Compound Interest

Percentage Accurate: 28.1% → 98.6%

Time: 16.1s

Alternatives: 20

Speedup: 38.0×

• could not determine a ground truth

  1. i = 1.0456249878475745e-102
  2. n = 3.82975919457679e+31

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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-164}:\\ \;\;\;\;t\_0 \cdot 100\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n}\right)}^{n} - 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 -5e-164)
     (* t_0 100.0)
     (if (<= t_0 0.0)
       (/ 100.0 (/ (/ i n) (expm1 (* n (log1p (/ i n))))))
       (if (<= t_0 INFINITY)
         (/ (* n (- (* 100.0 (pow (/ i n) n)) 100.0)) i)
         (/
          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) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= -5e-164) {
		tmp = t_0 * 100.0;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 / ((i / n) / expm1((n * log1p((i / n)))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (n * ((100.0 * pow((i / n), n)) - 100.0)) / i;
	} 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) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= -5e-164) {
		tmp = t_0 * 100.0;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 / ((i / n) / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (n * ((100.0 * Math.pow((i / n), n)) - 100.0)) / i;
	} 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) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= -5e-164:
		tmp = t_0 * 100.0
	elif t_0 <= 0.0:
		tmp = 100.0 / ((i / n) / math.expm1((n * math.log1p((i / n)))))
	elif t_0 <= math.inf:
		tmp = (n * ((100.0 * math.pow((i / n), n)) - 100.0)) / i
	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(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= -5e-164)
		tmp = Float64(t_0 * 100.0);
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 / Float64(Float64(i / n) / expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(n * Float64(Float64(100.0 * (Float64(i / n) ^ n)) - 100.0)) / i);
	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[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-164], N[(t$95$0 * 100.0), $MachinePrecision], If[LessEqual[t$95$0, 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$0, Infinity], N[(N[(n * N[(N[(100.0 * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]), $MachinePrecision] / i), $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 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -4.99999999999999962e-164

   1. Initial program 100.0%

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

   if -4.99999999999999962e-164 < (/.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 21.8%

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

      1. associate-/r/ 21.6%

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

      2. associate-*r* 21.6%

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

      3. *-commutative 21.6%

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

      4. associate-*r/ 21.6%

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

      5. sub-neg 21.6%

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

      6. distribute-lft-in 21.6%

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

      7. metadata-eval 21.6%

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

      8. metadata-eval 21.6%

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

      9. metadata-eval 21.6%

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

      10. fma-define 21.6%

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

      11. metadata-eval 21.6%

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

   3. Simplified 21.6%

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

      1. *-commutative 21.6%

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

      2. fma-undefine 21.6%

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

      3. *-commutative 21.6%

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

      4. associate-/r/ 21.8%

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

      5. metadata-eval 21.8%

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

      6. metadata-eval 21.8%

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

      7. distribute-rgt-in 21.8%

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

      8. sub-neg 21.8%

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

      9. associate-*r/ 21.8%

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

      10. clear-num 21.8%

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

      11. un-div-inv 21.8%

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

      12. add-exp-log 21.8%

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

      13. expm1-define 21.8%

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

      14. log-pow 37.4%

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

      15. log1p-define 99.5%

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

   6. Applied egg-rr 99.5%

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

   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 99.6%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in i around inf 99.8%

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

   6. Taylor expanded in n around inf 99.9%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      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 99.7%

      \[\leadsto \frac{100}{\color{blue}{i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-164}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n}\right)}^{n} - 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-175}:\\ \;\;\;\;t\_0 \cdot 100\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n}\right)}^{n} - 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 -1e-175)
     (* t_0 100.0)
     (if (<= t_0 0.0)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_0 INFINITY)
         (/ (* n (- (* 100.0 (pow (/ i n) n)) 100.0)) i)
         (/
          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) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= -1e-175) {
		tmp = t_0 * 100.0;
	} else if (t_0 <= 0.0) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (n * ((100.0 * pow((i / n), n)) - 100.0)) / i;
	} 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) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= -1e-175) {
		tmp = t_0 * 100.0;
	} else if (t_0 <= 0.0) {
		tmp = n * ((100.0 * Math.expm1((n * Math.log1p((i / n))))) / i);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (n * ((100.0 * Math.pow((i / n), n)) - 100.0)) / i;
	} 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) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= -1e-175:
		tmp = t_0 * 100.0
	elif t_0 <= 0.0:
		tmp = n * ((100.0 * math.expm1((n * math.log1p((i / n))))) / i)
	elif t_0 <= math.inf:
		tmp = (n * ((100.0 * math.pow((i / n), n)) - 100.0)) / i
	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(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= -1e-175)
		tmp = Float64(t_0 * 100.0);
	elseif (t_0 <= 0.0)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(n * Float64(Float64(100.0 * (Float64(i / n) ^ n)) - 100.0)) / i);
	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[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-175], N[(t$95$0 * 100.0), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(n * N[(N[(100.0 * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]), $MachinePrecision] / i), $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 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e-175

   1. Initial program 100.0%

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

   if -1e-175 < (/.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 21.4%

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

      1. associate-/r/ 21.2%

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

      2. associate-*r* 21.2%

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

      3. *-commutative 21.2%

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

      4. associate-*r/ 21.2%

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

      5. sub-neg 21.2%

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

      6. distribute-lft-in 21.2%

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

      7. metadata-eval 21.2%

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

      8. metadata-eval 21.2%

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

      9. metadata-eval 21.2%

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

      10. fma-define 21.2%

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

      11. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

      1. fma-undefine 21.2%

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

      2. metadata-eval 21.2%

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

      3. metadata-eval 21.2%

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

      4. distribute-lft-in 21.2%

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

      5. sub-neg 21.2%

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

      6. *-commutative 21.2%

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

      7. add-exp-log 21.2%

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

      8. expm1-define 21.2%

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

      9. log-pow 36.8%

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

      10. log1p-define 99.1%

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

   6. Applied egg-rr 99.1%

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

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

   1. Initial program 99.6%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in i around inf 99.8%

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

   6. Taylor expanded in n around inf 99.9%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      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 99.7%

