Compound Interest

Percentage Accurate: 28.1% → 94.7%

Time: 17.0s

Alternatives: 21

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n);
	} else {
		tmp = n * 100.0;
	}
	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 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n);
	} else {
		tmp = n * 100.0;
	}
	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 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n)
	else:
		tmp = n * 100.0
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.0%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 98.6%

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

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

   1. Initial program 96.1%

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

      1. frac-2neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. neg-sub0 N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. distribute-lft-neg-in N/A

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

      10. frac-2neg N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right), \color{blue}{\left(\frac{i}{n}\right)}\right) \]

   4. Applied egg-rr 96.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. *-lowering-*.f64 67.8%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 67.8%

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

4. Final simplification 93.4%

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

Alternative 2: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 3.4 \cdot 10^{+62}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}} - n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 3.4e+62)
   (* n (* 100.0 (/ (expm1 i) i)))
   (* 100.0 (/ (- (/ (pow (+ 1.0 (/ i n)) n) (/ 1.0 n)) n) i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 3.4e+62) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = 100.0 * (((pow((1.0 + (i / n)), n) / (1.0 / n)) - n) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 3.4e+62) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = 100.0 * (((Math.pow((1.0 + (i / n)), n) / (1.0 / n)) - n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 3.4e+62:
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = 100.0 * (((math.pow((1.0 + (i / n)), n) / (1.0 / n)) - n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 3.4e+62)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) / Float64(1.0 / n)) - n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 3.4e+62], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] / N[(1.0 / n), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 3.40000000000000014e62

   1. Initial program 18.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 69.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(\frac{e^{i} - 1}{i}\right), \color{blue}{100}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{i} - 1\right), i\right), 100\right)\right) \]

      7. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right), 100\right)\right) \]

      8. expm1-lowering-expm1.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right), 100\right)\right) \]

   7. Applied egg-rr 81.0%

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

   if 3.40000000000000014e62 < i

   1. Initial program 62.7%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 54.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 54.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{n}{i}}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{i}\right)}\right)\right)\right) \]

      3. /-lowering-/.f64 54.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{i}\right)\right)\right)\right) \]

   6. Applied egg-rr 54.9%

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{\frac{n}{i}}} - \color{blue}{\frac{1}{\frac{1}{\frac{n}{i}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n} \cdot i} - \frac{1}{\frac{1}{\frac{n}{i}}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}}}{i} - \frac{\color{blue}{1}}{\frac{1}{\frac{n}{i}}}\right)\right) \]

      4. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}}}{i} - \frac{n}{\color{blue}{i}}\right)\right) \]

      5. sub-div N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}} - n}{\color{blue}{i}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}} - n\right), \color{blue}{i}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}}\right), n\right), i\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(e^{n \cdot \log \left(1 + \frac{i}{n}\right)}\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n}\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), n\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{i}{n}\right)\right), n\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      14. /-lowering-/.f64 63.1%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \mathsf{/.f64}\left(1, n\right)\right), n\right), i\right)\right) \]

   8. Applied egg-rr 63.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.4 \cdot 10^{+62}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{1}{n}} - n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.1e+63)
   (* n (* 100.0 (/ (expm1 i) i)))
   (/ (+ -100.0 (* (pow (+ 1.0 (/ i n)) n) 100.0)) (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.1e+63) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = (-100.0 + (pow((1.0 + (i / n)), n) * 100.0)) / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.1e+63) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = (-100.0 + (Math.pow((1.0 + (i / n)), n) * 100.0)) / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.1e+63:
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = (-100.0 + (math.pow((1.0 + (i / n)), n) * 100.0)) / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.1e+63)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(-100.0 + Float64((Float64(1.0 + Float64(i / n)) ^ n) * 100.0)) / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.1e+63], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-100.0 + N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2.1000000000000002e63

   1. Initial program 18.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 69.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(\frac{e^{i} - 1}{i}\right), \color{blue}{100}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{i} - 1\right), i\right), 100\right)\right) \]

      7. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right), 100\right)\right) \]

