expm1((n * math.log1p((i / n)))) * 100.0) / i)
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	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(n * Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))));
	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[(n * N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 24.0%

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

      1. associate-/r/ 23.8%

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

      2. associate-*r* 23.8%

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

      3. *-commutative 23.8%

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

      4. associate-*r/ 23.8%

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

      5. sub-neg 23.8%

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

      6. distribute-lft-in 23.8%

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

      7. metadata-eval 23.8%

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

      8. metadata-eval 23.8%

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

      9. metadata-eval 23.8%

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

      10. fma-define 23.8%

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

      11. metadata-eval 23.8%

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

   3. Simplified 23.8%

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

      1. fma-undefine 23.8%

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

      2. metadata-eval 23.8%

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

      3. metadata-eval 23.8%

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

      4. distribute-lft-in 23.8%

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

      5. sub-neg 23.8%

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

      6. *-commutative 23.8%

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

      7. add-exp-log 23.8%

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

      8. expm1-define 23.8%

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

      9. log-pow 37.9%

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

      10. log1p-define 98.0%

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

   6. Applied egg-rr 98.0%

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

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

   1. Initial program 96.1%

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

      1. associate-*r/ 96.1%

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

      2. sub-neg 96.1%

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

      3. distribute-rgt-in 96.3%

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

      4. metadata-eval 96.3%

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

      5. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. associate-*r* 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r/ 1.8%

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

      5. sub-neg 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. metadata-eval 1.8%

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

      8. metadata-eval 1.8%

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

      9. metadata-eval 1.8%

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

      10. fma-define 1.8%

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

      11. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

      1. *-commutative 1.8%

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

      2. fma-undefine 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-/r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. sub-neg 0.0%

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

      9. associate-*r/ 0.0%

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

      10. clear-num 0.0%

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

      11. un-div-inv 0.0%

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

      12. add-exp-log 0.0%

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

      13. expm1-define 0.0%

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

      14. log-pow 0.0%

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

      15. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in i around 0 99.5%

      \[\leadsto \frac{100}{\color{blue}{i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

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

Alternative 4: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

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 (* n (/ (expm1 i) i)))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (/
        100.0
        (+
         (/ 1.0 n)
         (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n))))))))))

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 * (n * (expm1(i) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	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(n * Float64(expm1(i) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))));
	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 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 24.0%

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

   3. Taylor expanded in n around inf 30.2%

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

      1. *-commutative 30.2%

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

      2. associate-/l* 30.2%

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

      3. expm1-define 74.8%

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

   5. Simplified 74.8%

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

   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 96.1%

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

      1. associate-*r/ 96.1%

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

      2. sub-neg 96.1%

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

      3. distribute-rgt-in 96.3%

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

      4. metadata-eval 96.3%

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

      5. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. associate-*r* 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-*r/ 1.8%

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

      5. sub-neg 1.8%

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

      6. distribute-lft-in 1.8%

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

      7. metadata-eval 1.8%

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

      8. metadata-eval 1.8%

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

      9. metadata-eval 1.8%

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

      10. fma-define 1.8%

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

      11. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

      1. *-commutative 1.8%

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

      2. fma-undefine 1.8%

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

      3. *-commutative 1.8%

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

      4. associate-/r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. sub-neg 0.0%