      \[\leadsto \frac{100}{\color{blue}{i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-175}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n}\right)}^{n} - 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-175}:\\ \;\;\;\;t\_0 \cdot 100\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n}\right)}^{n} - 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 -1e-175)
     (* t_0 100.0)
     (if (<= t_0 0.0)
       (* n (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 i)))
       (if (<= t_0 INFINITY)
         (/ (* n (- (* 100.0 (pow (/ i n) n)) 100.0)) i)
         (/
          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) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= -1e-175) {
		tmp = t_0 * 100.0;
	} else if (t_0 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) * (100.0 / i));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (n * ((100.0 * pow((i / n), n)) - 100.0)) / i;
	} 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) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= -1e-175) {
		tmp = t_0 * 100.0;
	} else if (t_0 <= 0.0) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) * (100.0 / i));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (n * ((100.0 * Math.pow((i / n), n)) - 100.0)) / i;
	} 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) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= -1e-175:
		tmp = t_0 * 100.0
	elif t_0 <= 0.0:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) * (100.0 / i))
	elif t_0 <= math.inf:
		tmp = (n * ((100.0 * math.pow((i / n), n)) - 100.0)) / i
	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(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= -1e-175)
		tmp = Float64(t_0 * 100.0);
	elseif (t_0 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / i)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(n * Float64(Float64(100.0 * (Float64(i / n) ^ n)) - 100.0)) / i);
	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[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-175], N[(t$95$0 * 100.0), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(n * N[(N[(100.0 * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]), $MachinePrecision] / i), $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 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e-175

   1. Initial program 100.0%

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

   if -1e-175 < (/.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 21.4%

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

      1. associate-/r/ 21.2%

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

      2. associate-*r* 21.2%

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

      3. *-commutative 21.2%

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

      4. associate-*r/ 21.2%

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

      5. sub-neg 21.2%

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

      6. distribute-lft-in 21.2%

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

      7. metadata-eval 21.2%

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

      8. metadata-eval 21.2%

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

      9. metadata-eval 21.2%

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

      10. fma-define 21.2%

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

      11. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

      1. fma-undefine 21.2%

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

      2. metadata-eval 21.2%

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

      3. metadata-eval 21.2%

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

      4. distribute-lft-in 21.2%

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

      5. sub-neg 21.2%

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

      6. *-commutative 21.2%

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

      7. add-exp-log 21.2%

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

      8. expm1-define 21.2%

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

      9. log-pow 36.8%

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

      10. log1p-define 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. associate-/l* 99.1%

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

   8. Applied egg-rr 99.1%

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

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

   1. Initial program 99.6%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in i around inf 99.8%

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

   6. Taylor expanded in n around inf 99.9%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      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 99.7%

      \[\leadsto \frac{100}{\color{blue}{i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-175}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n}\right)}^{n} - 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-235}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8.2e-235)
   (* 100.0 (* n (/ (expm1 i) i)))
   (if (<= n 2.2e+16)
     (/
      100.0
      (+ (/ 1.0 n) (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n))))))
     (* n (/ (* 100.0 (expm1 i)) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8.2e-235) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else if (n <= 2.2e+16) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	} else {
		tmp = n * ((100.0 * expm1(i)) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8.2e-235) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else if (n <= 2.2e+16) {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	} else {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8.2e-235:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	elif n <= 2.2e+16:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	else:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8.2e-235)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	elseif (n <= 2.2e+16)
		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))))));
	else
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8.2e-235], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.2e+16], 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], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.19999999999999993e-235

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/l* 32.8%

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

      3. expm1-define 79.6%

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

   5. Simplified 79.6%

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

   if -8.19999999999999993e-235 < n < 2.2e16

   1. Initial program 39.2%

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

      1. associate-/r/ 38.9%

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

      2. associate-*r* 38.8%

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

      3. *-commutative 38.8%

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

      4. associate-*r/ 38.8%

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

      5. sub-neg 38.8%

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

      6. distribute-lft-in 38.8%

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

      7. metadata-eval 38.8%

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

      8. metadata-eval 38.8%

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

      9. metadata-eval 38.8%

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

      10. fma-define 38.8%

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

      11. metadata-eval 38.8%

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

   3. Simplified 38.8%

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

      1. *-commutative 38.8%

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

      2. fma-undefine 38.8%

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

      3. *-commutative 38.8%

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

      4. associate-/r/ 39.2%

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

      5. metadata-eval 39.2%

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

      6. metadata-eval 39.2%

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

      7. distribute-rgt-in 39.2%

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

      8. sub-neg 39.2%

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

      9. associate-*r/ 39.2%

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

      10. clear-num 39.2%

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

      11. un-div-inv 39.2%

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

      12. add-exp-log 39.0%

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

      13. expm1-define 39.0%

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

      14. log-pow 60.0%

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

      15. log1p-define 85.6%

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

   6. Applied egg-rr 85.6%

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

   7. Taylor expanded in i around 0 75.4%

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

   if 2.2e16 < n

   1. Initial program 18.2%

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

      1. associate-/r/ 18.7%

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

      2. associate-*r* 18.7%

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

      3. *-commutative 18.7%

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

      4. associate-*r/ 18.7%

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

      5. sub-neg 18.7%

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

      6. distribute-lft-in 18.7%

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

      7. metadata-eval 18.7%

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

      8. metadata-eval 18.7%

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

      9. metadata-eval 18.7%

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

      10. fma-define 18.7%

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

      11. metadata-eval 18.7%

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

   3. Simplified 18.7%

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

   5. Taylor expanded in n around inf 41.6%

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

      1. sub-neg 41.6%

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

      2. metadata-eval 41.6%

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

      3. metadata-eval 41.6%

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

      4. distribute-lft-in 41.6%

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

      5. metadata-eval 41.6%

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

      6. sub-neg 41.6%

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

      7. expm1-define 94.2%

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

   7. Simplified 94.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-235}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -7.8 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.3 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -7.8e-45)
     t_0
     (if (<= n -4.3e-234)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= n 4.6e-150) (/ 0.0 (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -7.8e-45) {
		tmp = t_0;
	} else if (n <= -4.3e-234) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 4.6e-150) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -7.8e-45) {
		tmp = t_0;
	} else if (n <= -4.3e-234) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 4.6e-150) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -7.8e-45:
		tmp = t_0
	elif n <= -4.3e-234:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 4.6e-150:
		tmp = 0.0 / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -7.8e-45)
		tmp = t_0;
	elseif (n <= -4.3e-234)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 4.6e-150)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7.8e-45], t$95$0, If[LessEqual[n, -4.3e-234], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.6e-150], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.7999999999999999e-45 or 4.60000000000000006e-150 < n

