Compound Interest

Percentage Accurate: 28.2% → 98.0%

Time: 31.5s

Alternatives: 18

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \frac{n \cdot t\_0 - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (/ (* (expm1 (* n (log1p (/ i n)))) 100.0) (/ i n))
     (if (<= t_1 INFINITY)
       (* 100.0 (/ (- (* n t_0) n) i))
       (* n (pow (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5))) -1.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (((n * t_0) - n) / i);
	} else {
		tmp = n * pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((n * t_0) - n) / i);
	} else {
		tmp = n * Math.pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) * 100.0) / (i / n)
	elif t_1 <= math.inf:
		tmp = 100.0 * (((n * t_0) - n) / i)
	else:
		tmp = n * math.pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * t_0) - n) / i));
	else
		tmp = Float64(n * (Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5))) ^ -1.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(N[(n * t$95$0), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[Power[N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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. associate-*r/ 25.5%

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

      2. *-commutative 25.5%

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

      3. add-exp-log 25.5%

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

      4. expm1-define 25.5%

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

      5. log-pow 36.3%

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

      6. log1p-define 98.5%

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

   4. Applied egg-rr 98.5%

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

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

   1. Initial program 95.8%

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

      1. div-sub 95.7%

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

      2. div-inv 95.7%

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

      3. clear-num 95.6%

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

      4. clear-num 95.9%

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

   4. Applied egg-rr 95.9%

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

      1. associate-*r/ 95.9%

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

      2. sub-div 96.5%

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

      3. +-commutative 96.5%

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

   6. Applied egg-rr 96.5%

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

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

   1. Initial program 0.0%

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

      1. add0 0.0%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 0.0%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 0.0%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 0.0%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 0.0%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 0.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 0.0%

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

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*l/ 1.9%

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

      6. associate-*r/ 1.9%

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

      7. associate-/l* 1.8%

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

   6. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. un-div-inv 1.9%

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

      3. div-inv 1.9%

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

      4. metadata-eval 1.9%

         \[\leadsto n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot \color{blue}{0.01}} \]

   8. Applied egg-rr 1.9%

      \[\leadsto n \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot 0.01}} \]
   9. Step-by-step derivation

      1. clear-num 1.9%

         \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. inv-pow 1.9%

         \[\leadsto n \cdot \color{blue}{{\left(\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]

   10. Applied egg-rr 1.9%

       \[\leadsto n \cdot \color{blue}{{\left(\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]

   11. Taylor expanded in i around 0 100.0%

       \[\leadsto n \cdot {\color{blue}{\left(0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)\right)}}^{-1} \]
   12. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto n \cdot {\left(0.01 + \color{blue}{\left(0.01 \cdot i\right) \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}\right)}^{-1} \]

       2. *-commutative 100.0%

          \[\leadsto n \cdot {\left(0.01 + \color{blue}{\left(i \cdot 0.01\right)} \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}^{-1} \]

       3. sub-neg 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}\right)}^{-1} \]

       4. associate-*r/ 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)\right)}^{-1} \]

       5. metadata-eval 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)\right)}^{-1} \]

       6. metadata-eval 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)\right)}^{-1} \]

   13. Simplified 100.0%

       \[\leadsto n \cdot {\color{blue}{\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}}^{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \frac{n \cdot t\_0 - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* n (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 i)))
     (if (<= t_1 INFINITY)
       (* 100.0 (/ (- (* n t_0) n) i))
       (* n (pow (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5))) -1.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (((n * t_0) - n) / i);
	} else {
		tmp = n * pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((n * t_0) - n) / i);
	} else {
		tmp = n * Math.pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) * (100.0 / i))
	elif t_1 <= math.inf:
		tmp = 100.0 * (((n * t_0) - n) / i)
	else:
		tmp = n * math.pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * t_0) - n) / i));
	else
		tmp = Float64(n * (Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5))) ^ -1.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(N[(n * t$95$0), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[Power[N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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. add0 25.5%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 25.5%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 25.4%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 25.0%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 25.0%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 25.0%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 25.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 35.8%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 95.6%

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

   4. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*l* 95.6%

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

      4. *-commutative 95.6%

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

      5. associate-*l/ 87.3%

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

      6. associate-*r/ 97.1%

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

      7. associate-/l* 97.1%

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

   6. Simplified 97.1%

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

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

   1. Initial program 95.8%

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

      1. div-sub 95.7%

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

      2. div-inv 95.7%

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

      3. clear-num 95.6%

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

      4. clear-num 95.9%

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

   4. Applied egg-rr 95.9%

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

      1. associate-*r/ 95.9%

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

      2. sub-div 96.5%

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

      3. +-commutative 96.5%

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

   6. Applied egg-rr 96.5%

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

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

   1. Initial program 0.0%

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

      1. add0 0.0%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 0.0%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 0.0%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 0.0%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 0.0%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 0.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 0.0%