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

      9. associate-*r/ 0.0%

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

      10. clear-num 0.0%

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

      11. un-div-inv 0.0%

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

      12. add-exp-log 0.0%

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

      13. expm1-define 0.0%

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

      14. log-pow 0.0%

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

      15. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in i around 0 99.5%

      \[\leadsto \frac{100}{\color{blue}{i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.4 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-92}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(\left(i \cdot 0.16666666666666666 - \frac{\frac{i}{n} \cdot -0.3333333333333333 + i \cdot 0.5}{n}\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)}{i}\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{+178}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -1.4e-67)
     t_0
     (if (<= i 1.4e-92)
       (*
        n
        (/
         (*
          i
          (+
           100.0
           (*
            i
            (*
             100.0
             (+
              (-
               (* i 0.16666666666666666)
               (/ (+ (* (/ i n) -0.3333333333333333) (* i 0.5)) n))
              (- 0.5 (/ 0.5 n)))))))
         i))
       (if (<= i 1.9e+178) t_0 (/ 0.0 (/ i n)))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -1.4e-67) {
		tmp = t_0;
	} else if (i <= 1.4e-92) {
		tmp = n * ((i * (100.0 + (i * (100.0 * (((i * 0.16666666666666666) - ((((i / n) * -0.3333333333333333) + (i * 0.5)) / n)) + (0.5 - (0.5 / n))))))) / i);
	} else if (i <= 1.9e+178) {
		tmp = t_0;
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -1.4e-67) {
		tmp = t_0;
	} else if (i <= 1.4e-92) {
		tmp = n * ((i * (100.0 + (i * (100.0 * (((i * 0.16666666666666666) - ((((i / n) * -0.3333333333333333) + (i * 0.5)) / n)) + (0.5 - (0.5 / n))))))) / i);
	} else if (i <= 1.9e+178) {
		tmp = t_0;
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -1.4e-67:
		tmp = t_0
	elif i <= 1.4e-92:
		tmp = n * ((i * (100.0 + (i * (100.0 * (((i * 0.16666666666666666) - ((((i / n) * -0.3333333333333333) + (i * 0.5)) / n)) + (0.5 - (0.5 / n))))))) / i)
	elif i <= 1.9e+178:
		tmp = t_0
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -1.4e-67)
		tmp = t_0;
	elseif (i <= 1.4e-92)
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(i * Float64(100.0 * Float64(Float64(Float64(i * 0.16666666666666666) - Float64(Float64(Float64(Float64(i / n) * -0.3333333333333333) + Float64(i * 0.5)) / n)) + Float64(0.5 - Float64(0.5 / n))))))) / i));
	elseif (i <= 1.9e+178)
		tmp = t_0;
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.4e-67], t$95$0, If[LessEqual[i, 1.4e-92], N[(n * N[(N[(i * N[(100.0 + N[(i * N[(100.0 * N[(N[(N[(i * 0.16666666666666666), $MachinePrecision] - N[(N[(N[(N[(i / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(i * 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.9e+178], t$95$0, N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.40000000000000005e-67 or 1.4e-92 < i < 1.89999999999999999e178

   1. Initial program 44.0%

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

   3. Taylor expanded in n around inf 49.7%

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

      1. expm1-define 63.6%

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

   5. Simplified 63.6%

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

   if -1.40000000000000005e-67 < i < 1.4e-92

   1. Initial program 5.8%

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

      1. associate-/r/ 6.2%

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

      2. associate-*r* 6.2%

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

      3. *-commutative 6.2%

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

      4. associate-*r/ 6.2%

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

      5. sub-neg 6.2%

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

      6. distribute-lft-in 6.2%

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

      7. metadata-eval 6.2%

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

      8. metadata-eval 6.2%

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

      9. metadata-eval 6.2%

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

      10. fma-define 6.2%

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

      11. metadata-eval 6.2%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in i around 0 77.4%

      \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}}{i} \]
   6. Step-by-step derivation

      1. distribute-lft-out 77.4%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \color{blue}{\left(100 \cdot \left(i \cdot \left(\left(0.16666666666666666 + 0.3333333333333333 \cdot \frac{1}{{n}^{2}}\right) - 0.5 \cdot \frac{1}{n}\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}\right)}{i} \]

      2. associate--l+ 77.4%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \color{blue}{\left(0.16666666666666666 + \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)} + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      3. associate-*r/ 77.4%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      4. metadata-eval 77.4%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      5. associate-*r/ 77.4%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      6. metadata-eval 77.4%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) + \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)}{i} \]

      7. associate-*r/ 77.4%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right)\right)}{i} \]

      8. metadata-eval 77.4%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right)\right)\right)}{i} \]

   7. Simplified 77.4%

      \[\leadsto n \cdot \frac{\color{blue}{i \cdot \left(100 + i \cdot \left(100 \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{{n}^{2}} - \frac{0.5}{n}\right)\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)}}{i} \]