      8. expm1-lowering-expm1.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right), 100\right)\right) \]

   7. Applied egg-rr 81.0%

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

   if 2.1000000000000002e63 < i

   1. Initial program 62.7%

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

      1. frac-2neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. neg-sub0 N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. distribute-lft-neg-in N/A

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

      10. frac-2neg N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-100 \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right), \color{blue}{\left(\frac{i}{n}\right)}\right) \]

   4. Applied egg-rr 62.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.35 \cdot 10^{+64}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.35e+64)
   (* n (* 100.0 (/ (expm1 i) i)))
   (* (/ n (/ i -100.0)) (- 1.0 (pow (+ 1.0 (/ i n)) n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.35e+64) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = (n / (i / -100.0)) * (1.0 - pow((1.0 + (i / n)), n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.35e+64) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = (n / (i / -100.0)) * (1.0 - Math.pow((1.0 + (i / n)), n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.35e+64:
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = (n / (i / -100.0)) * (1.0 - math.pow((1.0 + (i / n)), n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.35e+64)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(n / Float64(i / -100.0)) * Float64(1.0 - (Float64(1.0 + Float64(i / n)) ^ n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.35e+64], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n / N[(i / -100.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.35e64

   1. Initial program 18.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 69.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(\frac{e^{i} - 1}{i}\right), \color{blue}{100}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{i} - 1\right), i\right), 100\right)\right) \]

      7. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right), 100\right)\right) \]

      8. expm1-lowering-expm1.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right), 100\right)\right) \]

   7. Applied egg-rr 81.0%

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

   if 1.35e64 < i

   1. Initial program 62.7%

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

      1. frac-2neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-+l- N/A

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

      4. neg-sub0 N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-100}{\frac{i}{n}}\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

      12. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-100}{i} \cdot n\right), \left(\color{blue}{1} - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(n \cdot \frac{-100}{i}\right), \left(\color{blue}{1} - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \]

      14. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(n \cdot \frac{1}{\frac{i}{-100}}\right), \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \]

      15. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{n}{\frac{i}{-100}}\right), \left(\color{blue}{1} - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \left(\frac{i}{-100}\right)\right), \left(\color{blue}{1} - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(i, -100\right)\right), \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\right) \]

      18. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(i, -100\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}\right)\right) \]

      19. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(i, -100\right)\right), \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), \color{blue}{n}\right)\right)\right) \]

   4. Applied egg-rr 62.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.35 \cdot 10^{+64}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} \cdot \left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{+63}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 5e+63)
   (* n (* 100.0 (/ (expm1 i) i)))
   (* (- 1.0 (pow (+ 1.0 (/ i n)) n)) (/ -100.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 5e+63) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = (1.0 - pow((1.0 + (i / n)), n)) * (-100.0 / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 5e+63) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = (1.0 - Math.pow((1.0 + (i / n)), n)) * (-100.0 / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 5e+63:
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = (1.0 - math.pow((1.0 + (i / n)), n)) * (-100.0 / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 5e+63)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(1.0 - (Float64(1.0 + Float64(i / n)) ^ n)) * Float64(-100.0 / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 5e+63], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] * N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.00000000000000011e63

   1. Initial program 18.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 69.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(\frac{e^{i} - 1}{i}\right), \color{blue}{100}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{i} - 1\right), i\right), 100\right)\right) \]

      7. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right), 100\right)\right) \]

      8. expm1-lowering-expm1.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right), 100\right)\right) \]

   7. Applied egg-rr 81.0%

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

   if 5.00000000000000011e63 < i

   1. Initial program 62.7%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

         \[\leadsto \frac{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      4. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right) + 1\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left({\left(1 + \frac{i}{n}\right)}^{n}\right)\right)\right)\right)\right) \cdot 100}{\frac{i}{n}} \]

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 62.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{+63}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot \frac{-100}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.12 \cdot 10^{+64}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.12e+64)
   (* n (* 100.0 (/ (expm1 i) i)))
   (* (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) 100.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.12e+64) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = ((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.12e+64) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = ((Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.12e+64:
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = ((math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.12e+64)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n)) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.12e+64], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.11999999999999995e64