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

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*l/ 1.9%

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

      6. associate-*r/ 1.9%

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

      7. associate-/l* 1.8%

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

   6. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. un-div-inv 1.9%

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

      3. div-inv 1.9%

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

      4. metadata-eval 1.9%

         \[\leadsto n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot \color{blue}{0.01}} \]

   8. Applied egg-rr 1.9%

      \[\leadsto n \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot 0.01}} \]
   9. Step-by-step derivation

      1. clear-num 1.9%

         \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. inv-pow 1.9%

         \[\leadsto n \cdot \color{blue}{{\left(\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]

   10. Applied egg-rr 1.9%

       \[\leadsto n \cdot \color{blue}{{\left(\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]

   11. Taylor expanded in i around 0 100.0%

       \[\leadsto n \cdot {\color{blue}{\left(0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)\right)}}^{-1} \]
   12. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto n \cdot {\left(0.01 + \color{blue}{\left(0.01 \cdot i\right) \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}\right)}^{-1} \]

       2. *-commutative 100.0%

          \[\leadsto n \cdot {\left(0.01 + \color{blue}{\left(i \cdot 0.01\right)} \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}^{-1} \]

       3. sub-neg 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}\right)}^{-1} \]

       4. associate-*r/ 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)\right)}^{-1} \]

       5. metadata-eval 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)\right)}^{-1} \]

       6. metadata-eval 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)\right)}^{-1} \]

   13. Simplified 100.0%

       \[\leadsto n \cdot {\color{blue}{\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}}^{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \frac{n \cdot t\_0 - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (/ 100.0 (/ (/ i n) (expm1 (* n (log1p (/ i n))))))
     (if (<= t_1 INFINITY)
       (* 100.0 (/ (- (* n t_0) n) i))
       (* n (pow (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5))) -1.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 / ((i / n) / expm1((n * log1p((i / n)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (((n * t_0) - n) / i);
	} else {
		tmp = n * pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 / ((i / n) / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((n * t_0) - n) / i);
	} else {
		tmp = n * Math.pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 / ((i / n) / math.expm1((n * math.log1p((i / n)))))
	elif t_1 <= math.inf:
		tmp = 100.0 * (((n * t_0) - n) / i)
	else:
		tmp = n * math.pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 / Float64(Float64(i / n) / expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * t_0) - n) / i));
	else
		tmp = Float64(n * (Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5))) ^ -1.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(100.0 / N[(N[(i / n), $MachinePrecision] / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(N[(n * t$95$0), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[Power[N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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. clear-num 25.5%

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

      2. un-div-inv 25.5%

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

      3. add-exp-log 25.5%

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

      4. expm1-define 25.5%

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

      5. log-pow 36.2%

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

      6. log1p-define 98.5%

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

   4. Applied egg-rr 98.5%

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

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

   1. Initial program 95.8%

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

      1. div-sub 95.7%

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

      2. div-inv 95.7%

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

      3. clear-num 95.6%

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

      4. clear-num 95.9%

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

   4. Applied egg-rr 95.9%

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

      1. associate-*r/ 95.9%

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

      2. sub-div 96.5%

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

      3. +-commutative 96.5%

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

   6. Applied egg-rr 96.5%

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

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

   1. Initial program 0.0%

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

      1. add0 0.0%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 0.0%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 0.0%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 0.0%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 0.0%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 0.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 0.0%

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

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*l/ 1.9%

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

      6. associate-*r/ 1.9%

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

      7. associate-/l* 1.8%

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

   6. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. un-div-inv 1.9%

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

      3. div-inv 1.9%

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

      4. metadata-eval 1.9%

         \[\leadsto n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot \color{blue}{0.01}} \]

   8. Applied egg-rr 1.9%

      \[\leadsto n \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot 0.01}} \]
   9. Step-by-step derivation

      1. clear-num 1.9%

         \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. inv-pow 1.9%

         \[\leadsto n \cdot \color{blue}{{\left(\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]

   10. Applied egg-rr 1.9%

       \[\leadsto n \cdot \color{blue}{{\left(\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]

   11. Taylor expanded in i around 0 100.0%

       \[\leadsto n \cdot {\color{blue}{\left(0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)\right)}}^{-1} \]
   12. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto n \cdot {\left(0.01 + \color{blue}{\left(0.01 \cdot i\right) \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}\right)}^{-1} \]

       2. *-commutative 100.0%

          \[\leadsto n \cdot {\left(0.01 + \color{blue}{\left(i \cdot 0.01\right)} \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}^{-1} \]

       3. sub-neg 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}\right)}^{-1} \]

       4. associate-*r/ 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)\right)}^{-1} \]

       5. metadata-eval 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)\right)}^{-1} \]

       6. metadata-eval 100.0%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)\right)}^{-1} \]