   8. Taylor expanded in n around -inf 89.9%

      \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(\color{blue}{\left(-1 \cdot \frac{-0.3333333333333333 \cdot \frac{i}{n} + 0.5 \cdot i}{n} + 0.16666666666666666 \cdot i\right)} + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)}{i} \]

   if 1.89999999999999999e178 < i

   1. Initial program 58.9%

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

      1. associate-*r/ 58.9%

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

      2. sub-neg 58.9%

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

      3. distribute-rgt-in 59.1%

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

      4. metadata-eval 59.1%

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

      5. metadata-eval 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in i around 0 47.0%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.4 \cdot 10^{-67}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-92}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(100 \cdot \left(\left(i \cdot 0.16666666666666666 - \frac{\frac{i}{n} \cdot -0.3333333333333333 + i \cdot 0.5}{n}\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)}{i}\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{+178}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-249} \lor \neg \left(n \leq 3.3 \cdot 10^{-112}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-249) (not (<= n 3.3e-112)))
   (* 100.0 (* n (/ (expm1 i) i)))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-249) || !(n <= 3.3e-112)) {
		tmp = 100.0 * (n * (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-249) || !(n <= 3.3e-112)) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-249) or not (n <= 3.3e-112):
		tmp = 100.0 * (n * (math.expm1(i) / i))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-249) || !(n <= 3.3e-112))
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-249], N[Not[LessEqual[n, 3.3e-112]], $MachinePrecision]], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.9999999999999999e-249 or 3.3000000000000001e-112 < n

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf 31.3%

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

      1. *-commutative 31.3%

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

      2. associate-/l* 31.3%

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

      3. expm1-define 80.1%

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

   5. Simplified 80.1%

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

   if -4.9999999999999999e-249 < n < 3.3000000000000001e-112

   1. Initial program 31.8%

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

      1. associate-*r/ 31.8%

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

      2. sub-neg 31.8%

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

      3. distribute-rgt-in 31.8%

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

      4. metadata-eval 31.8%

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

      5. metadata-eval 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in i around 0 60.7%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-249} \lor \neg \left(n \leq 3.3 \cdot 10^{-112}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{-245} \lor \neg \left(n \leq 2.8 \cdot 10^{-112}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.7e-245) (not (<= n 2.8e-112)))
   (* n (/ (* 100.0 (expm1 i)) i))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.7e-245) || !(n <= 2.8e-112)) {
		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 <= -1.7e-245) || !(n <= 2.8e-112)) {
		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 <= -1.7e-245) or not (n <= 2.8e-112):
		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 <= -1.7e-245) || !(n <= 2.8e-112))
		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, -1.7e-245], N[Not[LessEqual[n, 2.8e-112]], $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 < -1.7e-245 or 2.80000000000000023e-112 < n

   1. Initial program 28.6%

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

      1. associate-/r/ 28.9%

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

      2. associate-*r* 28.9%

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

      3. *-commutative 28.9%

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

      4. associate-*r/ 28.9%

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

      5. sub-neg 28.9%

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

      6. distribute-lft-in 28.9%

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

      7. metadata-eval 28.9%

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

      8. metadata-eval 28.9%

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

      9. metadata-eval 28.9%

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

      10. fma-define 28.9%

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

      11. metadata-eval 28.9%

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

   3. Simplified 28.9%

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

   5. Taylor expanded in n around inf 31.3%

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

      1. sub-neg 31.3%

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

      2. metadata-eval 31.3%

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

      3. metadata-eval 31.3%

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

      4. distribute-lft-in 31.3%

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

      5. metadata-eval 31.3%

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

      6. sub-neg 31.3%

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

      7. expm1-define 80.0%

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

   7. Simplified 80.0%

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

   if -1.7e-245 < n < 2.80000000000000023e-112

   1. Initial program 31.8%

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

      1. associate-*r/ 31.8%

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

      2. sub-neg 31.8%

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

      3. distribute-rgt-in 31.8%

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

      4. metadata-eval 31.8%

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

      5. metadata-eval 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in i around 0 60.7%

