Compound Interest

Percentage Accurate: 28.9% → 99.1%

Time: 16.2s

Alternatives: 18

Speedup: 38.0×

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: 28.9% 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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{i}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-249}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 (- INFINITY))
     (*
      n
      (/
       (*
        i
        (+
         100.0
         (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
       i))
     (if (<= t_0 2e-249)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_0 INFINITY)
         (* t_0 100.0)
         (* n (/ 1.0 (+ 0.01 (* i -0.005)))))))))

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 <= -((double) INFINITY)) {
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i);
	} else if (t_0 <= 2e-249) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	}
	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 <= -Double.POSITIVE_INFINITY) {
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i);
	} else if (t_0 <= 2e-249) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	}
	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 <= -math.inf:
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i)
	elif t_0 <= 2e-249:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = n * (1.0 / (0.01 + (i * -0.005)))
	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 <= Float64(-Inf))
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667))))))) / i));
	elseif (t_0 <= 2e-249)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(i * -0.005))));
	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, (-Infinity)], N[(n * N[(N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-249], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(n * N[(1.0 / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

      1. associate-/r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-define 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in n around inf 0.8%

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

      1. sub-neg 0.8%

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

      2. metadata-eval 0.8%

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

      3. metadata-eval 0.8%

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

      4. distribute-lft-in 0.8%

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

      5. metadata-eval 0.8%

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

      6. sub-neg 0.8%

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

      7. expm1-define 0.8%

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

   7. Simplified 0.8%

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

   8. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 2.00000000000000011e-249

   1. Initial program 25.5%

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

      1. sub-neg 25.5%

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

      2. metadata-eval 25.5%

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

   4. Applied egg-rr 25.5%

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

      1. metadata-eval 25.5%

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

      2. sub-neg 25.5%

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

      3. exp-to-pow 25.5%

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

      4. log1p-undefine 48.3%

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

      5. *-commutative 48.3%

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

      6. expm1-undefine 99.7%

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

   6. Simplified 99.7%

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

   if 2.00000000000000011e-249 < (/.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 99.9%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 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. Taylor expanded in n around inf 1.9%

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

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-define 85.3%

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

   7. Simplified 85.3%

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

      1. clear-num 85.3%

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

      2. inv-pow 85.3%

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

      3. *-un-lft-identity 85.3%

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

      4. times-frac 85.4%

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

      5. metadata-eval 85.4%

         \[\leadsto n \cdot {\left(\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1} \]

   9. Applied egg-rr 85.4%

      \[\leadsto n \cdot \color{blue}{{\left(0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 85.4%

          \[\leadsto n \cdot \color{blue}{\frac{1}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

       2. associate-*r/ 85.3%

          \[\leadsto n \cdot \frac{1}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 85.3%

       \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 100.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   13. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto n \cdot \frac{1}{0.01 + \color{blue}{i \cdot -0.005}} \]

   14. Simplified 100.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + i \cdot -0.005}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-249}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.65 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-138}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-111}:\\ \;\;\;\;n \cdot \left(\left(n \cdot \log \left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* n (/ (expm1 i) i)))))
   (if (<= n -1.65e-201)
     t_0
     (if (<= n 2e-138)
       (/ 0.0 (/ i n))
       (if (<= n 8e-111) (* n (* (* n (log (/ i n))) (/ 100.0 i))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (expm1(i) / i));
	double tmp;
	if (n <= -1.65e-201) {
		tmp = t_0;
	} else if (n <= 2e-138) {
		tmp = 0.0 / (i / n);
	} else if (n <= 8e-111) {
		tmp = n * ((n * log((i / n))) * (100.0 / i));
	} 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 <= -1.65e-201) {
		tmp = t_0;
	} else if (n <= 2e-138) {
		tmp = 0.0 / (i / n);
	} else if (n <= 8e-111) {
		tmp = n * ((n * Math.log((i / n))) * (100.0 / i));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (math.expm1(i) / i))
	tmp = 0
	if n <= -1.65e-201:
		tmp = t_0
	elif n <= 2e-138:
		tmp = 0.0 / (i / n)
	elif n <= 8e-111:
		tmp = n * ((n * math.log((i / n))) * (100.0 / i))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -1.65e-201)
		tmp = t_0;
	elseif (n <= 2e-138)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 8e-111)
		tmp = Float64(n * Float64(Float64(n * log(Float64(i / n))) * Float64(100.0 / i)));
	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] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.65e-201], t$95$0, If[LessEqual[n, 2e-138], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8e-111], N[(n * N[(N[(n * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.6500000000000002e-201 or 8.00000000000000071e-111 < n

   1. Initial program 25.7%

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

   3. Taylor expanded in n around inf 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/l* 34.8%

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

      3. expm1-define 85.7%

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

   5. Simplified 85.7%

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

   if -1.6500000000000002e-201 < n < 2.00000000000000013e-138

   1. Initial program 56.3%

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

      1. associate-*r/ 56.3%

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

      2. sub-neg 56.3%

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

      3. distribute-rgt-in 56.3%

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

      4. metadata-eval 56.3%

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

      5. metadata-eval 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in i around 0 82.1%

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

   if 2.00000000000000013e-138 < n < 8.00000000000000071e-111

   1. Initial program 36.8%

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

      1. associate-/r/ 36.8%

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

      2. associate-*r* 36.8%

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

      3. *-commutative 36.8%

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

      4. associate-*r/ 36.8%

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

      5. sub-neg 36.8%

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

      6. distribute-lft-in 36.8%

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

      7. metadata-eval 36.8%

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

      8. metadata-eval 36.8%

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

      9. metadata-eval 36.8%

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

      10. fma-define 36.8%

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

      11. metadata-eval 36.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in n around 0 96.0%