   1. Initial program 19.6%

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

      1. associate-/r/ 19.9%

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

      2. associate-*r* 19.9%

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

      3. *-commutative 19.9%

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

      4. associate-*r/ 19.9%

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

      5. sub-neg 19.9%

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

      6. distribute-lft-in 19.9%

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

      7. metadata-eval 19.9%

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

      8. metadata-eval 19.9%

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

      9. metadata-eval 19.9%

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

      10. fma-define 19.9%

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

      11. metadata-eval 19.9%

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

   3. Simplified 19.9%

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

      1. fma-undefine 19.9%

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

      2. metadata-eval 19.9%

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

      3. metadata-eval 19.9%

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

      4. distribute-lft-in 19.9%

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

      5. sub-neg 19.9%

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

      6. *-commutative 19.9%

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

      7. add-exp-log 19.9%

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

      8. expm1-define 19.9%

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

      9. log-pow 20.0%

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

      10. log1p-define 73.9%

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

   6. Applied egg-rr 73.9%

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

   7. Taylor expanded in n around inf 34.1%

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

      1. expm1-define 86.7%

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

   9. Simplified 86.7%

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

   if -7.7999999999999999e-45 < n < -4.3000000000000001e-234

   1. Initial program 31.7%

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

   3. Taylor expanded in n around inf 17.2%

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

      1. expm1-define 49.0%

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

   5. Simplified 49.0%

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

   if -4.3000000000000001e-234 < n < 4.60000000000000006e-150

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.8 \cdot 10^{-45}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq -4.3 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq -1 \cdot 10^{-173}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right)\right)}{i}\\ \mathbf{elif}\;i \leq 5 \cdot 10^{-28}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(100 \cdot \left(\left(i \cdot 0.16666666666666666 - \frac{i \cdot 0.5 + \frac{i}{n} \cdot -0.3333333333333333}{n}\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -2.5e-5)
     t_0
     (if (<= i -1e-173)
       (* 100.0 (/ (* n (* i (+ 1.0 (* i 0.5)))) i))
       (if (<= i 5e-28)
         (*
          n
          (+
           100.0
           (*
            i
            (*
             100.0
             (+
              (-
               (* i 0.16666666666666666)
               (/ (+ (* i 0.5) (* (/ i n) -0.3333333333333333)) n))
              (- 0.5 (/ 0.5 n)))))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -2.5e-5) {
		tmp = t_0;
	} else if (i <= -1e-173) {
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i);
	} else if (i <= 5e-28) {
		tmp = n * (100.0 + (i * (100.0 * (((i * 0.16666666666666666) - (((i * 0.5) + ((i / n) * -0.3333333333333333)) / n)) + (0.5 - (0.5 / n))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -2.5e-5) {
		tmp = t_0;
	} else if (i <= -1e-173) {
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i);
	} else if (i <= 5e-28) {
		tmp = n * (100.0 + (i * (100.0 * (((i * 0.16666666666666666) - (((i * 0.5) + ((i / n) * -0.3333333333333333)) / n)) + (0.5 - (0.5 / n))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -2.5e-5:
		tmp = t_0
	elif i <= -1e-173:
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i)
	elif i <= 5e-28:
		tmp = n * (100.0 + (i * (100.0 * (((i * 0.16666666666666666) - (((i * 0.5) + ((i / n) * -0.3333333333333333)) / n)) + (0.5 - (0.5 / n))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -2.5e-5)
		tmp = t_0;
	elseif (i <= -1e-173)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * 0.5)))) / i));
	elseif (i <= 5e-28)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(100.0 * Float64(Float64(Float64(i * 0.16666666666666666) - Float64(Float64(Float64(i * 0.5) + Float64(Float64(i / n) * -0.3333333333333333)) / n)) + Float64(0.5 - Float64(0.5 / n)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.5e-5], t$95$0, If[LessEqual[i, -1e-173], N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5e-28], N[(n * N[(100.0 + N[(i * N[(100.0 * N[(N[(N[(i * 0.16666666666666666), $MachinePrecision] - N[(N[(N[(i * 0.5), $MachinePrecision] + N[(N[(i / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.50000000000000012e-5 or 5.0000000000000002e-28 < i

   1. Initial program 48.3%

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

   3. Taylor expanded in n around inf 64.1%

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

      1. expm1-define 66.9%

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

   5. Simplified 66.9%

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

   if -2.50000000000000012e-5 < i < -1e-173

   1. Initial program 17.7%

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

      1. associate-/r/ 17.9%

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

      2. associate-*r* 17.9%

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

      3. *-commutative 17.9%

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

      4. associate-*r/ 17.9%

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

      5. sub-neg 17.9%

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

      6. distribute-lft-in 17.9%

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

      7. metadata-eval 17.9%

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

      8. metadata-eval 17.9%

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

      9. metadata-eval 17.9%

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

      10. fma-define 17.9%

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

      11. metadata-eval 17.9%

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

   3. Simplified 17.9%

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

      1. fma-undefine 17.9%

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

      2. metadata-eval 17.9%

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

      3. metadata-eval 17.9%

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

      4. distribute-lft-in 17.9%

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

      5. sub-neg 17.9%

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

      6. *-commutative 17.9%

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

      7. add-exp-log 17.9%

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

      8. expm1-define 17.9%

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

      9. log-pow 31.4%

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

      10. log1p-define 89.3%

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

   6. Applied egg-rr 89.3%

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

   7. Taylor expanded in n around inf 17.1%

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

      1. expm1-define 84.8%

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

   9. Simplified 84.8%

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

   10. Taylor expanded in i around 0 84.8%

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

       1. *-commutative 84.8%

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

   12. Simplified 84.8%

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

   if -1e-173 < i < 5.0000000000000002e-28

   1. Initial program 7.6%

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

      1. associate-/r/ 8.2%

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

      2. associate-*r* 8.2%

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

      3. *-commutative 8.2%

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

      4. associate-*r/ 8.2%

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

      5. sub-neg 8.2%

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

      6. distribute-lft-in 8.2%

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

      7. metadata-eval 8.2%

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

      8. metadata-eval 8.2%

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

      9. metadata-eval 8.2%

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

      10. fma-define 8.2%

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

      11. metadata-eval 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in i around 0 72.5%