      \[\leadsto \frac{\color{blue}{1} \cdot 100 + -100}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{-245} \lor \neg \left(n \leq 2.8 \cdot 10^{-112}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1350000000000:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 + 100 \cdot {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1350000000000.0)
   (* 100.0 (* n (/ (expm1 i) i)))
   (/ (+ -100.0 (* 100.0 (pow (/ i n) n))) (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1350000000000.0) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else {
		tmp = (-100.0 + (100.0 * pow((i / n), n))) / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1350000000000.0) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else {
		tmp = (-100.0 + (100.0 * Math.pow((i / n), n))) / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1350000000000.0:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	else:
		tmp = (-100.0 + (100.0 * math.pow((i / n), n))) / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1350000000000.0)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(-100.0 + Float64(100.0 * (Float64(i / n) ^ n))) / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1350000000000.0], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-100.0 + N[(100.0 * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.35e12

   1. Initial program 17.7%

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

   3. Taylor expanded in n around inf 24.0%

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

      1. *-commutative 24.0%

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

      2. associate-/l* 24.0%

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

      3. expm1-define 82.1%

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

   5. Simplified 82.1%

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

   if 1.35e12 < i

   1. Initial program 59.8%

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

      1. associate-*r/ 59.8%

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

      2. sub-neg 59.8%

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

      3. distribute-rgt-in 59.9%

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

      4. metadata-eval 59.9%

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

      5. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in i around inf 68.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1350000000000:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 + 100 \cdot {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.35e+15)
   (* 100.0 (* n (/ (expm1 i) i)))
   (* 100.0 (/ (+ (pow (/ i n) n) -1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.35e+15) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else {
		tmp = 100.0 * ((pow((i / n), n) + -1.0) / (i / n));
	}
	return tmp;
}

Java

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

Python

def code(i, n):
	tmp = 0
	if i <= 1.35e+15:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	else:
		tmp = 100.0 * ((math.pow((i / n), n) + -1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.35e+15)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	else
		tmp = Float64(100.0 * Float64(Float64((Float64(i / n) ^ n) + -1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.35e+15], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.35e15

   1. Initial program 17.7%

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

   3. Taylor expanded in n around inf 24.0%

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

      1. *-commutative 24.0%

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

      2. associate-/l* 24.0%

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

      3. expm1-define 82.1%

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

   5. Simplified 82.1%

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

   if 1.35e15 < i

   1. Initial program 59.8%

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

   3. Taylor expanded in i around inf 68.1%

      \[\leadsto 100 \cdot \frac{{\color{blue}{\left(\frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

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

Alternative 10: 66.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e+22)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.42)
     (* 100.0 (/ i (/ i n)))
     (*
      n
      (/
       (*
        i
        (+
         100.0
         (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
       i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+22) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.6d+22)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * ((i * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+22) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.6e+22:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.6e+22)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667))))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.6e+22)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.6e+22], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6e22

   1. Initial program 38.3%

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

      1. associate-/r/ 38.7%

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

      2. associate-*r* 38.6%

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

      3. *-commutative 38.6%

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

      4. associate-*r/ 38.6%

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

      5. sub-neg 38.6%

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

      6. distribute-lft-in 38.6%

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

      7. metadata-eval 38.6%

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

      8. metadata-eval 38.6%

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

      9. metadata-eval 38.6%

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

      10. fma-define 38.6%

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

      11. metadata-eval 38.6%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in n around inf 42.9%

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

      1. sub-neg 42.9%

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

      2. metadata-eval 42.9%

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

      3. metadata-eval 42.9%

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

      4. distribute-lft-in 42.9%

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

      5. metadata-eval 42.9%

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

      6. sub-neg 42.9%

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

      7. expm1-define 81.3%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in i around 0 52.3%

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

      1. *-commutative 52.3%

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

   10. Simplified 52.3%

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

   if -3.6e22 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0 60.5%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