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

      1. associate-*r/ 95.9%

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

      2. *-commutative 95.9%

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

      3. mul-1-neg 95.9%

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

      4. unsub-neg 95.9%

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

   7. Simplified 95.9%

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

      1. associate-/l* 96.0%

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

      2. diff-log 96.2%

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

   9. Applied egg-rr 96.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-138}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-111}:\\ \;\;\;\;n \cdot \left(\left(n \cdot \log \left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-204}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\log \left(\frac{i}{n}\right) \cdot \frac{n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* n (/ (expm1 i) i)))))
   (if (<= n -5.2e-204)
     t_0
     (if (<= n 8.5e-139)
       (/ 0.0 (/ i n))
       (if (<= n 1.8e-112) (* n (* 100.0 (* (log (/ i n)) (/ n i)))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (expm1(i) / i));
	double tmp;
	if (n <= -5.2e-204) {
		tmp = t_0;
	} else if (n <= 8.5e-139) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.8e-112) {
		tmp = n * (100.0 * (log((i / n)) * (n / i)));
	} 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 <= -5.2e-204) {
		tmp = t_0;
	} else if (n <= 8.5e-139) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.8e-112) {
		tmp = n * (100.0 * (Math.log((i / n)) * (n / i)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (math.expm1(i) / i))
	tmp = 0
	if n <= -5.2e-204:
		tmp = t_0
	elif n <= 8.5e-139:
		tmp = 0.0 / (i / n)
	elif n <= 1.8e-112:
		tmp = n * (100.0 * (math.log((i / n)) * (n / i)))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -5.2e-204)
		tmp = t_0;
	elseif (n <= 8.5e-139)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.8e-112)
		tmp = Float64(n * Float64(100.0 * Float64(log(Float64(i / n)) * Float64(n / i))));
	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] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.2e-204], t$95$0, If[LessEqual[n, 8.5e-139], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-112], N[(n * N[(100.0 * N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.19999999999999965e-204 or 1.8e-112 < n

   1. Initial program 25.7%

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

   3. Taylor expanded in n around inf 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/l* 34.8%

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

      3. expm1-define 85.7%

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

   5. Simplified 85.7%

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

   if -5.19999999999999965e-204 < n < 8.5000000000000003e-139

   1. Initial program 56.3%

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

      1. associate-*r/ 56.3%

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

      2. sub-neg 56.3%

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

      3. distribute-rgt-in 56.3%

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

      4. metadata-eval 56.3%

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

      5. metadata-eval 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in i around 0 82.1%

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

   if 8.5000000000000003e-139 < n < 1.8e-112

   1. Initial program 36.8%

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

      1. associate-/r/ 36.8%

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

      2. associate-*r* 36.8%

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

      3. *-commutative 36.8%

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

      4. associate-*r/ 36.8%

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

      5. sub-neg 36.8%

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

      6. distribute-lft-in 36.8%

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

      7. metadata-eval 36.8%

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

      8. metadata-eval 36.8%

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

      9. metadata-eval 36.8%

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

      10. fma-define 36.8%

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

      11. metadata-eval 36.8%

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

   3. Simplified 36.8%

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

   5. Taylor expanded in n around 0 96.0%

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

      1. associate-*r/ 95.9%

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

      2. *-commutative 95.9%

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

      3. mul-1-neg 95.9%

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

      4. unsub-neg 95.9%

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

   7. Simplified 95.9%

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

      1. *-commutative 95.9%

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

      2. *-un-lft-identity 95.9%

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

      3. times-frac 96.0%

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

      4. metadata-eval 96.0%

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

      5. diff-log 96.2%

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

   9. Applied egg-rr 96.2%

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

       1. *-commutative 96.2%

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

       2. associate-/l* 96.1%

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

   11. Simplified 96.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-204}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;n \cdot \left(100 \cdot \left(\log \left(\frac{i}{n}\right) \cdot \frac{n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-208} \lor \neg \left(n \leq 4 \cdot 10^{-114}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9e-208) (not (<= n 4e-114)))
   (* 100.0 (* n (/ (expm1 i) i)))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9e-208) || !(n <= 4e-114)) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9e-208) || !(n <= 4e-114)) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9e-208) or not (n <= 4e-114):
		tmp = 100.0 * (n * (math.expm1(i) / i))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9e-208) || !(n <= 4e-114))
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9e-208], N[Not[LessEqual[n, 4e-114]], $MachinePrecision]], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.9999999999999992e-208 or 4.0000000000000002e-114 < n

   1. Initial program 25.6%

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

   3. Taylor expanded in n around inf 34.6%

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

      1. *-commutative 34.6%

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

      2. associate-/l* 34.7%

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

      3. expm1-define 85.3%

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

   5. Simplified 85.3%

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

   if -8.9999999999999992e-208 < n < 4.0000000000000002e-114

   1. Initial program 54.0%

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

      1. associate-*r/ 54.0%

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

      2. sub-neg 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in i around 0 75.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-208} \lor \neg \left(n \leq 4 \cdot 10^{-114}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-209} \lor \neg \left(n \leq 3.8 \cdot 10^{-114}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -7.5e-209) (not (<= n 3.8e-114)))
   (* n (/ (* 100.0 (expm1 i)) i))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -7.5e-209) || !(n <= 3.8e-114)) {
		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 <= -7.5e-209) || !(n <= 3.8e-114)) {
		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 <= -7.5e-209) or not (n <= 3.8e-114):
		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 <= -7.5e-209) || !(n <= 3.8e-114))
		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, -7.5e-209], N[Not[LessEqual[n, 3.8e-114]], $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 < -7.49999999999999965e-209 or 3.7999999999999998e-114 < n