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

      1. distribute-lft-out 72.5%

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

      2. associate--l+ 72.5%

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

      3. associate-*r/ 72.5%

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

      4. metadata-eval 72.5%

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

      5. associate-*r/ 72.5%

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

      6. metadata-eval 72.5%

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

      7. associate-*r/ 72.5%

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

      8. metadata-eval 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in n around -inf 84.4%

      \[\leadsto n \cdot \left(100 + i \cdot \left(100 \cdot \left(\color{blue}{\left(-1 \cdot \frac{-0.3333333333333333 \cdot \frac{i}{n} + 0.5 \cdot i}{n} + 0.16666666666666666 \cdot i\right)} + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -1 \cdot 10^{-173}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right)\right)}{i}\\ \mathbf{elif}\;i \leq 5 \cdot 10^{-28}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(100 \cdot \left(\left(i \cdot 0.16666666666666666 - \frac{i \cdot 0.5 + \frac{i}{n} \cdot -0.3333333333333333}{n}\right) + \left(0.5 - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-234} \lor \neg \left(n \leq 1.2 \cdot 10^{-153}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.4e-234) (not (<= n 1.2e-153)))
   (* n (/ (* 100.0 (expm1 i)) i))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.4e-234) || !(n <= 1.2e-153)) {
		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 <= -4.4e-234) || !(n <= 1.2e-153)) {
		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 <= -4.4e-234) or not (n <= 1.2e-153):
		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 <= -4.4e-234) || !(n <= 1.2e-153))
		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, -4.4e-234], N[Not[LessEqual[n, 1.2e-153]], $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 < -4.3999999999999998e-234 or 1.2000000000000001e-153 < n

   1. Initial program 21.0%

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

      1. associate-/r/ 21.3%

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

      2. associate-*r* 21.3%

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

      3. *-commutative 21.3%

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

      4. associate-*r/ 21.3%

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

      5. sub-neg 21.3%

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

      6. distribute-lft-in 21.3%

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

      7. metadata-eval 21.3%

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

      8. metadata-eval 21.3%

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

      9. metadata-eval 21.3%

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

      10. fma-define 21.3%

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

      11. metadata-eval 21.3%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in n around inf 32.1%

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

      1. sub-neg 32.1%

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

      2. metadata-eval 32.1%

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

      3. metadata-eval 32.1%

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

      4. distribute-lft-in 32.1%

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

      5. metadata-eval 32.1%

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

      6. sub-neg 32.1%

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

      7. expm1-define 82.4%

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

   7. Simplified 82.4%

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

   if -4.3999999999999998e-234 < n < 1.2000000000000001e-153

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-234} \lor \neg \left(n \leq 1.2 \cdot 10^{-153}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.4e-234)
   (* 100.0 (* n (/ (expm1 i) i)))
   (if (<= n 4.5e-154) (/ 0.0 (/ i n)) (* n (/ (* 100.0 (expm1 i)) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.4e-234) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else if (n <= 4.5e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * ((100.0 * expm1(i)) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.4e-234) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else if (n <= 4.5e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.4e-234:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	elif n <= 4.5e-154:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.4e-234)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	elseif (n <= 4.5e-154)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.4e-234], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.5e-154], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.3999999999999998e-234

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/l* 32.8%

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

      3. expm1-define 79.6%

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

   5. Simplified 79.6%

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

   if -4.3999999999999998e-234 < n < 4.4999999999999997e-154

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

   if 4.4999999999999997e-154 < n

   1. Initial program 15.3%

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

      1. associate-/r/ 15.7%

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

      2. associate-*r* 15.7%

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

      3. *-commutative 15.7%

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

      4. associate-*r/ 15.7%

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

      5. sub-neg 15.7%

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

      6. distribute-lft-in 15.7%

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

      7. metadata-eval 15.7%

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

      8. metadata-eval 15.7%

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

      9. metadata-eval 15.7%

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

      10. fma-define 15.7%

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

      11. metadata-eval 15.7%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in n around inf 31.4%

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

      1. sub-neg 31.4%

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

      2. metadata-eval 31.4%

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

      3. metadata-eval 31.4%

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

      4. distribute-lft-in 31.4%

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

      5. metadata-eval 31.4%

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

      6. sub-neg 31.4%

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

      7. expm1-define 85.8%

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

   7. Simplified 85.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-235}:\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8.5e-235)
   (* n (* (/ 100.0 i) (expm1 i)))
   (if (<= n 4.6e-150) (/ 0.0 (/ i n)) (* 100.0 (/ (* n (expm1 i)) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8.5e-235) {
		tmp = n * ((100.0 / i) * expm1(i));
	} else if (n <= 4.6e-150) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * ((n * expm1(i)) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8.5e-235) {
		tmp = n * ((100.0 / i) * Math.expm1(i));
	} else if (n <= 4.6e-150) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * ((n * Math.expm1(i)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8.5e-235:
		tmp = n * ((100.0 / i) * math.expm1(i))
	elif n <= 4.6e-150:
		tmp = 0.0 / (i / n)
	else:
		tmp = 100.0 * ((n * math.expm1(i)) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8.5e-235)
		tmp = Float64(n * Float64(Float64(100.0 / i) * expm1(i)));
	elseif (n <= 4.6e-150)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(Float64(n * expm1(i)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8.5e-235], N[(n * N[(N[(100.0 / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.6e-150], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.49999999999999964e-235

   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/ 26.1%

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

      2. associate-*r* 26.1%

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

      3. *-commutative 26.1%

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

      4. associate-*r/ 26.1%

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

      5. sub-neg 26.1%

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

      6. distribute-lft-in 26.1%

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

      7. metadata-eval 26.1%

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

      8. metadata-eval 26.1%

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

      9. metadata-eval 26.1%

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

      10. fma-define 26.1%

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

      11. metadata-eval 26.1%

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

   3. Simplified 26.1%

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

      1. fma-undefine 26.1%

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

      2. metadata-eval 26.1%

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

      3. metadata-eval 26.1%

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

      4. distribute-lft-in 26.1%

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

      5. sub-neg 26.1%

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

      6. *-commutative 26.1%

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

      7. add-exp-log 26.1%

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

      8. expm1-define 26.1%

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

      9. log-pow 31.5%

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

      10. log1p-define 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. associate-/l* 78.6%