      1. associate-/r/ 28.5%

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

      2. associate-*r* 28.4%

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

      3. *-commutative 28.4%

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

      4. associate-*r/ 28.5%

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

      5. sub-neg 28.5%

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

      6. distribute-lft-in 28.5%

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

      7. metadata-eval 28.5%

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

      8. metadata-eval 28.5%

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

      9. metadata-eval 28.5%

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

      10. fma-define 28.5%

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

      11. metadata-eval 28.5%

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

   3. Simplified 28.5%

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

   5. Taylor expanded in n around inf 44.9%

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

      1. sub-neg 44.9%

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

      2. metadata-eval 44.9%

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

      3. metadata-eval 44.9%

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

      4. distribute-lft-in 44.9%

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

      5. metadata-eval 44.9%

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

      6. sub-neg 44.9%

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

      7. expm1-define 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in i around 0 80.8%

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

      1. *-commutative 80.8%

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

   10. Simplified 80.8%

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

Alternative 11: 66.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e+22)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.42)
     (* 100.0 (/ i (/ 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 <= -3.6e+22) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (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 <= (-3.6d+22)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (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 <= -3.6e+22) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (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 <= -3.6e+22:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.42:
		tmp = 100.0 * (i / (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 <= -3.6e+22)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / 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 <= -3.6e+22)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (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, -3.6e+22], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6e22

   1. Initial program 38.3%

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

      1. associate-/r/ 38.7%

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

      2. associate-*r* 38.6%

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

      3. *-commutative 38.6%

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

      4. associate-*r/ 38.6%

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

      5. sub-neg 38.6%

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

      6. distribute-lft-in 38.6%

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

      7. metadata-eval 38.6%

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

      8. metadata-eval 38.6%

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

      9. metadata-eval 38.6%

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

      10. fma-define 38.6%

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

      11. metadata-eval 38.6%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in n around inf 42.9%

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

      1. sub-neg 42.9%

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

      2. metadata-eval 42.9%

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

      3. metadata-eval 42.9%

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

      4. distribute-lft-in 42.9%

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

      5. metadata-eval 42.9%

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

      6. sub-neg 42.9%

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

      7. expm1-define 81.3%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in i around 0 52.3%

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

      1. *-commutative 52.3%

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

   10. Simplified 52.3%

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

   if -3.6e22 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0 60.5%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

      1. associate-/r/ 28.5%

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

      2. associate-*r* 28.4%

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

      3. *-commutative 28.4%

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

      4. associate-*r/ 28.5%

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

      5. sub-neg 28.5%

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

      6. distribute-lft-in 28.5%

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

      7. metadata-eval 28.5%

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

      8. metadata-eval 28.5%

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

      9. metadata-eval 28.5%

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

      10. fma-define 28.5%

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

      11. metadata-eval 28.5%

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

   3. Simplified 28.5%

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

   5. Taylor expanded in n around inf 44.9%

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

      1. sub-neg 44.9%

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

      2. metadata-eval 44.9%

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

      3. metadata-eval 44.9%

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

      4. distribute-lft-in 44.9%

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

      5. metadata-eval 44.9%

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

      6. sub-neg 44.9%

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

      7. expm1-define 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in i around 0 79.6%

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

      1. *-commutative 79.6%

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

   10. Simplified 79.6%

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

Alternative 12: 65.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{+22} \lor \neg \left(n \leq 0.42\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.2e+22) (not (<= n 0.42)))
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.2e+22) || !(n <= 0.42)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} 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 <= (-5.2d+22)) .or. (.not. (n <= 0.42d0))) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    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 <= -5.2e+22) || !(n <= 0.42)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5.2e+22) or not (n <= 0.42):
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.2e+22) || !(n <= 0.42))
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	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 <= -5.2e+22) || ~((n <= 0.42)))
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.2e+22], N[Not[LessEqual[n, 0.42]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.2e22 or 0.419999999999999984 < n

   1. Initial program 33.1%

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

      1. associate-/r/ 33.5%

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

      2. associate-*r* 33.4%

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

      3. *-commutative 33.4%

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

      4. associate-*r/ 33.4%

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

      5. sub-neg 33.4%

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

      6. distribute-lft-in 33.4%

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

      7. metadata-eval 33.4%

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

      8. metadata-eval 33.4%

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

      9. metadata-eval 33.4%

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

      10. fma-define 33.4%

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

      11. metadata-eval 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in n around inf 43.9%