   1. Initial program 25.6%

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

      1. associate-/r/ 25.7%

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

      2. associate-*r* 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-*r/ 25.7%

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

      5. sub-neg 25.7%

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

      6. distribute-lft-in 25.7%

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

      7. metadata-eval 25.7%

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

      8. metadata-eval 25.7%

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

      9. metadata-eval 25.7%

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

      10. fma-define 25.7%

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

      11. metadata-eval 25.7%

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

   3. Simplified 25.7%

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

   5. Taylor expanded in n around inf 34.6%

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

      1. sub-neg 34.6%

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

      2. metadata-eval 34.6%

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

      3. metadata-eval 34.6%

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

      4. distribute-lft-in 34.7%

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

      5. metadata-eval 34.7%

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

      6. sub-neg 34.7%

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

      7. expm1-define 85.3%

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

   7. Simplified 85.3%

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

   if -7.49999999999999965e-209 < n < 3.7999999999999998e-114

   1. Initial program 54.0%

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

      1. associate-*r/ 54.0%

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

      2. sub-neg 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in i around 0 75.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-209} \lor \neg \left(n \leq 3.8 \cdot 10^{-114}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{+179}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -8.8 \cdot 10^{-209}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.05e+179)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n -8.8e-209)
     (* n (/ 1.0 (+ 0.01 (* i -0.005))))
     (if (<= n 4.4e-114)
       (/ 0.0 (/ i n))
       (*
        100.0
        (*
         n
         (/
          (*
           i
           (+
            1.0
            (*
             i
             (+
              0.5
              (* i (+ 0.16666666666666666 (* i 0.041666666666666664)))))))
          i)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.05e+179) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -8.8e-209) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 4.4e-114) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n * ((i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664))))))) / i));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.05d+179)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= (-8.8d-209)) then
        tmp = n * (1.0d0 / (0.01d0 + (i * (-0.005d0))))
    else if (n <= 4.4d-114) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = 100.0d0 * (n * ((i * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0))))))) / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.05e+179) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -8.8e-209) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 4.4e-114) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n * ((i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664))))))) / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.05e+179:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= -8.8e-209:
		tmp = n * (1.0 / (0.01 + (i * -0.005)))
	elif n <= 4.4e-114:
		tmp = 0.0 / (i / n)
	else:
		tmp = 100.0 * (n * ((i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664))))))) / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.05e+179)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= -8.8e-209)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(i * -0.005))));
	elseif (n <= 4.4e-114)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n * Float64(Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664))))))) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.05e+179)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= -8.8e-209)
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	elseif (n <= 4.4e-114)
		tmp = 0.0 / (i / n);
	else
		tmp = 100.0 * (n * ((i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664))))))) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.05e+179], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -8.8e-209], N[(n * N[(1.0 / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.4e-114], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], 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] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.0499999999999999e179

   1. Initial program 6.2%

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

   3. Taylor expanded in n around inf 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-/l* 35.0%

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

      3. expm1-define 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in i around 0 79.4%

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

      1. *-commutative 79.4%

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

   8. Simplified 79.4%

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

   if -1.0499999999999999e179 < n < -8.80000000000000039e-209

   1. Initial program 32.5%

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

      1. associate-/r/ 32.0%

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

      2. associate-*r* 31.9%

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

      3. *-commutative 31.9%

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

      4. associate-*r/ 32.0%

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

      5. sub-neg 32.0%

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

      6. distribute-lft-in 31.8%

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

      7. metadata-eval 31.8%

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

      8. metadata-eval 31.8%

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

      9. metadata-eval 31.8%

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

      10. fma-define 32.0%

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

      11. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in n around inf 31.2%

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

      1. sub-neg 31.2%

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

      2. metadata-eval 31.2%

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

      3. metadata-eval 31.2%

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

      4. distribute-lft-in 31.2%

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

      5. metadata-eval 31.2%

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

      6. sub-neg 31.2%

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

      7. expm1-define 77.3%

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

   7. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

      3. *-un-lft-identity 77.3%

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

      4. times-frac 77.4%

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

      5. metadata-eval 77.4%

         \[\leadsto n \cdot {\left(\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1} \]

   9. Applied egg-rr 77.4%

      \[\leadsto n \cdot \color{blue}{{\left(0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 77.4%

          \[\leadsto n \cdot \color{blue}{\frac{1}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

       2. associate-*r/ 77.4%

          \[\leadsto n \cdot \frac{1}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 77.4%

       \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   13. Step-by-step derivation

       1. *-commutative 62.0%

          \[\leadsto n \cdot \frac{1}{0.01 + \color{blue}{i \cdot -0.005}} \]

   14. Simplified 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -8.80000000000000039e-209 < n < 4.40000000000000022e-114

   1. Initial program 54.0%

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

      1. associate-*r/ 54.0%

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

      2. sub-neg 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in i around 0 75.6%

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

   if 4.40000000000000022e-114 < n

   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 n around inf 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-/l* 37.9%

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

      3. expm1-define 89.4%

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

   5. Simplified 89.4%

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

   6. Taylor expanded in i around 0 83.1%

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

      1. *-commutative 83.1%

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

   8. Simplified 83.1%

      \[\leadsto \left(n \cdot \frac{\color{blue}{i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)}}{i}\right) \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{+179}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -8.8 \cdot 10^{-209}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{+179}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -1.95 \cdot 10^{-209}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-113}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.4e+179)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n -1.95e-209)
     (* n (/ 1.0 (+ 0.01 (* i -0.005))))
     (if (<= n 1.65e-113)
       (/ 0.0 (/ i n))
       (*
        n
        (/
         (*
          i
          (+
           100.0
           (*
            i
            (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
         i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.4e+179) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -1.95e-209) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 1.65e-113) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.4d+179)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= (-1.95d-209)) then
        tmp = n * (1.0d0 / (0.01d0 + (i * (-0.005d0))))
    else if (n <= 1.65d-113) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * ((i * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.4e+179) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -1.95e-209) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 1.65e-113) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.4e+179:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= -1.95e-209:
		tmp = n * (1.0 / (0.01 + (i * -0.005)))
	elif n <= 1.65e-113:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.4e+179)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= -1.95e-209)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(i * -0.005))));
	elseif (n <= 1.65e-113)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(Float64(i * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667))))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.4e+179)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= -1.95e-209)
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	elseif (n <= 1.65e-113)
		tmp = 0.0 / (i / n);
	else
		tmp = n * ((i * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.4e+179], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.95e-209], N[(n * N[(1.0 / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e-113], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(i * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.40000000000000013e179

   1. Initial program 6.2%

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

   3. Taylor expanded in n around inf 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-/l* 35.0%

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

      3. expm1-define 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in i around 0 79.4%