   1. Initial program 18.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 69.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(\frac{e^{i} - 1}{i}\right), \color{blue}{100}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{i} - 1\right), i\right), 100\right)\right) \]

      7. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right), 100\right)\right) \]

      8. expm1-lowering-expm1.f64 81.0%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right), 100\right)\right) \]

   7. Applied egg-rr 81.0%

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

   if 1.11999999999999995e64 < i

   1. Initial program 62.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.12 \cdot 10^{+64}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.05 \cdot 10^{-247}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-112}:\\ \;\;\;\;100 \cdot \frac{n - n}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -1.05e-247)
     t_0
     (if (<= n 2.8e-112) (* 100.0 (/ (- n n) i)) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -1.05e-247) {
		tmp = t_0;
	} else if (n <= 2.8e-112) {
		tmp = 100.0 * ((n - n) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -1.05e-247) {
		tmp = t_0;
	} else if (n <= 2.8e-112) {
		tmp = 100.0 * ((n - n) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -1.05e-247:
		tmp = t_0
	elif n <= 2.8e-112:
		tmp = 100.0 * ((n - n) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -1.05e-247)
		tmp = t_0;
	elseif (n <= 2.8e-112)
		tmp = Float64(100.0 * Float64(Float64(n - n) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.05e-247], t$95$0, If[LessEqual[n, 2.8e-112], N[(100.0 * N[(N[(n - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.05000000000000007e-247 or 2.80000000000000023e-112 < n

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 71.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 71.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(\frac{e^{i} - 1}{i}\right), \color{blue}{100}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{i} - 1\right), i\right), 100\right)\right) \]

      7. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), i\right), 100\right)\right) \]

      8. expm1-lowering-expm1.f64 80.0%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), i\right), 100\right)\right) \]

   7. Applied egg-rr 80.0%

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

   if -1.05000000000000007e-247 < n < 2.80000000000000023e-112

   1. Initial program 31.8%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 70.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 70.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \left(\frac{1}{\color{blue}{\frac{n}{i}}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{i}\right)}\right)\right)\right) \]

      3. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{i}\right)\right)\right)\right) \]

   6. Applied egg-rr 71.0%

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{1}{\frac{n}{i}}}} \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{\frac{n}{i}}} - \color{blue}{\frac{1}{\frac{1}{\frac{n}{i}}}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n} \cdot i} - \frac{1}{\frac{1}{\frac{n}{i}}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}}}{i} - \frac{\color{blue}{1}}{\frac{1}{\frac{n}{i}}}\right)\right) \]

      4. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}}}{i} - \frac{n}{\color{blue}{i}}\right)\right) \]

      5. sub-div N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}} - n}{\color{blue}{i}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}} - n\right), \color{blue}{i}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{e^{n \cdot \log \left(1 + \frac{i}{n}\right)}}{\frac{1}{n}}\right), n\right), i\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(e^{n \cdot \log \left(1 + \frac{i}{n}\right)}\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n}\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n}\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), n\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{i}{n}\right)\right), n\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \left(\frac{1}{n}\right)\right), n\right), i\right)\right) \]

      14. /-lowering-/.f64 26.3%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \mathsf{/.f64}\left(1, n\right)\right), n\right), i\right)\right) \]

   8. Applied egg-rr 26.3%

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

   9. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{n}, n\right), i\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 60.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-247}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-112}:\\ \;\;\;\;100 \cdot \frac{n - n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -3.1 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -3.1e-67) t_0 (if (<= n 0.42) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -3.1e-67) {
		tmp = t_0;
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -3.1e-67) {
		tmp = t_0;
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -3.1e-67:
		tmp = t_0
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -3.1e-67)
		tmp = t_0;
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.1e-67], t$95$0, If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.1000000000000003e-67 or 0.419999999999999984 < n

   1. Initial program 30.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 86.6%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 86.6%