   13. Simplified 100.0%

       \[\leadsto n \cdot {\color{blue}{\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}}^{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(1 + \frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-5}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.15:\\ \;\;\;\;n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.8e-5)
   (* n (/ (* 100.0 (expm1 i)) i))
   (if (<= n 1.15)
     (* n (pow (+ 0.01 (* (* i 0.01) (+ (/ 0.5 n) -0.5))) -1.0))
     (* 100.0 (* n (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.8e-5) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else if (n <= 1.15) {
		tmp = n * pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0);
	} else {
		tmp = 100.0 * (n * (expm1(i) / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.8e-5) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else if (n <= 1.15) {
		tmp = n * Math.pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0);
	} else {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.8e-5:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	elif n <= 1.15:
		tmp = n * math.pow((0.01 + ((i * 0.01) * ((0.5 / n) + -0.5))), -1.0)
	else:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.8e-5)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	elseif (n <= 1.15)
		tmp = Float64(n * (Float64(0.01 + Float64(Float64(i * 0.01) * Float64(Float64(0.5 / n) + -0.5))) ^ -1.0));
	else
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.8e-5], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15], N[(n * N[Power[N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.8e-5

   1. Initial program 26.3%

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

      1. add0 26.3%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 26.3%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 26.3%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 26.3%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 26.3%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 26.3%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 26.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 17.2%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 67.5%

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

   4. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 67.5%

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

      2. *-commutative 67.5%

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

      3. associate-*l* 67.5%

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

      4. *-commutative 67.5%

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

      5. associate-*l/ 66.2%

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

      6. associate-*r/ 68.5%

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

      7. associate-/l* 68.4%

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

   6. Simplified 68.4%

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

   7. Taylor expanded in n around inf 40.4%

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

      1. associate-*r/ 40.4%

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

      2. expm1-define 89.7%

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

   9. Simplified 89.7%

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

   if -9.8e-5 < n < 1.1499999999999999

   1. Initial program 35.6%

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

      1. add0 35.6%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 35.6%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 35.6%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 34.7%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 34.7%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 34.7%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 34.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 60.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 93.0%

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

   4. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 93.0%

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

      2. *-commutative 93.0%

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

      3. associate-*l* 93.2%

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

      4. *-commutative 93.2%

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

      5. associate-*l/ 74.7%

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

      6. associate-*r/ 93.3%

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

      7. associate-/l* 93.3%

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

   6. Simplified 93.3%

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

      1. clear-num 93.2%

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

      2. un-div-inv 93.2%

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

      3. div-inv 93.3%

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

      4. metadata-eval 93.3%

         \[\leadsto n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot \color{blue}{0.01}} \]

   8. Applied egg-rr 93.3%

      \[\leadsto n \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot 0.01}} \]
   9. Step-by-step derivation

      1. clear-num 93.4%

         \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}} \]

      2. inv-pow 93.4%

         \[\leadsto n \cdot \color{blue}{{\left(\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]

   10. Applied egg-rr 93.4%

       \[\leadsto n \cdot \color{blue}{{\left(\frac{i \cdot 0.01}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}\right)}^{-1}} \]

   11. Taylor expanded in i around 0 77.5%

       \[\leadsto n \cdot {\color{blue}{\left(0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)\right)}}^{-1} \]
   12. Step-by-step derivation

       1. associate-*r* 77.5%

          \[\leadsto n \cdot {\left(0.01 + \color{blue}{\left(0.01 \cdot i\right) \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)}\right)}^{-1} \]

       2. *-commutative 77.5%

          \[\leadsto n \cdot {\left(0.01 + \color{blue}{\left(i \cdot 0.01\right)} \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}^{-1} \]

       3. sub-neg 77.5%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}\right)}^{-1} \]

       4. associate-*r/ 77.5%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)\right)}^{-1} \]

       5. metadata-eval 77.5%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)\right)}^{-1} \]

       6. metadata-eval 77.5%

          \[\leadsto n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)\right)}^{-1} \]

   13. Simplified 77.5%

       \[\leadsto n \cdot {\color{blue}{\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}}^{-1} \]

   if 1.1499999999999999 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-/l* 43.8%

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

      3. expm1-define 97.3%

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

   5. Simplified 97.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-5}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.15:\\ \;\;\;\;n \cdot {\left(0.01 + \left(i \cdot 0.01\right) \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.8 \cdot 10^{-23} \lor \neg \left(i \leq 7.5 \cdot 10^{-79}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 100\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.8e-23) (not (<= i 7.5e-79)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (* n (+ 100.0 (* (* i 100.0) (- 0.5 (/ 0.5 n)))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.8e-23) || !(i <= 7.5e-79)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.8e-23) || !(i <= 7.5e-79)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.8e-23) or not (i <= 7.5e-79):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.8e-23) || !(i <= 7.5e-79))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i * 100.0) * Float64(0.5 - Float64(0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.8e-23], N[Not[LessEqual[i, 7.5e-79]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(N[(i * 100.0), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.7999999999999999e-23 or 7.49999999999999969e-79 < i