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

   8. Applied egg-rr 78.6%

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

   9. Taylor expanded in n around inf 32.7%

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

       1. expm1-define 79.5%

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

   11. Simplified 79.5%

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

   if -8.49999999999999964e-235 < n < 4.60000000000000006e-150

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

   if 4.60000000000000006e-150 < n

   1. Initial program 15.3%

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

      1. associate-/r/ 15.7%

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

      2. associate-*r* 15.7%

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

      3. *-commutative 15.7%

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

      4. associate-*r/ 15.7%

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

      5. sub-neg 15.7%

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

      6. distribute-lft-in 15.7%

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

      7. metadata-eval 15.7%

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

      8. metadata-eval 15.7%

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

      9. metadata-eval 15.7%

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

      10. fma-define 15.7%

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

      11. metadata-eval 15.7%

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

   3. Simplified 15.7%

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

      1. fma-undefine 15.7%

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

      2. metadata-eval 15.7%

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

      3. metadata-eval 15.7%

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

      4. distribute-lft-in 15.7%

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

      5. sub-neg 15.7%

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

      6. *-commutative 15.7%

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

      7. add-exp-log 15.6%

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

      8. expm1-define 15.6%

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

      9. log-pow 19.3%

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

      10. log1p-define 75.0%

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

   6. Applied egg-rr 75.0%

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

   7. Taylor expanded in n around inf 31.4%

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

      1. expm1-define 85.3%

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

   9. Simplified 85.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.5 \cdot 10^{-235}:\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \mathbf{if}\;n \leq -1.05 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.3 \cdot 10^{-235}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          100.0
          (/
           (*
            n
            (*
             i
             (+
              1.0
              (*
               i
               (+
                0.5
                (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))))))
           i))))
   (if (<= n -1.05e+48)
     t_0
     (if (<= n -2.3e-235)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 4.6e-150) (/ 0.0 (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	double tmp;
	if (n <= -1.05e+48) {
		tmp = t_0;
	} else if (n <= -2.3e-235) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 4.6e-150) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * ((n * (i * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))) / i)
    if (n <= (-1.05d+48)) then
        tmp = t_0
    else if (n <= (-2.3d-235)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 4.6d-150) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	double tmp;
	if (n <= -1.05e+48) {
		tmp = t_0;
	} else if (n <= -2.3e-235) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 4.6e-150) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i)
	tmp = 0
	if n <= -1.05e+48:
		tmp = t_0
	elif n <= -2.3e-235:
		tmp = 100.0 * (i / (i / n))
	elif n <= 4.6e-150:
		tmp = 0.0 / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))))) / i))
	tmp = 0.0
	if (n <= -1.05e+48)
		tmp = t_0;
	elseif (n <= -2.3e-235)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 4.6e-150)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	tmp = 0.0;
	if (n <= -1.05e+48)
		tmp = t_0;
	elseif (n <= -2.3e-235)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 4.6e-150)
		tmp = 0.0 / (i / n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.05e+48], t$95$0, If[LessEqual[n, -2.3e-235], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.6e-150], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.0499999999999999e48 or 4.60000000000000006e-150 < n

   1. Initial program 19.1%

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

      1. associate-/r/ 19.6%

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

      2. associate-*r* 19.6%

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

      3. *-commutative 19.6%

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

      4. associate-*r/ 19.6%

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

      5. sub-neg 19.6%

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

      6. distribute-lft-in 19.6%

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

      7. metadata-eval 19.6%

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

      8. metadata-eval 19.6%

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

      9. metadata-eval 19.6%

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

      10. fma-define 19.6%

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

      11. metadata-eval 19.6%

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

   3. Simplified 19.6%

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

      1. fma-undefine 19.6%

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

      2. metadata-eval 19.6%

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

      3. metadata-eval 19.6%

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

      4. distribute-lft-in 19.6%

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

      5. sub-neg 19.6%

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

      6. *-commutative 19.6%

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

      7. add-exp-log 19.5%

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

      8. expm1-define 19.5%

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

      9. log-pow 18.6%

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

      10. log1p-define 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in n around inf 37.5%

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

      1. expm1-define 87.9%

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

   9. Simplified 87.9%

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

   10. Taylor expanded in i around 0 68.6%

       \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot i\right)\right)\right)\right)}}{i} \]
   11. Step-by-step derivation

       1. *-commutative 68.6%

          \[\leadsto 100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + \color{blue}{i \cdot 0.041666666666666664}\right)\right)\right)\right)}{i} \]

   12. Simplified 68.6%

       \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}}{i} \]

   if -1.0499999999999999e48 < n < -2.29999999999999997e-235

   1. Initial program 26.7%

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

   3. Taylor expanded in i around 0 60.8%

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

   if -2.29999999999999997e-235 < n < 4.60000000000000006e-150

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{+48}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \mathbf{elif}\;n \leq -2.3 \cdot 10^{-235}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{+47}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{-235}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.6e+47)
   (*
    n
    (+
     100.0
     (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
   (if (<= n -7.5e-235)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 2.7e-151)
       (/ 0.0 (/ i n))
       (*
        100.0
        (+
         n
         (*
          i
          (+
           (* n 0.5)
           (*
            i
            (+
             (* 0.041666666666666664 (* i n))
             (* n 0.16666666666666666)))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.6e+47) {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else if (n <= -7.5e-235) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.7e-151) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.6d+47)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    else if (n <= (-7.5d-235)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 2.7d-151) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = 100.0d0 * (n + (i * ((n * 0.5d0) + (i * ((0.041666666666666664d0 * (i * n)) + (n * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.6e+47) {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else if (n <= -7.5e-235) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.7e-151) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.6e+47:
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	elif n <= -7.5e-235:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.7e-151:
		tmp = 0.0 / (i / n)
	else:
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.6e+47)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	elseif (n <= -7.5e-235)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.7e-151)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.6e+47)
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	elseif (n <= -7.5e-235)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 2.7e-151)
		tmp = 0.0 / (i / n);
	else
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.6e+47], 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], If[LessEqual[n, -7.5e-235], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.7e-151], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.60000000000000075e47