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

   if -3.1000000000000003e-67 < n < 0.419999999999999984

   1. Initial program 26.7%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.0%

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

Alternative 9: 66.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+21}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.5e+21)
   (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.42)
     (* 100.0 (/ i (/ i n)))
     (*
      100.0
      (/
       (*
        n
        (*
         i
         (+
          1.0
          (*
           i
           (+ 0.5 (* i (+ 0.16666666666666666 (* i 0.041666666666666664))))))))
       i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.5e+21) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.5d+21)) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * ((n * (i * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0)))))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.5e+21) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.5e+21:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.5e+21)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664)))))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.5e+21)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664)))))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.5e+21], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.500000000000001e21

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 52.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 52.3%

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

   if -9.500000000000001e21 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.5%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 96.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{24} \cdot i\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \left(i \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

      8. *-lowering-*.f64 82.2%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(i, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 82.2%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+21}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{+22}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\left(i \cdot i\right) \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right) + \left(100 + i \cdot 50\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.2e+22)
   (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.42)
     (* 100.0 (/ i (/ i n)))
     (*
      n
      (+
       (* (* i i) (+ 16.666666666666668 (* i 4.166666666666667)))
       (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.2e+22) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (((i * i) * (16.666666666666668 + (i * 4.166666666666667))) + (100.0 + (i * 50.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-7.2d+22)) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (((i * i) * (16.666666666666668d0 + (i * 4.166666666666667d0))) + (100.0d0 + (i * 50.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.2e+22) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (((i * i) * (16.666666666666668 + (i * 4.166666666666667))) + (100.0 + (i * 50.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.2e+22:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (((i * i) * (16.666666666666668 + (i * 4.166666666666667))) + (100.0 + (i * 50.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.2e+22)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(Float64(Float64(i * i) * Float64(16.666666666666668 + Float64(i * 4.166666666666667))) + Float64(100.0 + Float64(i * 50.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -7.2e+22)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (((i * i) * (16.666666666666668 + (i * 4.166666666666667))) + (100.0 + (i * 50.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.2e+22], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(N[(i * i), $MachinePrecision] * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.2e22

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 52.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 52.3%

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

   if -7.2e22 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.5%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 96.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto 100 \cdot n + \left(i \cdot \left(50 \cdot n\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right), \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(\left(i \cdot i\right) \cdot \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left({i}^{2} \cdot \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left({i}^{2}\right), \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left(i \cdot i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right) \]

   8. Simplified 79.6%

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

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{n \cdot \left(100 + \left(50 \cdot i + {i}^{2} \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + \left(50 \cdot i + {i}^{2} \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \left(\left(50 \cdot i + {i}^{2} \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + \color{blue}{100}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \left(\left({i}^{2} \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + 50 \cdot i\right) + 100\right)\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{*.f64}\left(n, \left({i}^{2} \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + \color{blue}{\left(50 \cdot i + 100\right)}\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \left({i}^{2} \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + \left(100 + \color{blue}{50 \cdot i}\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left({i}^{2} \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right), \color{blue}{\left(100 + 50 \cdot i\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right), \left(\color{blue}{100} + 50 \cdot i\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right), \left(100 + 50 \cdot i\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right), \left(100 + 50 \cdot i\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{50}{3}, \left(\frac{25}{6} \cdot i\right)\right)\right), \left(100 + 50 \cdot i\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{50}{3}, \left(i \cdot \frac{25}{6}\right)\right)\right), \left(100 + 50 \cdot i\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(i, \frac{25}{6}\right)\right)\right), \left(100 + 50 \cdot i\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(i, \frac{25}{6}\right)\right)\right), \mathsf{+.f64}\left(100, \color{blue}{\left(50 \cdot i\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(i, \frac{25}{6}\right)\right)\right), \mathsf{+.f64}\left(100, \left(i \cdot \color{blue}{50}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 79.6%

           \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(i, \frac{25}{6}\right)\right)\right), \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{50}\right)\right)\right)\right) \]