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

      1. *-commutative 79.4%

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

   8. Simplified 79.4%

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

   if -2.40000000000000013e179 < n < -1.95e-209

   1. Initial program 32.5%

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

      1. associate-/r/ 32.0%

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

      2. associate-*r* 31.9%

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

      3. *-commutative 31.9%

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

      4. associate-*r/ 32.0%

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

      5. sub-neg 32.0%

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

      6. distribute-lft-in 31.8%

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

      7. metadata-eval 31.8%

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

      8. metadata-eval 31.8%

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

      9. metadata-eval 31.8%

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

      10. fma-define 32.0%

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

      11. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in n around inf 31.2%

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

      1. sub-neg 31.2%

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

      2. metadata-eval 31.2%

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

      3. metadata-eval 31.2%

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

      4. distribute-lft-in 31.2%

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

      5. metadata-eval 31.2%

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

      6. sub-neg 31.2%

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

      7. expm1-define 77.3%

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

   7. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

      3. *-un-lft-identity 77.3%

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

      4. times-frac 77.4%

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

      5. metadata-eval 77.4%

         \[\leadsto n \cdot {\left(\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1} \]

   9. Applied egg-rr 77.4%

      \[\leadsto n \cdot \color{blue}{{\left(0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 77.4%

          \[\leadsto n \cdot \color{blue}{\frac{1}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

       2. associate-*r/ 77.4%

          \[\leadsto n \cdot \frac{1}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 77.4%

       \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   13. Step-by-step derivation

       1. *-commutative 62.0%

          \[\leadsto n \cdot \frac{1}{0.01 + \color{blue}{i \cdot -0.005}} \]

   14. Simplified 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -1.95e-209 < n < 1.6500000000000001e-113

   1. Initial program 54.0%

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

      1. associate-*r/ 54.0%

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

      2. sub-neg 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in i around 0 75.6%

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

   if 1.6500000000000001e-113 < n

   1. Initial program 26.7%

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

      1. associate-/r/ 27.1%

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

      2. associate-*r* 27.1%

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

      3. *-commutative 27.1%

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

      4. associate-*r/ 27.1%

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

      5. sub-neg 27.1%

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

      6. distribute-lft-in 27.1%

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

      7. metadata-eval 27.1%

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

      8. metadata-eval 27.1%

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

      9. metadata-eval 27.1%

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

      10. fma-define 27.1%

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

      11. metadata-eval 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in n around inf 37.9%

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

      1. sub-neg 37.9%

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

      2. metadata-eval 37.9%

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

      3. metadata-eval 37.9%

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

      4. distribute-lft-in 37.9%

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

      5. metadata-eval 37.9%

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

      6. sub-neg 37.9%

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

      7. expm1-define 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in i around 0 83.0%

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

      1. *-commutative 83.0%

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

   10. Simplified 83.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{+179}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -1.95 \cdot 10^{-209}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-113}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{+181}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -5.1 \cdot 10^{-201}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.1e+181)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n -5.1e-201)
     (* n (/ 1.0 (+ 0.01 (* i -0.005))))
     (if (<= n 6e-114)
       (/ 0.0 (/ i n))
       (*
        n
        (+
         100.0
         (*
          i
          (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.1e+181) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -5.1e-201) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 6e-114) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.1d+181)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= (-5.1d-201)) then
        tmp = n * (1.0d0 / (0.01d0 + (i * (-0.005d0))))
    else if (n <= 6d-114) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.1e+181) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -5.1e-201) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 6e-114) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.1e+181:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= -5.1e-201:
		tmp = n * (1.0 / (0.01 + (i * -0.005)))
	elif n <= 6e-114:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.1e+181)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= -5.1e-201)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(i * -0.005))));
	elseif (n <= 6e-114)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.1e+181)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= -5.1e-201)
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	elseif (n <= 6e-114)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.1e+181], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5.1e-201], N[(n * N[(1.0 / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6e-114], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.09999999999999989e181

   1. Initial program 6.2%

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

   3. Taylor expanded in n around inf 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-/l* 35.0%

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

      3. expm1-define 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in i around 0 79.4%

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

      1. *-commutative 79.4%

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

   8. Simplified 79.4%

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

   if -3.09999999999999989e181 < n < -5.1000000000000001e-201

   1. Initial program 32.5%

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

      1. associate-/r/ 32.0%

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

      2. associate-*r* 31.9%

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

      3. *-commutative 31.9%

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

      4. associate-*r/ 32.0%

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

      5. sub-neg 32.0%

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

      6. distribute-lft-in 31.8%

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

      7. metadata-eval 31.8%

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

      8. metadata-eval 31.8%

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

      9. metadata-eval 31.8%

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

      10. fma-define 32.0%

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

      11. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in n around inf 31.2%

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

      1. sub-neg 31.2%

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

      2. metadata-eval 31.2%

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

      3. metadata-eval 31.2%

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

      4. distribute-lft-in 31.2%

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

      5. metadata-eval 31.2%

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

      6. sub-neg 31.2%

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

      7. expm1-define 77.3%

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

   7. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

      3. *-un-lft-identity 77.3%

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

      4. times-frac 77.4%

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

      5. metadata-eval 77.4%

         \[\leadsto n \cdot {\left(\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1} \]

   9. Applied egg-rr 77.4%

      \[\leadsto n \cdot \color{blue}{{\left(0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 77.4%

          \[\leadsto n \cdot \color{blue}{\frac{1}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

       2. associate-*r/ 77.4%

          \[\leadsto n \cdot \frac{1}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 77.4%

       \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   13. Step-by-step derivation

       1. *-commutative 62.0%

          \[\leadsto n \cdot \frac{1}{0.01 + \color{blue}{i \cdot -0.005}} \]

   14. Simplified 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -5.1000000000000001e-201 < n < 6.0000000000000003e-114

   1. Initial program 54.0%

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

      1. associate-*r/ 54.0%

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

      2. sub-neg 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in i around 0 75.6%

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

   if 6.0000000000000003e-114 < n

   1. Initial program 26.7%

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

      1. associate-/r/ 27.1%

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

      2. associate-*r* 27.1%

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

      3. *-commutative 27.1%

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

      4. associate-*r/ 27.1%

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

      5. sub-neg 27.1%

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

      6. distribute-lft-in 27.1%

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

      7. metadata-eval 27.1%

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

      8. metadata-eval 27.1%

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

      9. metadata-eval 27.1%

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

      10. fma-define 27.1%

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

      11. metadata-eval 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in n around inf 37.9%

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

      1. sub-neg 37.9%

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

      2. metadata-eval 37.9%

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

      3. metadata-eval 37.9%

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

      4. distribute-lft-in 37.9%

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

      5. metadata-eval 37.9%

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

      6. sub-neg 37.9%

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

      7. expm1-define 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in i around 0 82.1%