   1. Initial program 46.8%

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

   3. Taylor expanded in n around inf 58.5%

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

      1. expm1-define 68.2%

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

   5. Simplified 68.2%

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

   if -1.7999999999999999e-23 < i < 7.49999999999999969e-79

   1. Initial program 5.8%

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

      1. add0 5.8%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 5.8%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 5.7%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 5.7%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 5.7%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 5.7%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 5.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 15.1%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 66.4%

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

   4. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 66.4%

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

      2. *-commutative 66.4%

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

      3. associate-*l* 66.5%

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

      4. *-commutative 66.5%

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

      5. associate-*l/ 54.3%

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

      6. associate-*r/ 68.6%

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

      7. associate-/l* 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in i around 0 89.5%

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

      1. associate-*r* 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*r/ 89.5%

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

      4. metadata-eval 89.5%

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

   9. Simplified 89.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.8 \cdot 10^{-23} \lor \neg \left(i \leq 7.5 \cdot 10^{-79}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 100\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105} \lor \neg \left(n \leq 7.6 \cdot 10^{-143}\right):\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{i \cdot 0.01}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.3e-105) (not (<= n 7.6e-143)))
   (* n (/ (expm1 i) (* i 0.01)))
   (* 0.0 (/ n i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-105) || !(n <= 7.6e-143)) {
		tmp = n * (expm1(i) / (i * 0.01));
	} else {
		tmp = 0.0 * (n / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-105) || !(n <= 7.6e-143)) {
		tmp = n * (Math.expm1(i) / (i * 0.01));
	} else {
		tmp = 0.0 * (n / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.3e-105) or not (n <= 7.6e-143):
		tmp = n * (math.expm1(i) / (i * 0.01))
	else:
		tmp = 0.0 * (n / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.3e-105) || !(n <= 7.6e-143))
		tmp = Float64(n * Float64(expm1(i) / Float64(i * 0.01)));
	else
		tmp = Float64(0.0 * Float64(n / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.3e-105], N[Not[LessEqual[n, 7.6e-143]], $MachinePrecision]], N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i * 0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2999999999999999e-105 or 7.59999999999999962e-143 < n

   1. Initial program 22.0%

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

      1. add0 22.0%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 22.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 22.0%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 21.9%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 21.9%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 21.9%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 21.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 24.6%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 76.1%

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

   4. Applied egg-rr 76.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 76.1%

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

      2. *-commutative 76.1%

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

      3. associate-*l* 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*l/ 74.2%

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

      6. associate-*r/ 77.7%

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

      7. associate-/l* 77.8%

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

   6. Simplified 77.8%

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

      1. clear-num 77.7%

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

      2. un-div-inv 77.7%

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

      3. div-inv 77.7%

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

      4. metadata-eval 77.7%

         \[\leadsto n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot \color{blue}{0.01}} \]

   8. Applied egg-rr 77.7%

      \[\leadsto n \cdot \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i \cdot 0.01}} \]

   9. Taylor expanded in n around inf 85.5%

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

   if -1.2999999999999999e-105 < n < 7.59999999999999962e-143

   1. Initial program 58.1%

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

      1. associate-*r/ 58.1%

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

      2. associate-/r/ 57.0%

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

      3. associate-*l/ 57.0%

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

      4. associate-/l* 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-lft-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. metadata-eval 56.8%

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

      9. metadata-eval 56.8%

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

      10. fma-define 56.8%

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

      11. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. fma-undefine 56.8%

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

      2. *-commutative 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in i around 0 69.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105} \lor \neg \left(n \leq 7.6 \cdot 10^{-143}\right):\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{i \cdot 0.01}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105} \lor \neg \left(n \leq 7.8 \cdot 10^{-143}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.3e-105) (not (<= n 7.8e-143)))
   (* n (/ (* 100.0 (expm1 i)) i))
   (* 0.0 (/ n i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-105) || !(n <= 7.8e-143)) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else {
		tmp = 0.0 * (n / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-105) || !(n <= 7.8e-143)) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else {
		tmp = 0.0 * (n / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.3e-105) or not (n <= 7.8e-143):
		tmp = n * ((100.0 * math.expm1(i)) / i)
	else:
		tmp = 0.0 * (n / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.3e-105) || !(n <= 7.8e-143))
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	else
		tmp = Float64(0.0 * Float64(n / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.3e-105], N[Not[LessEqual[n, 7.8e-143]], $MachinePrecision]], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(0.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2999999999999999e-105 or 7.80000000000000007e-143 < n

   1. Initial program 22.0%

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

      1. add0 22.0%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 22.0%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 22.0%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 21.9%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 21.9%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 21.9%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 21.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 24.6%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 76.1%

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

   4. Applied egg-rr 76.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 76.1%

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

      2. *-commutative 76.1%

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

      3. associate-*l* 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*l/ 74.2%

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

      6. associate-*r/ 77.7%

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

      7. associate-/l* 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in n around inf 33.9%