   11. Simplified 79.6%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{+22}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\left(i \cdot i\right) \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right) + \left(100 + i \cdot 50\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+22}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.7e+22)
   (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.42)
     (* 100.0 (/ i (/ i n)))
     (*
      100.0
      (/ (* n (* i (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666)))))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.7e+22) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.7d+22)) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * ((n * (i * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0)))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.7e+22) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.7e+22:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.7e+22)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666)))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.7e+22)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * ((n * (i * (1.0 + (i * (0.5 + (i * 0.16666666666666666)))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.7e+22], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6999999999999998e22

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 52.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 52.3%

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

   if -3.6999999999999998e22 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.5%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 96.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)\right)\right), i\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)\right)\right), i\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot i\right)\right)\right)\right)\right)\right), i\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right), i\right)\right) \]

      6. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \frac{1}{6}\right)\right)\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 78.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+22}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{+23}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot i\right) \cdot \left(n \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.35e+23)
   (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.42)
     (* 100.0 (/ i (/ i n)))
     (+
      (* n 100.0)
      (* (* i i) (* n (+ 16.666666666666668 (* i 4.166666666666667))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.35e+23) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) + ((i * i) * (n * (16.666666666666668 + (i * 4.166666666666667))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.35d+23)) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) + ((i * i) * (n * (16.666666666666668d0 + (i * 4.166666666666667d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.35e+23) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) + ((i * i) * (n * (16.666666666666668 + (i * 4.166666666666667))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.35e+23:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) + ((i * i) * (n * (16.666666666666668 + (i * 4.166666666666667))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.35e+23)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(i * i) * Float64(n * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.35e+23)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) + ((i * i) * (n * (16.666666666666668 + (i * 4.166666666666667))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.35e+23], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.3499999999999999e23

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 52.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 52.3%

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

   if -1.3499999999999999e23 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.5%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 96.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto 100 \cdot n + \left(i \cdot \left(50 \cdot n\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right), \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(\left(i \cdot i\right) \cdot \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left({i}^{2} \cdot \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left({i}^{2}\right), \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left(i \cdot i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right) \]

   8. Simplified 79.6%

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

   9. Taylor expanded in i around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{100}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 78.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{+23}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot i\right) \cdot \left(n \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e+22)
   (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.42)
     (* 100.0 (/ i (/ i n)))
     (* 100.0 (/ (* n (* i (+ 1.0 (* i 0.5)))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+22) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.4d+22)) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * ((n * (i * (1.0d0 + (i * 0.5d0)))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e+22) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e+22:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e+22)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * 0.5)))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.4e+22)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e+22], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.4e22

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 52.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 52.3%

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

   if -3.4e22 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.5%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 96.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + \frac{1}{2} \cdot i\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot i\right)\right)\right)\right), i\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \frac{1}{2}\right)\right)\right)\right), i\right)\right) \]

      4. *-lowering-*.f64 76.2%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \frac{1}{2}\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 76.2%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+22}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{+23}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot \left(1 + i \cdot 0.5\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.6e+23)
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 0.42)
     (* 100.0 (/ i (/ i n)))
     (* 100.0 (/ (* n (* i (+ 1.0 (* i 0.5)))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.6e+23) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.6d+23)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * ((n * (i * (1.0d0 + (i * 0.5d0)))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.6e+23) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.6e+23:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.6e+23)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * Float64(1.0 + Float64(i * 0.5)))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.6e+23)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * ((n * (i * (1.0 + (i * 0.5)))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.6e+23], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.59999999999999992e23

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 81.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 52.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 52.3%

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

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + \frac{50}{3} \cdot i\right)}\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(\frac{50}{3} \cdot i\right)}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \left(i \cdot \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 52.3%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

   11. Simplified 52.3%

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

   if -2.59999999999999992e23 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.5%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 96.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)}\right), i\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + \frac{1}{2} \cdot i\right)\right)\right), i\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot i\right)\right)\right)\right), i\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \frac{1}{2}\right)\right)\right)\right), i\right)\right) \]

      4. *-lowering-*.f64 76.2%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \frac{1}{2}\right)\right)\right)\right), i\right)\right) \]