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

      1. *-commutative 82.1%

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

   10. Simplified 82.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{+181}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -5.1 \cdot 10^{-201}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+180}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-207}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.9e+180)
   (* 100.0 (* n (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n -3e-207)
     (* n (/ 1.0 (+ 0.01 (* i -0.005))))
     (if (<= n 3.8e-114)
       (/ 0.0 (/ i n))
       (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.9e+180) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -3e-207) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 3.8e-114) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.9d+180)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= (-3d-207)) then
        tmp = n * (1.0d0 / (0.01d0 + (i * (-0.005d0))))
    else if (n <= 3.8d-114) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.9e+180) {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -3e-207) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 3.8e-114) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.9e+180:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= -3e-207:
		tmp = n * (1.0 / (0.01 + (i * -0.005)))
	elif n <= 3.8e-114:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.9e+180)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= -3e-207)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(i * -0.005))));
	elseif (n <= 3.8e-114)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.9e+180)
		tmp = 100.0 * (n * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= -3e-207)
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	elseif (n <= 3.8e-114)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.9e+180], N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -3e-207], N[(n * N[(1.0 / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.8e-114], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.9e180

   1. Initial program 6.2%

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

   3. Taylor expanded in n around inf 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-/l* 35.0%

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

      3. expm1-define 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in i around 0 79.4%

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

      1. *-commutative 79.4%

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

   8. Simplified 79.4%

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

   if -1.9e180 < n < -2.9999999999999999e-207

   1. Initial program 32.5%

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

      1. associate-/r/ 32.0%

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

      2. associate-*r* 31.9%

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

      3. *-commutative 31.9%

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

      4. associate-*r/ 32.0%

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

      5. sub-neg 32.0%

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

      6. distribute-lft-in 31.8%

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

      7. metadata-eval 31.8%

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

      8. metadata-eval 31.8%

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

      9. metadata-eval 31.8%

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

      10. fma-define 32.0%

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

      11. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in n around inf 31.2%

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

      1. sub-neg 31.2%

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

      2. metadata-eval 31.2%

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

      3. metadata-eval 31.2%

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

      4. distribute-lft-in 31.2%

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

      5. metadata-eval 31.2%

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

      6. sub-neg 31.2%

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

      7. expm1-define 77.3%

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

   7. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

      3. *-un-lft-identity 77.3%

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

      4. times-frac 77.4%

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

      5. metadata-eval 77.4%

         \[\leadsto n \cdot {\left(\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1} \]

   9. Applied egg-rr 77.4%

      \[\leadsto n \cdot \color{blue}{{\left(0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 77.4%

          \[\leadsto n \cdot \color{blue}{\frac{1}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

       2. associate-*r/ 77.4%

          \[\leadsto n \cdot \frac{1}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 77.4%

       \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   13. Step-by-step derivation

       1. *-commutative 62.0%

          \[\leadsto n \cdot \frac{1}{0.01 + \color{blue}{i \cdot -0.005}} \]

   14. Simplified 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -2.9999999999999999e-207 < n < 3.7999999999999998e-114

   1. Initial program 54.0%

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

      1. associate-*r/ 54.0%

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

      2. sub-neg 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in i around 0 75.6%

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

   if 3.7999999999999998e-114 < n

   1. Initial program 26.7%

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

      1. associate-/r/ 27.1%

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

      2. associate-*r* 27.1%

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

      3. *-commutative 27.1%

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

      4. associate-*r/ 27.1%

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

      5. sub-neg 27.1%

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

      6. distribute-lft-in 27.1%

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

      7. metadata-eval 27.1%

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

      8. metadata-eval 27.1%

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

      9. metadata-eval 27.1%

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

      10. fma-define 27.1%

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

      11. metadata-eval 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in n around inf 37.9%

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

      1. sub-neg 37.9%

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

      2. metadata-eval 37.9%

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

      3. metadata-eval 37.9%

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

      4. distribute-lft-in 37.9%

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

      5. metadata-eval 37.9%

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

      6. sub-neg 37.9%

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

      7. expm1-define 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in i around 0 78.0%

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

      1. *-commutative 78.0%

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

   10. Simplified 78.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+180}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-207}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{if}\;n \leq -7.8 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -5.8 \cdot 10^{-211}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))))
   (if (<= n -7.8e+183)
     t_0
     (if (<= n -5.8e-211)
       (* n (/ 1.0 (+ 0.01 (* i -0.005))))
       (if (<= n 3.8e-114) (/ 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))));
	double tmp;
	if (n <= -7.8e+183) {
		tmp = t_0;
	} else if (n <= -5.8e-211) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 3.8e-114) {
		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))))
    if (n <= (-7.8d+183)) then
        tmp = t_0
    else if (n <= (-5.8d-211)) then
        tmp = n * (1.0d0 / (0.01d0 + (i * (-0.005d0))))
    else if (n <= 3.8d-114) 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))));
	double tmp;
	if (n <= -7.8e+183) {
		tmp = t_0;
	} else if (n <= -5.8e-211) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 3.8e-114) {
		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))))
	tmp = 0
	if n <= -7.8e+183:
		tmp = t_0
	elif n <= -5.8e-211:
		tmp = n * (1.0 / (0.01 + (i * -0.005)))
	elif n <= 3.8e-114:
		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 * 16.666666666666668)))))
	tmp = 0.0
	if (n <= -7.8e+183)
		tmp = t_0;
	elseif (n <= -5.8e-211)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(i * -0.005))));
	elseif (n <= 3.8e-114)
		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))));
	tmp = 0.0;
	if (n <= -7.8e+183)
		tmp = t_0;
	elseif (n <= -5.8e-211)
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	elseif (n <= 3.8e-114)
		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 * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7.8e+183], t$95$0, If[LessEqual[n, -5.8e-211], N[(n * N[(1.0 / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.8e-114], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.7999999999999998e183 or 3.7999999999999998e-114 < n