   1. Initial program 25.2%

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

   3. Taylor expanded in n around inf 46.5%

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

      1. expm1-define 65.5%

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

   5. Simplified 65.5%

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

   6. Taylor expanded in i around 0 60.2%

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

   7. Taylor expanded in n around 0 60.3%

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

      1. *-commutative 60.3%

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

   9. Simplified 60.3%

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

   if -9.60000000000000075e47 < n < -7.49999999999999968e-235

   1. Initial program 26.7%

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

   3. Taylor expanded in i around 0 60.8%

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

   if -7.49999999999999968e-235 < n < 2.70000000000000007e-151

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

   if 2.70000000000000007e-151 < n

   1. Initial program 15.3%

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

   3. Taylor expanded in n around inf 31.0%

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

      1. expm1-define 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in i around 0 72.6%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{+47}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{-235}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{if}\;n \leq -1 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (+
           100.0
           (*
            i
            (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))))
   (if (<= n -1e+48)
     t_0
     (if (<= n -4.4e-234)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 3.8e-154) (/ 0.0 (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	double tmp;
	if (n <= -1e+48) {
		tmp = t_0;
	} else if (n <= -4.4e-234) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3.8e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    if (n <= (-1d+48)) then
        tmp = t_0
    else if (n <= (-4.4d-234)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 3.8d-154) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	double tmp;
	if (n <= -1e+48) {
		tmp = t_0;
	} else if (n <= -4.4e-234) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3.8e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	tmp = 0
	if n <= -1e+48:
		tmp = t_0
	elif n <= -4.4e-234:
		tmp = 100.0 * (i / (i / n))
	elif n <= 3.8e-154:
		tmp = 0.0 / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))))
	tmp = 0.0
	if (n <= -1e+48)
		tmp = t_0;
	elseif (n <= -4.4e-234)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 3.8e-154)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	tmp = 0.0;
	if (n <= -1e+48)
		tmp = t_0;
	elseif (n <= -4.4e-234)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 3.8e-154)
		tmp = 0.0 / (i / n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[n, -1e+48], t$95$0, If[LessEqual[n, -4.4e-234], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.8e-154], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.00000000000000004e48 or 3.8000000000000001e-154 < n

   1. Initial program 19.1%

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

   3. Taylor expanded in n around inf 37.0%

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

      1. expm1-define 67.2%

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

   5. Simplified 67.2%

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

   6. Taylor expanded in i around 0 67.8%

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

   7. Taylor expanded in n around 0 67.8%

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

      1. *-commutative 67.8%

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

   9. Simplified 67.8%

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

   if -1.00000000000000004e48 < n < -4.3999999999999998e-234

   1. Initial program 26.7%

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

   3. Taylor expanded in i around 0 60.8%

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

   if -4.3999999999999998e-234 < n < 3.8000000000000001e-154

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+48}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;n \leq -1 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.75 \cdot 10^{-235}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (+ n (* i (* n (+ 0.5 (* i 0.16666666666666666))))))))
   (if (<= n -1e+48)
     t_0
     (if (<= n -2.75e-235)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 5.8e-154) (/ 0.0 (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n + (i * (n * (0.5 + (i * 0.16666666666666666)))));
	double tmp;
	if (n <= -1e+48) {
		tmp = t_0;
	} else if (n <= -2.75e-235) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.8e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n + (i * (n * (0.5d0 + (i * 0.16666666666666666d0)))))
    if (n <= (-1d+48)) then
        tmp = t_0
    else if (n <= (-2.75d-235)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 5.8d-154) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n + (i * (n * (0.5 + (i * 0.16666666666666666)))));
	double tmp;
	if (n <= -1e+48) {
		tmp = t_0;
	} else if (n <= -2.75e-235) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.8e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n + (i * (n * (0.5 + (i * 0.16666666666666666)))))
	tmp = 0
	if n <= -1e+48:
		tmp = t_0
	elif n <= -2.75e-235:
		tmp = 100.0 * (i / (i / n))
	elif n <= 5.8e-154:
		tmp = 0.0 / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(0.5 + Float64(i * 0.16666666666666666))))))
	tmp = 0.0
	if (n <= -1e+48)
		tmp = t_0;
	elseif (n <= -2.75e-235)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 5.8e-154)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n + (i * (n * (0.5 + (i * 0.16666666666666666)))));
	tmp = 0.0;
	if (n <= -1e+48)
		tmp = t_0;
	elseif (n <= -2.75e-235)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 5.8e-154)
		tmp = 0.0 / (i / n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n + N[(i * N[(n * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1e+48], t$95$0, If[LessEqual[n, -2.75e-235], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-154], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.00000000000000004e48 or 5.8e-154 < n

   1. Initial program 19.1%

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

   3. Taylor expanded in n around inf 37.0%

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

      1. expm1-define 67.2%

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

   5. Simplified 67.2%

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

   6. Taylor expanded in i around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. associate-*r* 66.0%

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

      3. distribute-rgt-in 66.0%

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

      4. associate-*r* 66.0%

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

      5. *-commutative 66.0%

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

      6. *-commutative 66.0%

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

   8. Simplified 66.0%

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

   9. Taylor expanded in n around 0 66.0%

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

   if -1.00000000000000004e48 < n < -2.7499999999999999e-235

   1. Initial program 26.7%

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

   3. Taylor expanded in i around 0 60.8%

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

   if -2.7499999999999999e-235 < n < 5.8e-154

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+48}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -2.75 \cdot 10^{-235}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.95 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e-19)
   (* 100.0 (/ (* i n) i))
   (if (<= n -1.95e-234)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 3.3e-154) (/ 0.0 (/ i n)) (+ (* 50.0 (* i n)) (* n 100.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e-19) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -1.95e-234) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3.3e-154) {
		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 (n <= (-5d-19)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= (-1.95d-234)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 3.3d-154) 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 (n <= -5e-19) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -1.95e-234) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3.3e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5e-19:
		tmp = 100.0 * ((i * n) / i)
	elif n <= -1.95e-234:
		tmp = 100.0 * (i / (i / n))
	elif n <= 3.3e-154:
		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 (n <= -5e-19)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -1.95e-234)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 3.3e-154)
		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 (n <= -5e-19)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= -1.95e-234)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 3.3e-154)
		tmp = 0.0 / (i / n);
	else
		tmp = (50.0 * (i * n)) + (n * 100.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e-19], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.95e-234], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.3e-154], 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 4 regimes