   8. Simplified 76.2%

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

Alternative 15: 65.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))))
   (if (<= n -2.8e+22) t_0 (if (<= n 0.42) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -2.8e+22) {
		tmp = t_0;
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    if (n <= (-2.8d+22)) then
        tmp = t_0
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -2.8e+22) {
		tmp = t_0;
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	tmp = 0
	if n <= -2.8e+22:
		tmp = t_0
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))))
	tmp = 0.0
	if (n <= -2.8e+22)
		tmp = t_0;
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	tmp = 0.0;
	if (n <= -2.8e+22)
		tmp = t_0;
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.8e+22], t$95$0, If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.8e22 or 0.419999999999999984 < n

   1. Initial program 33.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 89.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 89.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 64.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 64.3%

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

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \color{blue}{\left(50 + \frac{50}{3} \cdot i\right)}\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \color{blue}{\left(\frac{50}{3} \cdot i\right)}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \left(i \cdot \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 64.3%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(50, \mathsf{*.f64}\left(i, \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

   11. Simplified 64.3%

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

   if -2.8e22 < n < 0.419999999999999984

   1. Initial program 25.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.5%

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

Alternative 16: 62.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-60)
   (* 100.0 (/ (* i n) i))
   (if (<= n 0.42) (* 100.0 (/ i (/ i n))) (+ (* n 100.0) (* i (* n 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-60) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) + (i * (n * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.12d-60)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 0.42d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) + (i * (n * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-60) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) + (i * (n * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.12e-60:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 0.42:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) + (i * (n * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-60)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 0.42)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.12e-60)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 0.42)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) + (i * (n * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-60], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.12e-60

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 80.2%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 80.2%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(i \cdot n\right)}, i\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot i\right), i\right)\right) \]

      2. *-lowering-*.f64 53.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, i\right), i\right)\right) \]

   8. Simplified 53.4%

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

   if -1.12e-60 < n < 0.419999999999999984

   1. Initial program 26.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 59.4%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 96.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 75.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 75.7%

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

   9. Taylor expanded in i around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(50 \cdot n\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{50}\right)\right)\right) \]

       2. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{50}\right)\right)\right) \]

   11. Simplified 69.3%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-60)
   (* 100.0 (/ (* i n) i))
   (if (<= n 0.42) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-60) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 0.42) {
		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 (n <= (-1.12d-60)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 0.42d0) 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 (n <= -1.12e-60) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 0.42) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.12e-60:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 0.42:
		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 (n <= -1.12e-60)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 0.42)
		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 (n <= -1.12e-60)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 0.42)
		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[n, -1.12e-60], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.42], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.12e-60

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 80.2%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 80.2%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(i \cdot n\right)}, i\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot i\right), i\right)\right) \]

      2. *-lowering-*.f64 53.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, i\right), i\right)\right) \]

   8. Simplified 53.4%

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

   if -1.12e-60 < n < 0.419999999999999984

   1. Initial program 26.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 59.4%

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

   if 0.419999999999999984 < n

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 96.9%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + 50 \cdot i\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(50 \cdot i\right)}\right)\right) \]

      6. *-lowering-*.f64 69.3%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   8. Simplified 69.3%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 0.42:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -1.12e-60) t_0 (if (<= n 5e-19) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double tmp;
	if (n <= -1.12e-60) {
		tmp = t_0;
	} else if (n <= 5e-19) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * ((i * n) / i)
    if (n <= (-1.12d-60)) then
        tmp = t_0
    else if (n <= 5d-19) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double tmp;
	if (n <= -1.12e-60) {
		tmp = t_0;
	} else if (n <= 5e-19) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((i * n) / i)
	tmp = 0
	if n <= -1.12e-60:
		tmp = t_0
	elif n <= 5e-19:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(i * n) / i))
	tmp = 0.0
	if (n <= -1.12e-60)
		tmp = t_0;
	elseif (n <= 5e-19)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * ((i * n) / i);
	tmp = 0.0;
	if (n <= -1.12e-60)
		tmp = t_0;
	elseif (n <= 5e-19)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.12e-60], t$95$0, If[LessEqual[n, 5e-19], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.12e-60 or 5.0000000000000004e-19 < n