   1. Initial program 20.8%

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

      1. associate-/r/ 21.4%

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

      2. associate-*r* 21.4%

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

      3. *-commutative 21.4%

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

      4. associate-*r/ 21.4%

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

      5. sub-neg 21.4%

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

      6. distribute-lft-in 21.4%

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

      7. metadata-eval 21.4%

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

      8. metadata-eval 21.4%

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

      9. metadata-eval 21.4%

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

      10. fma-define 21.4%

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

      11. metadata-eval 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in n around inf 37.1%

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

      1. sub-neg 37.1%

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

      2. metadata-eval 37.1%

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

      3. metadata-eval 37.1%

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

      4. distribute-lft-in 37.1%

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

      5. metadata-eval 37.1%

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

      6. sub-neg 37.1%

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

      7. expm1-define 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in i around 0 78.4%

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

      1. *-commutative 78.4%

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

   10. Simplified 78.4%

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

   if -7.7999999999999998e183 < n < -5.80000000000000029e-211

   1. Initial program 32.5%

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

      1. associate-/r/ 32.0%

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

      2. associate-*r* 31.9%

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

      3. *-commutative 31.9%

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

      4. associate-*r/ 32.0%

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

      5. sub-neg 32.0%

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

      6. distribute-lft-in 31.8%

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

      7. metadata-eval 31.8%

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

      8. metadata-eval 31.8%

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

      9. metadata-eval 31.8%

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

      10. fma-define 32.0%

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

      11. metadata-eval 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in n around inf 31.2%

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

      1. sub-neg 31.2%

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

      2. metadata-eval 31.2%

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

      3. metadata-eval 31.2%

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

      4. distribute-lft-in 31.2%

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

      5. metadata-eval 31.2%

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

      6. sub-neg 31.2%

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

      7. expm1-define 77.3%

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

   7. Simplified 77.3%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

      3. *-un-lft-identity 77.3%

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

      4. times-frac 77.4%

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

      5. metadata-eval 77.4%

         \[\leadsto n \cdot {\left(\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1} \]

   9. Applied egg-rr 77.4%

      \[\leadsto n \cdot \color{blue}{{\left(0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 77.4%

          \[\leadsto n \cdot \color{blue}{\frac{1}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

       2. associate-*r/ 77.4%

          \[\leadsto n \cdot \frac{1}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 77.4%

       \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   13. Step-by-step derivation

       1. *-commutative 62.0%

          \[\leadsto n \cdot \frac{1}{0.01 + \color{blue}{i \cdot -0.005}} \]

   14. Simplified 62.0%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -5.80000000000000029e-211 < n < 3.7999999999999998e-114

   1. Initial program 54.0%

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

      1. associate-*r/ 54.0%

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

      2. sub-neg 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in i around 0 75.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.8 \cdot 10^{+183}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -5.8 \cdot 10^{-211}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+39} \lor \neg \left(n \leq 3.7 \cdot 10^{-17}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.2e+39) (not (<= n 3.7e-17)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.2e+39) || !(n <= 3.7e-17)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-4.2d+39)) .or. (.not. (n <= 3.7d-17))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.2e+39) || !(n <= 3.7e-17)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.2e+39) or not (n <= 3.7e-17):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.2e+39) || !(n <= 3.7e-17))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -4.2e+39) || ~((n <= 3.7e-17)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.2e+39], N[Not[LessEqual[n, 3.7e-17]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.1999999999999997e39 or 3.6999999999999997e-17 < n

   1. Initial program 24.5%

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

   3. Taylor expanded in n around inf 41.6%

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

      1. *-commutative 41.6%

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

      2. associate-/l* 41.6%

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

      3. expm1-define 90.6%

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

   5. Simplified 90.6%

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

   6. Taylor expanded in i around 0 64.9%

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

      1. +-commutative 64.9%

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

      2. associate-*r* 64.9%

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

      3. distribute-rgt-in 65.0%

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

      4. *-commutative 65.0%

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

   8. Simplified 65.0%

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

   if -4.1999999999999997e39 < n < 3.6999999999999997e-17

   1. Initial program 37.0%

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

   3. Taylor expanded in i around 0 58.1%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+39} \lor \neg \left(n \leq 3.7 \cdot 10^{-17}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -6.5 \cdot 10^{+158} \lor \neg \left(i \leq 3.3 \cdot 10^{-35}\right):\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -6.5e+158) (not (<= i 3.3e-35)))
   (* 100.0 (* i (/ n i)))
   (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -6.5e+158) || !(i <= 3.3e-35)) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-6.5d+158)) .or. (.not. (i <= 3.3d-35))) then
        tmp = 100.0d0 * (i * (n / i))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -6.5e+158) || !(i <= 3.3e-35)) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -6.5e+158) or not (i <= 3.3e-35):
		tmp = 100.0 * (i * (n / i))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -6.5e+158) || !(i <= 3.3e-35))
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -6.5e+158) || ~((i <= 3.3e-35)))
		tmp = 100.0 * (i * (n / i));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -6.5e+158], N[Not[LessEqual[i, 3.3e-35]], $MachinePrecision]], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -6.5000000000000001e158 or 3.3e-35 < i

   1. Initial program 57.9%

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

   3. Taylor expanded in i around 0 23.4%

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

      1. clear-num 23.4%

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

      2. associate-/r/ 23.4%

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

      3. clear-num 22.3%

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

   5. Applied egg-rr 22.3%

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

   if -6.5000000000000001e158 < i < 3.3e-35

   1. Initial program 14.9%

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

      1. associate-/r/ 15.5%

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

      2. associate-*r* 15.5%

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

      3. *-commutative 15.5%

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

      4. associate-*r/ 15.5%

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

      5. sub-neg 15.5%

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

      6. distribute-lft-in 15.4%

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

      7. metadata-eval 15.4%

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

      8. metadata-eval 15.4%

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

      9. metadata-eval 15.4%

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

      10. fma-define 15.5%

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

      11. metadata-eval 15.5%

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

   3. Simplified 15.5%

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

   5. Taylor expanded in i around 0 71.7%

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

      1. *-commutative 71.7%

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

   7. Simplified 71.7%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6.5 \cdot 10^{+158} \lor \neg \left(i \leq 3.3 \cdot 10^{-35}\right):\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.06 \cdot 10^{-203}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.06e-203)
   (* n (/ 1.0 (+ 0.01 (* i -0.005))))
   (if (<= n 4.1e-114) (/ 0.0 (/ i n)) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.06e-203) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 4.1e-114) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.06d-203)) then
        tmp = n * (1.0d0 / (0.01d0 + (i * (-0.005d0))))
    else if (n <= 4.1d-114) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.06e-203) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else if (n <= 4.1e-114) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.06e-203:
		tmp = n * (1.0 / (0.01 + (i * -0.005)))
	elif n <= 4.1e-114:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.06e-203)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(i * -0.005))));
	elseif (n <= 4.1e-114)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.06e-203)
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	elseif (n <= 4.1e-114)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.06e-203], N[(n * N[(1.0 / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.1e-114], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.0599999999999999e-203