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

      1. sub-neg 43.9%

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

      2. metadata-eval 43.9%

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

      3. metadata-eval 43.9%

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

      4. distribute-lft-in 43.9%

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

      5. metadata-eval 43.9%

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

      6. sub-neg 43.9%

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

      7. expm1-define 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in i around 0 64.3%

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

      1. *-commutative 64.3%

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

   10. Simplified 64.3%

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

   if -5.2e22 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0 60.5%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{+22} \lor \neg \left(n \leq 0.42\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.35 \cdot 10^{-17} \lor \neg \left(i \leq 9.8 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2.35e-17) (not (<= i 9.8e+17)))
   (/ 0.0 (/ i n))
   (+ (* 50.0 (* i n)) (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2.35e-17) || !(i <= 9.8e+17)) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-2.35d-17)) .or. (.not. (i <= 9.8d+17))) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = (50.0d0 * (i * n)) + (n * 100.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2.35e-17) || !(i <= 9.8e+17)) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2.35e-17) or not (i <= 9.8e+17):
		tmp = 0.0 / (i / n)
	else:
		tmp = (50.0 * (i * n)) + (n * 100.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2.35e-17) || !(i <= 9.8e+17))
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(Float64(50.0 * Float64(i * n)) + Float64(n * 100.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -2.35e-17) || ~((i <= 9.8e+17)))
		tmp = 0.0 / (i / n);
	else
		tmp = (50.0 * (i * n)) + (n * 100.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2.35e-17], N[Not[LessEqual[i, 9.8e+17]], $MachinePrecision]], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2.35e-17 or 9.8e17 < i

   1. Initial program 57.0%

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

      1. associate-*r/ 57.0%

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

      2. sub-neg 57.0%

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

      3. distribute-rgt-in 57.0%

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

      4. metadata-eval 57.0%

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

      5. metadata-eval 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in i around 0 30.8%

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

   if -2.35e-17 < i < 9.8e17

   1. Initial program 6.2%

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

      1. associate-/r/ 6.6%

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

      2. associate-*r* 6.6%

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

      3. *-commutative 6.6%

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

      4. associate-*r/ 6.6%

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

      5. sub-neg 6.6%

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

      6. distribute-lft-in 6.6%

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

      7. metadata-eval 6.6%

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

      8. metadata-eval 6.6%

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

      9. metadata-eval 6.6%

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

      10. fma-define 6.6%

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

      11. metadata-eval 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in n around inf 8.6%

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

      1. sub-neg 8.6%

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

      2. metadata-eval 8.6%

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

      3. metadata-eval 8.6%

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

      4. distribute-lft-in 8.6%

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

      5. metadata-eval 8.6%

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

      6. sub-neg 8.6%

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

      7. expm1-define 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in i around 0 84.0%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.35 \cdot 10^{-17} \lor \neg \left(i \leq 9.8 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{+133}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.3e+133)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n 0.42) (* 100.0 (/ i (/ i n))) (+ (* 50.0 (* i n)) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.3e+133) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.3d+133)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (50.0d0 * (i * n)) + (n * 100.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.3e+133) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.3e+133:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (50.0 * (i * n)) + (n * 100.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.3e+133)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(50.0 * Float64(i * n)) + Float64(n * 100.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.3e+133)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (50.0 * (i * n)) + (n * 100.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.3e+133], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.29999999999999994e133

   1. Initial program 25.9%

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

      1. associate-/r/ 26.4%

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

      2. associate-*r* 26.4%

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

      3. *-commutative 26.4%

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

      4. associate-*r/ 26.4%

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

      5. sub-neg 26.4%

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

      6. distribute-lft-in 26.4%

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

      7. metadata-eval 26.4%

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

      8. metadata-eval 26.4%

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

      9. metadata-eval 26.4%

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

      10. fma-define 26.4%

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

      11. metadata-eval 26.4%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in i around 0 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*r/ 56.0%

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

      3. metadata-eval 56.0%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in n around inf 56.0%