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

      1. associate-*r/ 34.0%

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

      2. expm1-define 85.6%

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

   9. Simplified 85.6%

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

   if -1.2999999999999999e-105 < n < 7.80000000000000007e-143

   1. Initial program 58.1%

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

      1. associate-*r/ 58.1%

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

      2. associate-/r/ 57.0%

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

      3. associate-*l/ 57.0%

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

      4. associate-/l* 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-lft-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. metadata-eval 56.8%

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

      9. metadata-eval 56.8%

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

      10. fma-define 56.8%

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

      11. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. fma-undefine 56.8%

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

      2. *-commutative 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in i around 0 69.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105} \lor \neg \left(n \leq 7.8 \cdot 10^{-143}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-142}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.3e-105)
   (* n (/ (* 100.0 (expm1 i)) i))
   (if (<= n 1.65e-142) (* 0.0 (/ n i)) (* 100.0 (* n (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.3e-105) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else if (n <= 1.65e-142) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = 100.0 * (n * (expm1(i) / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.3e-105) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else if (n <= 1.65e-142) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.3e-105:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	elif n <= 1.65e-142:
		tmp = 0.0 * (n / i)
	else:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.3e-105)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	elseif (n <= 1.65e-142)
		tmp = Float64(0.0 * Float64(n / i));
	else
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.3e-105], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e-142], N[(0.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2999999999999999e-105

   1. Initial program 23.3%

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

      1. add0 23.3%

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} + 0} \]

      2. *-commutative 23.3%

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \cdot 100} + 0 \]

      3. div-inv 23.2%

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)} \cdot 100 + 0 \]

      4. clear-num 23.0%

         \[\leadsto \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \cdot 100 + 0 \]

      5. associate-*l* 23.0%

         \[\leadsto \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right)} + 0 \]

      6. add-exp-log 23.0%

         \[\leadsto \left(\color{blue}{e^{\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)}} - 1\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      7. expm1-define 23.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)} \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      8. log-pow 24.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0 \]

      9. log1p-define 73.4%

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

   4. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right) + 0} \]
   5. Step-by-step derivation

      1. add0 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*l* 73.5%

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

      4. *-commutative 73.5%

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

      5. associate-*l/ 69.1%

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

      6. associate-*r/ 74.3%

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

      7. associate-/l* 74.3%

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

   6. Simplified 74.3%

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

   7. Taylor expanded in n around inf 34.1%

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

      1. associate-*r/ 34.2%

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

      2. expm1-define 83.3%

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

   9. Simplified 83.3%

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

   if -1.2999999999999999e-105 < n < 1.6499999999999998e-142

   1. Initial program 58.1%

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

      1. associate-*r/ 58.1%

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

      2. associate-/r/ 57.0%

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

      3. associate-*l/ 57.0%

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

      4. associate-/l* 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-lft-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. metadata-eval 56.8%

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

      9. metadata-eval 56.8%

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

      10. fma-define 56.8%

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

      11. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. fma-undefine 56.8%

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

      2. *-commutative 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in i around 0 69.7%

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

   if 1.6499999999999998e-142 < n

   1. Initial program 20.8%

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

   3. Taylor expanded in n around inf 33.8%

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

      1. *-commutative 33.8%

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

      2. associate-/l* 33.8%

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

      3. expm1-define 88.0%

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

   5. Simplified 88.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-142}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-102}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.16 \cdot 10^{-142}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.45e-102)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n 1.16e-142) (* 0.0 (/ n i)) (* (* n 100.0) (+ 1.0 (* i 0.5))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.45e-102) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 1.16e-142) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.45d-102)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= 1.16d-142) then
        tmp = 0.0d0 * (n / i)
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.45e-102) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 1.16e-142) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.45e-102:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= 1.16e-142:
		tmp = 0.0 * (n / i)
	else:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.45e-102)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= 1.16e-142)
		tmp = Float64(0.0 * Float64(n / i));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.45e-102)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= 1.16e-142)
		tmp = 0.0 * (n / i);
	else
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.45e-102], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.16e-142], N[(0.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.44999999999999993e-102

   1. Initial program 23.3%

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

   3. Taylor expanded in i around 0 55.1%

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

   4. Taylor expanded in n around inf 55.0%

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

      1. associate-*r* 55.0%

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

      2. *-commutative 55.0%

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

      3. *-commutative 55.0%

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

   6. Simplified 55.0%

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

   7. Taylor expanded in i around 0 55.0%

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

      1. associate-*r* 55.0%

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

      2. distribute-rgt-out 55.0%

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

   9. Simplified 55.0%

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

   if -1.44999999999999993e-102 < n < 1.16e-142

   1. Initial program 58.1%

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

      1. associate-*r/ 58.1%

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

      2. associate-/r/ 57.0%

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

      3. associate-*l/ 57.0%

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

      4. associate-/l* 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-lft-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. metadata-eval 56.8%