   1. Initial program 24.8%

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

      1. associate-/r/ 24.7%

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

      2. associate-*r* 24.7%

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

      3. *-commutative 24.7%

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

      4. associate-*r/ 24.7%

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

      5. sub-neg 24.7%

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

      6. distribute-lft-in 24.6%

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

      7. metadata-eval 24.6%

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

      8. metadata-eval 24.6%

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

      9. metadata-eval 24.6%

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

      10. fma-define 24.7%

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

      11. metadata-eval 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in n around inf 32.3%

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

      1. sub-neg 32.3%

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

      2. metadata-eval 32.3%

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

      3. metadata-eval 32.3%

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

      4. distribute-lft-in 32.3%

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

      5. metadata-eval 32.3%

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

      6. sub-neg 32.3%

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

      7. expm1-define 82.4%

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

   7. Simplified 82.4%

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

      1. clear-num 82.3%

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

      2. inv-pow 82.3%

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

      3. *-un-lft-identity 82.3%

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

      4. times-frac 82.4%

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

      5. metadata-eval 82.4%

         \[\leadsto n \cdot {\left(\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1} \]

   9. Applied egg-rr 82.4%

      \[\leadsto n \cdot \color{blue}{{\left(0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 82.4%

          \[\leadsto n \cdot \color{blue}{\frac{1}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

       2. associate-*r/ 82.4%

          \[\leadsto n \cdot \frac{1}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 82.4%

       \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 63.2%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   13. Step-by-step derivation

       1. *-commutative 63.2%

          \[\leadsto n \cdot \frac{1}{0.01 + \color{blue}{i \cdot -0.005}} \]

   14. Simplified 63.2%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -1.0599999999999999e-203 < n < 4.0999999999999997e-114

   1. Initial program 54.0%

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

      1. associate-*r/ 54.0%

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

      2. sub-neg 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in i around 0 75.6%

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

   if 4.0999999999999997e-114 < n

   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 n around inf 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-/l* 37.9%

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

      3. expm1-define 89.4%

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

   5. Simplified 89.4%

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

   6. Taylor expanded in i around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. associate-*r* 71.1%

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

      3. distribute-rgt-in 71.1%

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

      4. *-commutative 71.1%

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

   8. Simplified 71.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.06 \cdot 10^{-203}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-114}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1500000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50 - 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1e-10)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 1500000000.0) (* n 100.0) (* i (- (* n 50.0) 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1e-10) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1500000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = i * ((n * 50.0) - 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1d-10)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 1500000000.0d0) then
        tmp = n * 100.0d0
    else
        tmp = i * ((n * 50.0d0) - 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1e-10) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1500000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = i * ((n * 50.0) - 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1e-10:
		tmp = 100.0 * (i / (i / n))
	elif i <= 1500000000.0:
		tmp = n * 100.0
	else:
		tmp = i * ((n * 50.0) - 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1e-10)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 1500000000.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(i * Float64(Float64(n * 50.0) - 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1e-10)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 1500000000.0)
		tmp = n * 100.0;
	else
		tmp = i * ((n * 50.0) - 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1e-10], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1500000000.0], N[(n * 100.0), $MachinePrecision], N[(i * N[(N[(n * 50.0), $MachinePrecision] - 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.00000000000000004e-10

   1. Initial program 54.0%

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

   3. Taylor expanded in i around 0 22.9%

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

   if -1.00000000000000004e-10 < i < 1.5e9

   1. Initial program 10.6%

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

      1. associate-/r/ 11.2%

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

      2. associate-*r* 11.2%

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

      3. *-commutative 11.2%

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

      4. associate-*r/ 11.2%

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

      5. sub-neg 11.2%

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

      6. distribute-lft-in 11.2%

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

      7. metadata-eval 11.2%

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

      8. metadata-eval 11.2%

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

      9. metadata-eval 11.2%

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

      10. fma-define 11.2%

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

      11. metadata-eval 11.2%

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

   3. Simplified 11.2%

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

   5. Taylor expanded in i around 0 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if 1.5e9 < i

   1. Initial program 53.2%

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

      1. associate-/r/ 53.3%

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

      2. associate-*r* 53.2%

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

      3. *-commutative 53.2%

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

      4. associate-*r/ 53.3%

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

      5. sub-neg 53.3%

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

      6. distribute-lft-in 53.3%

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

      7. metadata-eval 53.3%

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

      8. metadata-eval 53.3%

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

      9. metadata-eval 53.3%

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

      10. fma-define 53.3%

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

      11. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in i around 0 31.8%

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

      1. *-commutative 31.8%

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

      2. associate-*r/ 31.8%

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

      3. metadata-eval 31.8%

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

   7. Simplified 31.8%

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

   8. Taylor expanded in n around 0 31.8%

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

   9. Taylor expanded in i around inf 31.8%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1500000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50 - 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 3 \cdot 10^{-35}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1e-11)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 3e-35) (* n 100.0) (* 100.0 (* i (/ n i))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1e-11) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 3e-35) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i * (n / i));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1d-11)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 3d-35) then
        tmp = n * 100.0d0
    else
        tmp = 100.0d0 * (i * (n / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1e-11) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 3e-35) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i * (n / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1e-11:
		tmp = 100.0 * (i / (i / n))
	elif i <= 3e-35:
		tmp = n * 100.0
	else:
		tmp = 100.0 * (i * (n / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1e-11)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 3e-35)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1e-11)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 3e-35)
		tmp = n * 100.0;
	else
		tmp = 100.0 * (i * (n / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1e-11], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3e-35], N[(n * 100.0), $MachinePrecision], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -9.99999999999999939e-12