2. if n < -5.0000000000000004e-19

   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/ 24.4%

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

      2. associate-*r* 24.3%

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

      3. *-commutative 24.3%

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

      4. associate-*r/ 24.4%

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

      5. sub-neg 24.4%

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

      6. distribute-lft-in 24.4%

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

      7. metadata-eval 24.4%

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

      8. metadata-eval 24.4%

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

      9. metadata-eval 24.4%

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

      10. fma-define 24.4%

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

      11. metadata-eval 24.4%

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

   3. Simplified 24.4%

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

      1. fma-undefine 24.4%

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

      2. metadata-eval 24.4%

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

      3. metadata-eval 24.4%

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

      4. distribute-lft-in 24.4%

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

      5. sub-neg 24.4%

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

      6. *-commutative 24.4%

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

      7. add-exp-log 24.4%

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

      8. expm1-define 24.4%

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

      9. log-pow 17.0%

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

      10. log1p-define 71.5%

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

   6. Applied egg-rr 71.5%

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

   7. Taylor expanded in n around inf 37.5%

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

      1. expm1-define 90.6%

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

   9. Simplified 90.6%

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

   10. Taylor expanded in i around 0 64.3%

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

       1. *-commutative 64.3%

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

   12. Simplified 64.3%

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

   if -5.0000000000000004e-19 < n < -1.9500000000000001e-234

   1. Initial program 31.3%

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

   3. Taylor expanded in i around 0 46.3%

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

   if -1.9500000000000001e-234 < n < 3.30000000000000027e-154

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

   if 3.30000000000000027e-154 < n

   1. Initial program 15.3%

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

   3. Taylor expanded in n around inf 31.0%

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

      1. expm1-define 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in i around 0 65.7%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.95 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e-19)
   (* 100.0 (/ (* i n) i))
   (if (<= n -4.4e-234)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 5.8e-154) (/ 0.0 (/ i n)) (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e-19) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -4.4e-234) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.8e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5d-19)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= (-4.4d-234)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 5.8d-154) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5e-19) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -4.4e-234) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.8e-154) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5e-19:
		tmp = 100.0 * ((i * n) / i)
	elif n <= -4.4e-234:
		tmp = 100.0 * (i / (i / n))
	elif n <= 5.8e-154:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e-19)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -4.4e-234)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 5.8e-154)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -5e-19)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= -4.4e-234)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 5.8e-154)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e-19], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -4.4e-234], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-154], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.0000000000000004e-19

   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/ 24.4%

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

      2. associate-*r* 24.3%

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

      3. *-commutative 24.3%

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

      4. associate-*r/ 24.4%

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

      5. sub-neg 24.4%

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

      6. distribute-lft-in 24.4%

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

      7. metadata-eval 24.4%

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

      8. metadata-eval 24.4%

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

      9. metadata-eval 24.4%

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

      10. fma-define 24.4%

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

      11. metadata-eval 24.4%

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

   3. Simplified 24.4%

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

      1. fma-undefine 24.4%

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

      2. metadata-eval 24.4%

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

      3. metadata-eval 24.4%

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

      4. distribute-lft-in 24.4%

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

      5. sub-neg 24.4%

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

      6. *-commutative 24.4%

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

      7. add-exp-log 24.4%

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

      8. expm1-define 24.4%

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

      9. log-pow 17.0%

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

      10. log1p-define 71.5%

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

   6. Applied egg-rr 71.5%

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

   7. Taylor expanded in n around inf 37.5%

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

      1. expm1-define 90.6%

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

   9. Simplified 90.6%

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

   10. Taylor expanded in i around 0 64.3%

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

       1. *-commutative 64.3%

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

   12. Simplified 64.3%

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

   if -5.0000000000000004e-19 < n < -4.3999999999999998e-234

   1. Initial program 31.3%

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

   3. Taylor expanded in i around 0 46.3%

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

   if -4.3999999999999998e-234 < n < 5.8e-154

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in i around inf 57.3%

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

   6. Taylor expanded in n around 0 75.3%

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

   if 5.8e-154 < n

   1. Initial program 15.3%

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

      1. associate-/r/ 15.7%

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

      2. associate-*r* 15.7%

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

      3. *-commutative 15.7%

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

      4. associate-*r/ 15.7%

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

      5. sub-neg 15.7%

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

      6. distribute-lft-in 15.7%

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

      7. metadata-eval 15.7%

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

      8. metadata-eval 15.7%

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

      9. metadata-eval 15.7%

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

      10. fma-define 15.7%

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

      11. metadata-eval 15.7%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in i around 0 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*r/ 66.2%

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

      3. metadata-eval 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in n around inf 65.7%

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

      1. *-commutative 65.7%

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

   10. Simplified 65.7%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-234} \lor \neg \left(n \leq 1.2 \cdot 10^{-142}\right):\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.4e-234) (not (<= n 1.2e-142)))
   (+ (* i -50.0) (* n (+ 100.0 (* i 50.0))))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.4e-234) || !(n <= 1.2e-142)) {
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-4.4d-234)) .or. (.not. (n <= 1.2d-142))) then
        tmp = (i * (-50.0d0)) + (n * (100.0d0 + (i * 50.0d0)))
    else
        tmp = 0.0d0 / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.4e-234) || !(n <= 1.2e-142)) {
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.4e-234) or not (n <= 1.2e-142):
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.4e-234) || !(n <= 1.2e-142))
		tmp = Float64(Float64(i * -50.0) + Float64(n * Float64(100.0 + Float64(i * 50.0))));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -4.4e-234) || ~((n <= 1.2e-142)))
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)));
	else
		tmp = 0.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.4e-234], N[Not[LessEqual[n, 1.2e-142]], $MachinePrecision]], N[(N[(i * -50.0), $MachinePrecision] + N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.3999999999999998e-234 or 1.19999999999999994e-142 < n