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

      9. metadata-eval 56.8%

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

      10. fma-define 56.8%

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

      11. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. fma-undefine 56.8%

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

      2. *-commutative 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in i around 0 69.7%

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

   if 1.16e-142 < n

   1. Initial program 20.8%

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

   3. Taylor expanded in i around 0 69.7%

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

   4. Taylor expanded in n around inf 70.1%

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

      1. associate-*r* 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Simplified 70.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-102}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.16 \cdot 10^{-142}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 7.8 \cdot 10^{-143}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.3e-105)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n 7.8e-143) (* 0.0 (/ n i)) (+ (* 50.0 (* i n)) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.3e-105) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 7.8e-143) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.3d-105)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= 7.8d-143) then
        tmp = 0.0d0 * (n / i)
    else
        tmp = (50.0d0 * (i * n)) + (n * 100.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.3e-105) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 7.8e-143) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.3e-105:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= 7.8e-143:
		tmp = 0.0 * (n / i)
	else:
		tmp = (50.0 * (i * n)) + (n * 100.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.3e-105)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= 7.8e-143)
		tmp = Float64(0.0 * Float64(n / i));
	else
		tmp = Float64(Float64(50.0 * Float64(i * n)) + Float64(n * 100.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.3e-105)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= 7.8e-143)
		tmp = 0.0 * (n / i);
	else
		tmp = (50.0 * (i * n)) + (n * 100.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.3e-105], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.8e-143], N[(0.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2999999999999999e-105

   1. Initial program 23.3%

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

   3. Taylor expanded in i around 0 55.1%

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

   4. Taylor expanded in n around inf 55.0%

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

      1. associate-*r* 55.0%

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

      2. *-commutative 55.0%

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

      3. *-commutative 55.0%

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

   6. Simplified 55.0%

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

   7. Taylor expanded in i around 0 55.0%

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

      1. associate-*r* 55.0%

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

      2. distribute-rgt-out 55.0%

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

   9. Simplified 55.0%

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

   if -1.2999999999999999e-105 < n < 7.80000000000000007e-143

   1. Initial program 58.1%

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

      1. associate-*r/ 58.1%

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

      2. associate-/r/ 57.0%

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

      3. associate-*l/ 57.0%

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

      4. associate-/l* 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-lft-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. metadata-eval 56.8%

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

      9. metadata-eval 56.8%

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

      10. fma-define 56.8%

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

      11. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. fma-undefine 56.8%

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

      2. *-commutative 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in i around 0 69.7%

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

   if 7.80000000000000007e-143 < n

   1. Initial program 20.8%

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

   3. Taylor expanded in i around 0 69.7%

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

   4. Taylor expanded in n around inf 70.1%

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

      1. associate-*r* 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in i around 0 70.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 7.8 \cdot 10^{-143}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 1.02 \cdot 10^{-142}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.35e-105)
   (* 100.0 (+ n (* (- 0.5 (/ 0.5 n)) (* i n))))
   (if (<= n 1.02e-142) (* 0.0 (/ n i)) (+ (* 50.0 (* i n)) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.35e-105) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else if (n <= 1.02e-142) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.35d-105)) then
        tmp = 100.0d0 * (n + ((0.5d0 - (0.5d0 / n)) * (i * n)))
    else if (n <= 1.02d-142) then
        tmp = 0.0d0 * (n / i)
    else
        tmp = (50.0d0 * (i * n)) + (n * 100.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.35e-105) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else if (n <= 1.02e-142) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.35e-105:
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)))
	elif n <= 1.02e-142:
		tmp = 0.0 * (n / i)
	else:
		tmp = (50.0 * (i * n)) + (n * 100.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.35e-105)
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * n))));
	elseif (n <= 1.02e-142)
		tmp = Float64(0.0 * Float64(n / i));
	else
		tmp = Float64(Float64(50.0 * Float64(i * n)) + Float64(n * 100.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.35e-105)
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	elseif (n <= 1.02e-142)
		tmp = 0.0 * (n / i);
	else
		tmp = (50.0 * (i * n)) + (n * 100.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.35e-105], N[(100.0 * N[(n + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.02e-142], N[(0.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.34999999999999996e-105

   1. Initial program 23.3%

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

   3. Taylor expanded in i around 0 55.1%

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

      1. associate-*r* 55.1%

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

      2. associate-*r/ 55.1%

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

      3. metadata-eval 55.1%

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

   5. Simplified 55.1%

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

   if -1.34999999999999996e-105 < n < 1.0200000000000001e-142

   1. Initial program 58.1%

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

      1. associate-*r/ 58.1%

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

      2. associate-/r/ 57.0%

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

      3. associate-*l/ 57.0%

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

      4. associate-/l* 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-lft-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. metadata-eval 56.8%

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

      9. metadata-eval 56.8%

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

      10. fma-define 56.8%

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

      11. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. fma-undefine 56.8%

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

      2. *-commutative 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in i around 0 69.7%