   1. Initial program 31.7%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 86.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 86.1%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(i \cdot n\right)}, i\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot i\right), i\right)\right) \]

      2. *-lowering-*.f64 59.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, i\right), i\right)\right) \]

   8. Simplified 59.4%

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

   if -1.12e-60 < n < 5.0000000000000004e-19

   1. Initial program 25.3%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-60}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -5e+28)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 1.4e+24) (* n 100.0) (* 16.666666666666668 (* n (* i i))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -5e+28) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1.4e+24) {
		tmp = n * 100.0;
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-5d+28)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 1.4d+24) then
        tmp = n * 100.0d0
    else
        tmp = 16.666666666666668d0 * (n * (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -5e+28) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1.4e+24) {
		tmp = n * 100.0;
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -5e+28:
		tmp = 100.0 * (i / (i / n))
	elif i <= 1.4e+24:
		tmp = n * 100.0
	else:
		tmp = 16.666666666666668 * (n * (i * i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -5e+28)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 1.4e+24)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -5e+28)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 1.4e+24)
		tmp = n * 100.0;
	else
		tmp = 16.666666666666668 * (n * (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -5e+28], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.4e+24], N[(n * 100.0), $MachinePrecision], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.99999999999999957e28

   1. Initial program 57.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 15.8%

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

   if -4.99999999999999957e28 < i < 1.4000000000000001e24

   1. Initial program 7.4%

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

   3. Taylor expanded in i around 0

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

      1. *-lowering-*.f64 80.8%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 80.8%

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

   if 1.4000000000000001e24 < i

   1. Initial program 58.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 39.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 39.7%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 31.5%

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

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \color{blue}{\left({i}^{2} \cdot n\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \left(n \cdot \color{blue}{{i}^{2}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(n, \color{blue}{\left({i}^{2}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(n, \left(i \cdot \color{blue}{i}\right)\right)\right) \]

       5. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right) \]

   11. Simplified 31.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.85 \cdot 10^{+24}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.85e+24) (* n 100.0) (* 16.666666666666668 (* n (* i i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.85e+24) {
		tmp = n * 100.0;
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.85d+24) then
        tmp = n * 100.0d0
    else
        tmp = 16.666666666666668d0 * (n * (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.85e+24) {
		tmp = n * 100.0;
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.85e+24:
		tmp = n * 100.0
	else:
		tmp = 16.666666666666668 * (n * (i * i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.85e+24)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.85e+24)
		tmp = n * 100.0;
	else
		tmp = 16.666666666666668 * (n * (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.85e+24], N[(n * 100.0), $MachinePrecision], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.85e24

   1. Initial program 19.0%

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

   3. Taylor expanded in i around 0

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

      1. *-lowering-*.f64 63.2%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 63.2%

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

   if 1.85e24 < i

   1. Initial program 58.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{i}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right)\right) \]

      3. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right)\right) \]

      4. expm1-lowering-expm1.f64 39.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right)\right) \]

   5. Simplified 39.7%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(n \cdot i\right) + 50 \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(\frac{50}{3} \cdot n\right) \cdot i + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\left(\frac{50}{3} \cdot n\right) \cdot i + 50 \cdot n\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(n \cdot i\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   8. Simplified 31.5%

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

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \color{blue}{\left({i}^{2} \cdot n\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \left(n \cdot \color{blue}{{i}^{2}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(n, \color{blue}{\left({i}^{2}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(n, \left(i \cdot \color{blue}{i}\right)\right)\right) \]

       5. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{50}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right) \]

   11. Simplified 31.5%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.85 \cdot 10^{+24}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 50.0% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

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

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 29.2%

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

3. Taylor expanded in i around 0

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

   1. *-lowering-*.f64 47.9%

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

5. Simplified 47.9%

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

6. Final simplification 47.9%

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

Developer Target 1: 33.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

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