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

      1. *-commutative 56.0%

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

   10. Simplified 56.0%

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

   if -4.29999999999999994e133 < n < 0.419999999999999984

   1. Initial program 30.6%

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

   3. Taylor expanded in i around 0 56.6%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

      1. associate-/r/ 28.5%

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

      2. associate-*r* 28.4%

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

      3. *-commutative 28.4%

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

      4. associate-*r/ 28.5%

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

      5. sub-neg 28.5%

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

      6. distribute-lft-in 28.5%

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

      7. metadata-eval 28.5%

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

      8. metadata-eval 28.5%

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

      9. metadata-eval 28.5%

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

      10. fma-define 28.5%

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

      11. metadata-eval 28.5%

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

   3. Simplified 28.5%

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

   5. Taylor expanded in n around inf 44.9%

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

      1. sub-neg 44.9%

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

      2. metadata-eval 44.9%

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

      3. metadata-eval 44.9%

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

      4. distribute-lft-in 44.9%

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

      5. metadata-eval 44.9%

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

      6. sub-neg 44.9%

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

      7. expm1-define 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in i around 0 69.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{+133}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{+133} \lor \neg \left(n \leq 0.42\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.3e+133) (not (<= n 0.42)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.3e+133) || !(n <= 0.42)) {
		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 <= (-4.3d+133)) .or. (.not. (n <= 0.42d0))) 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 <= -4.3e+133) || !(n <= 0.42)) {
		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 <= -4.3e+133) or not (n <= 0.42):
		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 <= -4.3e+133) || !(n <= 0.42))
		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 <= -4.3e+133) || ~((n <= 0.42)))
		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, -4.3e+133], N[Not[LessEqual[n, 0.42]], $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 < -4.29999999999999994e133 or 0.419999999999999984 < n

   1. Initial program 27.2%

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

      1. associate-/r/ 27.7%

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

      2. associate-*r* 27.7%

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

      3. *-commutative 27.7%

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

      4. associate-*r/ 27.7%

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

      5. sub-neg 27.7%

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

      6. distribute-lft-in 27.7%

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

      7. metadata-eval 27.7%

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

      8. metadata-eval 27.7%

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

      9. metadata-eval 27.7%

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

      10. fma-define 27.7%

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

      11. metadata-eval 27.7%

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

   3. Simplified 27.7%

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

   5. Taylor expanded in i around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-*r/ 64.3%

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

      3. metadata-eval 64.3%

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

   7. Simplified 64.3%

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

   8. Taylor expanded in n around inf 64.3%

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

      1. *-commutative 64.3%

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

   10. Simplified 64.3%

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

   if -4.29999999999999994e133 < n < 0.419999999999999984

   1. Initial program 30.6%

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

   3. Taylor expanded in i around 0 56.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{+133} \lor \neg \left(n \leq 0.42\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+28} \lor \neg \left(i \leq 5 \cdot 10^{+33}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -5e+28) (not (<= i 5e+33)))
   (* 100.0 (/ i (/ i n)))
   (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -5e+28) || !(i <= 5e+33)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-5d+28)) .or. (.not. (i <= 5d+33))) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -5e+28) || !(i <= 5e+33)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -5e+28) or not (i <= 5e+33):
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -5e+28) || !(i <= 5e+33))
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -5e+28) || ~((i <= 5e+33)))
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -5e+28], N[Not[LessEqual[i, 5e+33]], $MachinePrecision]], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -4.99999999999999957e28 or 4.99999999999999973e33 < i

   1. Initial program 58.6%

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

   3. Taylor expanded in i around 0 21.2%

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

   if -4.99999999999999957e28 < i < 4.99999999999999973e33

   1. Initial program 7.4%

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

      1. associate-/r/ 7.7%

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

      2. associate-*r* 7.7%

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

      3. *-commutative 7.7%

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

      4. associate-*r/ 7.7%

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

      5. sub-neg 7.7%

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

      6. distribute-lft-in 7.7%

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

      7. metadata-eval 7.7%

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

      8. metadata-eval 7.7%

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

      9. metadata-eval 7.7%

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

      10. fma-define 7.7%

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

      11. metadata-eval 7.7%

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

   3. Simplified 7.7%

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

   5. Taylor expanded in i around 0 79.8%

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

      1. *-commutative 79.8%

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

   7. Simplified 79.8%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+28} \lor \neg \left(i \leq 5 \cdot 10^{+33}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.4e+24) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.4e+24) {
		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 <= 1.4d+24) 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 <= 1.4e+24) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.4e+24:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.4e+24)
		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 <= 1.4e+24)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.4e+24], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.4000000000000001e24