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

   if 1.0200000000000001e-142 < n

   1. Initial program 20.8%

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

   3. Taylor expanded in i around 0 69.7%

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

   4. Taylor expanded in n around inf 70.1%

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

      1. associate-*r* 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in i around 0 70.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 1.02 \cdot 10^{-142}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;i \cdot -50 + 100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{-143}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.3e-105)
   (+ (* i -50.0) (* 100.0 (* n (+ 1.0 (* i 0.5)))))
   (if (<= n 7.6e-143) (* 0.0 (/ n i)) (+ (* 50.0 (* i n)) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.3e-105) {
		tmp = (i * -50.0) + (100.0 * (n * (1.0 + (i * 0.5))));
	} else if (n <= 7.6e-143) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.3d-105)) then
        tmp = (i * (-50.0d0)) + (100.0d0 * (n * (1.0d0 + (i * 0.5d0))))
    else if (n <= 7.6d-143) then
        tmp = 0.0d0 * (n / i)
    else
        tmp = (50.0d0 * (i * n)) + (n * 100.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.3e-105) {
		tmp = (i * -50.0) + (100.0 * (n * (1.0 + (i * 0.5))));
	} else if (n <= 7.6e-143) {
		tmp = 0.0 * (n / i);
	} else {
		tmp = (50.0 * (i * n)) + (n * 100.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.3e-105:
		tmp = (i * -50.0) + (100.0 * (n * (1.0 + (i * 0.5))))
	elif n <= 7.6e-143:
		tmp = 0.0 * (n / i)
	else:
		tmp = (50.0 * (i * n)) + (n * 100.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.3e-105)
		tmp = Float64(Float64(i * -50.0) + Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * 0.5)))));
	elseif (n <= 7.6e-143)
		tmp = Float64(0.0 * Float64(n / i));
	else
		tmp = Float64(Float64(50.0 * Float64(i * n)) + Float64(n * 100.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.3e-105)
		tmp = (i * -50.0) + (100.0 * (n * (1.0 + (i * 0.5))));
	elseif (n <= 7.6e-143)
		tmp = 0.0 * (n / i);
	else
		tmp = (50.0 * (i * n)) + (n * 100.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.3e-105], N[(N[(i * -50.0), $MachinePrecision] + N[(100.0 * N[(n * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.6e-143], N[(0.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2999999999999999e-105

   1. Initial program 23.3%

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

      1. associate-*r/ 23.3%

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

      2. sub-neg 23.3%

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

      3. distribute-lft-in 23.3%

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

      4. metadata-eval 23.3%

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

      5. metadata-eval 23.3%

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

      6. metadata-eval 23.3%

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

      7. fma-define 23.3%

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

      8. metadata-eval 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in i around 0 38.0%

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

      1. distribute-lft-out 38.0%

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

      2. associate-*r/ 38.0%

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

      3. metadata-eval 38.0%

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

   7. Simplified 38.0%

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

   8. Taylor expanded in n around 0 55.1%

      \[\leadsto \color{blue}{-50 \cdot i + 100 \cdot \left(n \cdot \left(1 + 0.5 \cdot i\right)\right)} \]

   if -1.2999999999999999e-105 < n < 7.59999999999999962e-143

   1. Initial program 58.1%

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

      1. associate-*r/ 58.1%

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

      2. associate-/r/ 57.0%

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

      3. associate-*l/ 57.0%

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

      4. associate-/l* 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-lft-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. metadata-eval 56.8%

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

      9. metadata-eval 56.8%

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

      10. fma-define 56.8%

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

      11. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. fma-undefine 56.8%

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

      2. *-commutative 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in i around 0 69.7%

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

   if 7.59999999999999962e-143 < n

   1. Initial program 20.8%

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

   3. Taylor expanded in i around 0 69.7%

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

   4. Taylor expanded in n around inf 70.1%

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

      1. associate-*r* 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in i around 0 70.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105}:\\ \;\;\;\;i \cdot -50 + 100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{-143}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right) + n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105} \lor \neg \left(n \leq 1.26 \cdot 10^{-142}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.3e-105) (not (<= n 1.26e-142)))
   (* n (+ 100.0 (* i 50.0)))
   (* 0.0 (/ n i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-105) || !(n <= 1.26e-142)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0 * (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 ((n <= (-1.3d-105)) .or. (.not. (n <= 1.26d-142))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0 * (n / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-105) || !(n <= 1.26e-142)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0 * (n / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.3e-105) or not (n <= 1.26e-142):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0 * (n / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.3e-105) || !(n <= 1.26e-142))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(0.0 * Float64(n / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.3e-105) || ~((n <= 1.26e-142)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0 * (n / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.3e-105], N[Not[LessEqual[n, 1.26e-142]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2999999999999999e-105 or 1.26000000000000007e-142 < n