   1. Initial program 21.2%

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

      1. associate-/r/ 21.5%

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

      2. associate-*r* 21.5%

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

      3. *-commutative 21.5%

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

      4. associate-*r/ 21.5%

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

      5. sub-neg 21.5%

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

      6. distribute-lft-in 21.5%

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

      7. metadata-eval 21.5%

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

      8. metadata-eval 21.5%

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

      9. metadata-eval 21.5%

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

      10. fma-define 21.5%

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

      11. metadata-eval 21.5%

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

   3. Simplified 21.5%

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

   5. Taylor expanded in i around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*r/ 61.4%

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

      3. metadata-eval 61.4%

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

   7. Simplified 61.4%

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

   8. Taylor expanded in n around 0 61.4%

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

   if -4.3999999999999998e-234 < n < 1.19999999999999994e-142

   1. Initial program 55.3%

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

      1. associate-*r/ 55.3%

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

      2. sub-neg 55.3%

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

      3. distribute-rgt-in 55.3%

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

      4. metadata-eval 55.3%

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

      5. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in i around inf 55.1%

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

   6. Taylor expanded in n around 0 72.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-234} \lor \neg \left(n \leq 1.2 \cdot 10^{-142}\right):\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-24} \lor \neg \left(n \leq 1.4 \cdot 10^{+17}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2e-24) (not (<= n 1.4e+17)))
   (* 100.0 (/ (* i n) i))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2e-24) || !(n <= 1.4e+17)) {
		tmp = 100.0 * ((i * n) / i);
	} 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 <= (-2d-24)) .or. (.not. (n <= 1.4d+17))) then
        tmp = 100.0d0 * ((i * n) / i)
    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 <= -2e-24) || !(n <= 1.4e+17)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2e-24) or not (n <= 1.4e+17):
		tmp = 100.0 * ((i * n) / i)
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2e-24) || !(n <= 1.4e+17))
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	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 <= -2e-24) || ~((n <= 1.4e+17)))
		tmp = 100.0 * ((i * n) / i);
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2e-24], N[Not[LessEqual[n, 1.4e+17]], $MachinePrecision]], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.99999999999999985e-24 or 1.4e17 < n

   1. Initial program 20.9%

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

      1. associate-/r/ 21.3%

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

      2. associate-*r* 21.3%

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

      3. *-commutative 21.3%

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

      4. associate-*r/ 21.3%

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

      5. sub-neg 21.3%

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

      6. distribute-lft-in 21.3%

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

      7. metadata-eval 21.3%

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

      8. metadata-eval 21.3%

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

      9. metadata-eval 21.3%

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

      10. fma-define 21.3%

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

      11. metadata-eval 21.3%

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

   3. Simplified 21.3%

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

      1. fma-undefine 21.3%

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

      2. metadata-eval 21.3%

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

      3. metadata-eval 21.3%

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

      4. distribute-lft-in 21.3%

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

      5. sub-neg 21.3%

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

      6. *-commutative 21.3%

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

      7. add-exp-log 21.3%

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

      8. expm1-define 21.3%

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

      9. log-pow 14.7%

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

      10. log1p-define 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in n around inf 39.0%

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

      1. expm1-define 92.2%

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

   9. Simplified 92.2%

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

   10. Taylor expanded in i around 0 65.8%

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

       1. *-commutative 65.8%

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

   12. Simplified 65.8%

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

   if -1.99999999999999985e-24 < n < 1.4e17

   1. Initial program 37.5%

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

   3. Taylor expanded in i around 0 55.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-24} \lor \neg \left(n \leq 1.4 \cdot 10^{+17}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-19} \lor \neg \left(n \leq 2 \cdot 10^{-20}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-19) (not (<= n 2e-20)))
   (* n 100.0)
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-19) || !(n <= 2e-20)) {
		tmp = n * 100.0;
	} 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 <= (-5d-19)) .or. (.not. (n <= 2d-20))) then
        tmp = n * 100.0d0
    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 <= -5e-19) || !(n <= 2e-20)) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-19) or not (n <= 2e-20):
		tmp = n * 100.0
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-19) || !(n <= 2e-20))
		tmp = Float64(n * 100.0);
	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 <= -5e-19) || ~((n <= 2e-20)))
		tmp = n * 100.0;
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-19], N[Not[LessEqual[n, 2e-20]], $MachinePrecision]], N[(n * 100.0), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.0000000000000004e-19 or 1.99999999999999989e-20 < n

   1. Initial program 21.5%

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

      1. associate-/r/ 21.9%

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

      2. associate-*r* 21.9%

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

      3. *-commutative 21.9%

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

      4. associate-*r/ 21.9%

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

      5. sub-neg 21.9%

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

      6. distribute-lft-in 21.9%

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

      7. metadata-eval 21.9%

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

      8. metadata-eval 21.9%

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

      9. metadata-eval 21.9%

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

      10. fma-define 21.9%

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

      11. metadata-eval 21.9%

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

   3. Simplified 21.9%

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

   5. Taylor expanded in i around 0 57.7%

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

      1. *-commutative 57.7%

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

   7. Simplified 57.7%

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

   if -5.0000000000000004e-19 < n < 1.99999999999999989e-20

   1. Initial program 37.3%

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

   3. Taylor expanded in i around 0 55.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-19} \lor \neg \left(n \leq 2 \cdot 10^{-20}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 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 27.4%

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

   1. associate-/r/ 27.6%

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

   2. associate-*r* 27.6%

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

   3. *-commutative 27.6%

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

   4. associate-*r/ 27.6%

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

   5. sub-neg 27.6%

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

   6. distribute-lft-in 27.6%

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

   7. metadata-eval 27.6%

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

   8. metadata-eval 27.6%

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

   9. metadata-eval 27.6%

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

   10. fma-define 27.6%

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

   11. metadata-eval 27.6%

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

3. Simplified 27.6%

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

5. Taylor expanded in i around 0 48.0%

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

   1. *-commutative 48.0%

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

7. Simplified 48.0%

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

Alternative 20: 2.8% 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 27.4%

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

   1. associate-/r/ 27.6%

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

   2. associate-*r* 27.6%

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

   3. *-commutative 27.6%

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

   4. associate-*r/ 27.6%

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

   5. sub-neg 27.6%

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

   6. distribute-lft-in 27.6%

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

   7. metadata-eval 27.6%

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

   8. metadata-eval 27.6%

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

   9. metadata-eval 27.6%

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

   10. fma-define 27.6%

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

   11. metadata-eval 27.6%

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

3. Simplified 27.6%

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

5. Taylor expanded in i around 0 52.5%

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

   1. *-commutative 52.5%

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

   2. associate-*r/ 52.5%

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

   3. metadata-eval 52.5%

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

7. Simplified 52.5%

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

8. Taylor expanded in n around 0 2.9%

   \[\leadsto \color{blue}{-50 \cdot i} \]
9. Step-by-step derivation

   1. *-commutative 2.9%

      \[\leadsto \color{blue}{i \cdot -50} \]

10. Simplified 2.9%

    \[\leadsto \color{blue}{i \cdot -50} \]
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 2024185 
(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))))