   1. Initial program 19.0%

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

      1. associate-/r/ 19.1%

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

      2. associate-*r* 19.1%

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

      3. *-commutative 19.1%

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

      4. associate-*r/ 19.1%

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

      5. sub-neg 19.1%

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

      6. distribute-lft-in 19.1%

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

      7. metadata-eval 19.1%

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

      8. metadata-eval 19.1%

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

      9. metadata-eval 19.1%

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

      10. fma-define 19.1%

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

      11. metadata-eval 19.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in i around 0 63.2%

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

      1. *-commutative 63.2%

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

   7. Simplified 63.2%

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

   if 1.4000000000000001e24 < i

   1. Initial program 58.0%

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

      1. associate-/r/ 58.3%

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

      2. associate-*r* 58.2%

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

      3. *-commutative 58.2%

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

      4. associate-*r/ 58.2%

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

      5. sub-neg 58.2%

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

      6. distribute-lft-in 58.3%

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

      7. metadata-eval 58.3%

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

      8. metadata-eval 58.3%

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

      9. metadata-eval 58.3%

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

      10. fma-define 58.2%

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

      11. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in n around inf 39.7%

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

      1. sub-neg 39.7%

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

      2. metadata-eval 39.7%

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

      3. metadata-eval 39.7%

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

      4. distribute-lft-in 39.7%

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

      5. metadata-eval 39.7%

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

      6. sub-neg 39.7%

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

      7. expm1-define 39.7%

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

   7. Simplified 39.7%

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

   8. Taylor expanded in i around 0 26.1%

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

      1. *-commutative 26.1%

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

   10. Simplified 26.1%

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

   11. Taylor expanded in i around inf 23.5%

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

       1. *-commutative 23.5%

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

   13. Simplified 23.5%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.0% 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 29.2%

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

   1. associate-/r/ 29.3%

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

   2. associate-*r* 29.3%

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

   3. *-commutative 29.3%

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

   4. associate-*r/ 29.3%

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

   5. sub-neg 29.3%

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

   6. distribute-lft-in 29.3%

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

   7. metadata-eval 29.3%

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

   8. metadata-eval 29.3%

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

   9. metadata-eval 29.3%

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

   10. fma-define 29.3%

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

   11. metadata-eval 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in i around 0 47.9%

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

   1. *-commutative 47.9%

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

7. Simplified 47.9%

   \[\leadsto \color{blue}{n \cdot 100} \]
8. Add Preprocessing

Alternative 19: 2.9% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

1. Initial program 29.2%

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

   1. associate-/r/ 29.3%

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

   2. associate-*r* 29.3%

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

   3. *-commutative 29.3%

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

   4. associate-*r/ 29.3%

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

   5. sub-neg 29.3%

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

   6. distribute-lft-in 29.3%

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

   7. metadata-eval 29.3%

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

   8. metadata-eval 29.3%

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

   9. metadata-eval 29.3%

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

   10. fma-define 29.3%

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

   11. metadata-eval 29.3%

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

3. Simplified 29.3%

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

5. Taylor expanded in i around 0 52.3%

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

   1. *-commutative 52.3%

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

   2. associate-*r/ 52.3%

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

   3. metadata-eval 52.3%

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

7. Simplified 52.3%

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

8. Taylor expanded in n around 0 3.0%

   \[\leadsto \color{blue}{-50 \cdot i} \]
9. Step-by-step derivation

   1. *-commutative 3.0%

      \[\leadsto \color{blue}{i \cdot -50} \]

10. Simplified 3.0%

    \[\leadsto \color{blue}{i \cdot -50} \]
11. Add Preprocessing

Developer Target 1: 33.8% 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 2024139 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

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