   1. Initial program 22.0%

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

   3. Taylor expanded in i around 0 62.3%

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

   4. Taylor expanded in n around inf 62.4%

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

      1. associate-*r* 62.4%

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

      2. *-commutative 62.4%

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

      3. *-commutative 62.4%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in i around 0 62.4%

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

      1. associate-*r* 62.4%

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

      2. distribute-rgt-out 62.4%

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

   9. Simplified 62.4%

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

   if -1.2999999999999999e-105 < n < 1.26000000000000007e-142

   1. Initial program 58.1%

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

      1. associate-*r/ 58.1%

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

      2. associate-/r/ 57.0%

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

      3. associate-*l/ 57.0%

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

      4. associate-/l* 56.8%

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

      5. sub-neg 56.8%

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

      6. distribute-lft-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. metadata-eval 56.8%

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

      9. metadata-eval 56.8%

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

      10. fma-define 56.8%

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

      11. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. fma-undefine 56.8%

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

      2. *-commutative 56.8%

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

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in i around 0 69.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-105} \lor \neg \left(n \leq 1.26 \cdot 10^{-142}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \frac{n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -9 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.7:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -9e-23)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 1.7) (* 100.0 (+ n (* i -0.5))) (* 50.0 (* i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -9e-23) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1.7) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-9d-23)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 1.7d0) then
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -9e-23) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1.7) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -9e-23:
		tmp = 100.0 * (i / (i / n))
	elif i <= 1.7:
		tmp = 100.0 * (n + (i * -0.5))
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -9e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 1.7)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -9e-23)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 1.7)
		tmp = 100.0 * (n + (i * -0.5));
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -9e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.7], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -8.9999999999999995e-23

   1. Initial program 55.1%

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

   3. Taylor expanded in i around 0 21.7%

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

   if -8.9999999999999995e-23 < i < 1.69999999999999996

   1. Initial program 7.3%

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

   3. Taylor expanded in i around 0 86.2%

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

   4. Taylor expanded in n around 0 85.6%

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

   if 1.69999999999999996 < i

   1. Initial program 51.5%

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

   3. Taylor expanded in i around 0 33.7%

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

   4. Taylor expanded in n around inf 34.2%

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

      1. associate-*r* 34.2%

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

      2. *-commutative 34.2%

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

      3. *-commutative 34.2%

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

   6. Simplified 34.2%

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

   7. Taylor expanded in i around inf 34.2%

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

      1. *-commutative 34.2%

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

   9. Simplified 34.2%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -9 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.7:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.2% accurate, 9.5× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.4) {
		tmp = 100.0 * (i / (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 (i <= (-1.4d0)) then
        tmp = 100.0d0 * (i / (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 (i <= -1.4) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

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

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.4)
		tmp = Float64(100.0 * Float64(i / 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 (i <= -1.4)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < -1.3999999999999999

   1. Initial program 56.9%

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

   3. Taylor expanded in i around 0 19.5%

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

   if -1.3999999999999999 < i

   1. Initial program 19.2%

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

   3. Taylor expanded in i around 0 71.9%

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

   4. Taylor expanded in n around inf 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

      3. *-commutative 71.6%

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

   6. Simplified 71.6%

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

   7. Taylor expanded in i around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. distribute-rgt-out 71.6%

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

   9. Simplified 71.6%

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

4. Final simplification 59.4%

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

Alternative 16: 54.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1060:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1060.0) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1060.0) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1060.0d0) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1060.0) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1060.0:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1060.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1060.0)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1060.0], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1060

   1. Initial program 22.2%

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

   3. Taylor expanded in i around 0 60.8%

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

      1. *-commutative 60.8%

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

   5. Simplified 60.8%

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

   if 1060 < i

   1. Initial program 52.4%

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

   3. Taylor expanded in i around 0 34.4%

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

   4. Taylor expanded in n around inf 34.8%

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

      1. associate-*r* 34.8%

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

      2. *-commutative 34.8%

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

      3. *-commutative 34.8%

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

   6. Simplified 34.8%

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

   7. Taylor expanded in i around inf 34.8%

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

      1. *-commutative 34.8%

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

   9. Simplified 34.8%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1060:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 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 28.1%

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

3. Taylor expanded in i around 0 55.3%

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

4. Taylor expanded in n around 0 2.7%

   \[\leadsto \color{blue}{-50 \cdot i} \]
5. Step-by-step derivation

   1. *-commutative 2.7%

      \[\leadsto \color{blue}{i \cdot -50} \]

6. Simplified 2.7%

   \[\leadsto \color{blue}{i \cdot -50} \]

7. Final simplification 2.7%

   \[\leadsto i \cdot -50 \]
8. Add Preprocessing

Alternative 18: 48.5% 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 28.1%

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

3. Taylor expanded in i around 0 49.9%

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

   1. *-commutative 49.9%

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

5. Simplified 49.9%

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

6. Final simplification 49.9%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer target: 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 2024046 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

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

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