   1. Initial program 53.1%

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

   3. Taylor expanded in i around 0 23.9%

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

   if -9.99999999999999939e-12 < i < 2.99999999999999989e-35

   1. Initial program 9.4%

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

      1. associate-/r/ 10.1%

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

      2. associate-*r* 10.1%

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

      3. *-commutative 10.1%

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

      4. associate-*r/ 10.1%

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

      5. sub-neg 10.1%

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

      6. distribute-lft-in 10.1%

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

      7. metadata-eval 10.1%

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

      8. metadata-eval 10.1%

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

      9. metadata-eval 10.1%

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

      10. fma-define 10.1%

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

      11. metadata-eval 10.1%

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

   3. Simplified 10.1%

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

   5. Taylor expanded in i around 0 86.1%

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

      1. *-commutative 86.1%

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

   7. Simplified 86.1%

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

   if 2.99999999999999989e-35 < i

   1. Initial program 52.5%

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

   3. Taylor expanded in i around 0 16.6%

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

      1. clear-num 16.6%

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

      2. associate-/r/ 16.6%

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

      3. clear-num 16.6%

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

   5. Applied egg-rr 16.6%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 3 \cdot 10^{-35}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.65 \cdot 10^{-17}:\\ \;\;\;\;n \cdot \frac{1}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 2.65e-17)
   (* n (/ 1.0 (+ 0.01 (* i -0.005))))
   (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 2.65e-17) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} 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 <= 2.65d-17) then
        tmp = n * (1.0d0 / (0.01d0 + (i * (-0.005d0))))
    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 <= 2.65e-17) {
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 2.65e-17:
		tmp = n * (1.0 / (0.01 + (i * -0.005)))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 2.65e-17)
		tmp = Float64(n * Float64(1.0 / Float64(0.01 + Float64(i * -0.005))));
	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 <= 2.65e-17)
		tmp = n * (1.0 / (0.01 + (i * -0.005)));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 2.65e-17], N[(n * N[(1.0 / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.6499999999999999e-17

   1. Initial program 29.2%

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

      1. associate-/r/ 29.2%

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

      2. associate-*r* 29.2%

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

      3. *-commutative 29.2%

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

      4. associate-*r/ 29.2%

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

      5. sub-neg 29.2%

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

      6. distribute-lft-in 29.1%

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

      7. metadata-eval 29.1%

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

      8. metadata-eval 29.1%

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

      9. metadata-eval 29.1%

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

      10. fma-define 29.2%

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

      11. metadata-eval 29.2%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in n around inf 29.9%

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

      1. sub-neg 29.9%

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

      2. metadata-eval 29.9%

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

      3. metadata-eval 29.9%

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

      4. distribute-lft-in 29.9%

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

      5. metadata-eval 29.9%

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

      6. sub-neg 29.9%

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

      7. expm1-define 67.3%

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

   7. Simplified 67.3%

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

      1. clear-num 67.3%

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

      2. inv-pow 67.3%

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

      3. *-un-lft-identity 67.3%

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

      4. times-frac 67.3%

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

      5. metadata-eval 67.3%

         \[\leadsto n \cdot {\left(\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1} \]

   9. Applied egg-rr 67.3%

      \[\leadsto n \cdot \color{blue}{{\left(0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 67.3%

          \[\leadsto n \cdot \color{blue}{\frac{1}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]

       2. associate-*r/ 67.3%

          \[\leadsto n \cdot \frac{1}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 67.3%

       \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 58.3%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   13. Step-by-step derivation

       1. *-commutative 58.3%

          \[\leadsto n \cdot \frac{1}{0.01 + \color{blue}{i \cdot -0.005}} \]

   14. Simplified 58.3%

       \[\leadsto n \cdot \frac{1}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if 2.6499999999999999e-17 < n

   1. Initial program 30.9%

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

   3. Taylor expanded in n around inf 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-/l* 44.6%

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

      3. expm1-define 93.5%

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

   5. Simplified 93.5%

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

   6. Taylor expanded in i around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. associate-*r* 71.6%

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

      3. distribute-rgt-in 71.6%

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

      4. *-commutative 71.6%

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

   8. Simplified 71.6%

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

Alternative 17: 49.2% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

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

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 29.7%

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

   1. associate-/r/ 29.9%

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

   2. associate-*r* 29.9%

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

   3. *-commutative 29.9%

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

   4. associate-*r/ 29.9%

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

   5. sub-neg 29.9%

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

   6. distribute-lft-in 29.8%

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

   7. metadata-eval 29.8%

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

   8. metadata-eval 29.8%

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

   9. metadata-eval 29.8%

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

   10. fma-define 29.9%

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

   11. metadata-eval 29.9%

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

3. Simplified 29.9%

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

5. Taylor expanded in i around 0 48.8%

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

   1. *-commutative 48.8%

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

7. Simplified 48.8%

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

Alternative 18: 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 29.7%

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

   1. associate-/r/ 29.9%

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

   2. associate-*r* 29.9%

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

   3. *-commutative 29.9%

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

   4. associate-*r/ 29.9%

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

   5. sub-neg 29.9%

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

   6. distribute-lft-in 29.8%

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

   7. metadata-eval 29.8%

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

   8. metadata-eval 29.8%

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

   9. metadata-eval 29.8%

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

   10. fma-define 29.9%

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

   11. metadata-eval 29.9%

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

3. Simplified 29.9%

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

5. Taylor expanded in i around 0 55.4%

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

   1. *-commutative 55.4%

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

   2. associate-*r/ 55.4%

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

   3. metadata-eval 55.4%

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

7. Simplified 55.4%

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

8. Taylor expanded in n around 0 2.6%

   \[\leadsto \color{blue}{-50 \cdot i} \]
9. Step-by-step derivation

   1. *-commutative 2.6%

      \[\leadsto \color{blue}{i \cdot -50} \]

10. Simplified 2.6%

    \[\leadsto \color{blue}{i \cdot -50} \]
11. Add Preprocessing

Developer Target 1: 33.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

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

Reproduce

herbie shell --seed 2024172 
(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))))