Compound Interest

Percentage Accurate: 27.6% → 95.8%

Time: 19.3s

Alternatives: 17

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: 27.6% 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: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-104}:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -4e-104)
     (* n (/ (+ (* t_0 100.0) -100.0) i))
     (if (<= t_2 0.0)
       (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) (/ i n))
       (if (<= t_2 INFINITY) (* 100.0 (* (/ n i) t_1)) (/ 100.0 (/ 1.0 n)))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -4e-104) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else if (t_2 <= 0.0) {
		tmp = (100.0 * expm1((n * log1p((i / n))))) / (i / n);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 100.0 * ((n / i) * t_1);
	} else {
		tmp = 100.0 / (1.0 / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -4e-104) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else if (t_2 <= 0.0) {
		tmp = (100.0 * Math.expm1((n * Math.log1p((i / n))))) / (i / n);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((n / i) * t_1);
	} else {
		tmp = 100.0 / (1.0 / n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = t_0 + -1.0
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -4e-104:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	elif t_2 <= 0.0:
		tmp = (100.0 * math.expm1((n * math.log1p((i / n))))) / (i / n)
	elif t_2 <= math.inf:
		tmp = 100.0 * ((n / i) * t_1)
	else:
		tmp = 100.0 / (1.0 / n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -4e-104)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / Float64(i / n));
	elseif (t_2 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(n / i) * t_1));
	else
		tmp = Float64(100.0 / Float64(1.0 / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-104], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(1.0 / n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -3.99999999999999971e-104

   1. Initial program 99.7%

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

      1. associate-/r/ 99.7%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. fma-def 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -3.99999999999999971e-104 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 20.5%

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

      1. associate-*r/ 20.5%

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

      2. *-commutative 20.5%

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

      3. pow-to-exp 20.5%

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

      4. expm1-def 31.4%

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

      5. add-log-exp 20.5%

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

      6. pow-to-exp 20.5%

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

      7. log-pow 31.4%

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

      8. log1p-udef 99.6%

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

   3. Applied egg-rr 99.6%

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

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

   1. Initial program 96.0%

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

      1. div-sub 95.9%

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

      2. clear-num 96.1%

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

      3. sub-neg 96.1%

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

      4. div-inv 96.1%

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

      5. clear-num 96.1%

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

   3. Applied egg-rr 96.1%

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

      1. +-commutative 96.1%

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

      2. neg-mul-1 96.1%

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

      3. distribute-rgt-out 96.1%

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

   5. Simplified 96.1%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. fma-udef 0.0%

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

      7. div-inv 0.0%

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

      8. associate-/r* 1.8%

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

   3. Applied egg-rr 1.8%

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

   4. Taylor expanded in i around 0 66.9%

      \[\leadsto \frac{\color{blue}{100}}{\frac{1}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-104}:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n}}\\ \end{array} \]

Alternative 2: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ t_2 := 100 \cdot \left(\frac{n}{i} \cdot t_0\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* 100.0 (* (/ n i) t_0))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 0.0)
       (* n (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 i)))
       (if (<= t_1 INFINITY) t_2 (/ 100.0 (/ 1.0 n)))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = 100.0 * ((n / i) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 / (1.0 / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = 100.0 * ((n / i) * t_0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 / (1.0 / n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	t_2 = 100.0 * ((n / i) * t_0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) * (100.0 / i))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 / (1.0 / n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(100.0 * Float64(Float64(n / i) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / i)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 / Float64(1.0 / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 / N[(1.0 / n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.9%

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

      1. div-sub 96.8%

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

      2. clear-num 96.9%

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

      3. sub-neg 96.9%

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

      4. div-inv 96.9%

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

      5. clear-num 96.9%

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

   3. Applied egg-rr 96.9%

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

      1. +-commutative 96.9%

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

      2. neg-mul-1 96.9%

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

      3. distribute-rgt-out 97.0%

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

   5. Simplified 97.0%

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

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

   1. Initial program 23.4%

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

      1. associate-/r/ 23.4%

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

      2. associate-*r* 23.4%

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

      3. *-commutative 23.4%

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

      4. associate-*r/ 23.4%

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

      5. sub-neg 23.4%

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

      6. distribute-lft-in 23.3%

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

      7. fma-def 23.4%

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

      8. metadata-eval 23.4%

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

      9. metadata-eval 23.4%

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

   3. Simplified 23.4%

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

      1. expm1-log1p-u 23.4%

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

      2. expm1-udef 18.9%

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

   5. Applied egg-rr 73.3%

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

      1. expm1-def 97.8%

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

      2. expm1-log1p 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. associate-*r/ 98.6%

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

      5. associate-*l* 98.6%

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

      6. associate-*r/ 98.7%

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

      7. metadata-eval 98.7%

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

   7. Simplified 98.7%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. fma-udef 0.0%

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

      7. div-inv 0.0%

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

      8. associate-/r* 1.8%

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

   3. Applied egg-rr 1.8%

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

   4. Taylor expanded in i around 0 66.9%

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

4. Final simplification 91.7%

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

Alternative 3: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+153}:\\ \;\;\;\;n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 8200:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.2e+153)
   (* n (+ (+ 100.0 (* 16.666666666666668 (* i i))) (* i 50.0)))
   (if (<= n 8200.0)
     (* 100.0 (/ (* n n) (+ n (* (* i n) (- (/ 0.5 n) 0.5)))))
     (if (<= n 3.6e+186)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.2e+153) {
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	} else if (n <= 8200.0) {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	} else if (n <= 3.6e+186) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.2e+153) {
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	} else if (n <= 8200.0) {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	} else if (n <= 3.6e+186) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.2e+153:
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0))
	elif n <= 8200.0:
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))))
	elif n <= 3.6e+186:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.2e+153)
		tmp = Float64(n * Float64(Float64(100.0 + Float64(16.666666666666668 * Float64(i * i))) + Float64(i * 50.0)));
	elseif (n <= 8200.0)
		tmp = Float64(100.0 * Float64(Float64(n * n) / Float64(n + Float64(Float64(i * n) * Float64(Float64(0.5 / n) - 0.5)))));
	elseif (n <= 3.6e+186)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.2e+153], N[(n * N[(N[(100.0 + N[(16.666666666666668 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8200.0], N[(100.0 * N[(N[(n * n), $MachinePrecision] / N[(n + N[(N[(i * n), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.6e+186], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.2000000000000001e153

   1. Initial program 16.2%

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

      1. associate-/r/ 16.9%

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

      2. associate-*r* 16.9%

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

      3. *-commutative 16.9%

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

      4. associate-*r/ 17.0%

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

      5. sub-neg 17.0%

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

      6. distribute-lft-in 17.0%

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

      7. fma-def 17.0%

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

      8. metadata-eval 17.0%

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

      9. metadata-eval 17.0%

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

   3. Simplified 17.0%

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

   4. Taylor expanded in i around 0 68.0%

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

      1. +-commutative 68.0%

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

      2. distribute-lft-out 68.0%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in n around inf 68.0%

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

      1. distribute-lft-in 68.0%

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

      2. associate-+r+ 68.0%

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

      3. associate-*r* 68.0%

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

      4. metadata-eval 68.0%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 68.0%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 68.0%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 68.0%

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

   9. Simplified 68.0%

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

   if -3.2000000000000001e153 < n < 8200

   1. Initial program 29.6%

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

   2. Taylor expanded in i around 0 46.0%

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

      1. associate-*r* 46.4%

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

      2. *-commutative 46.4%

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

      3. associate-*r/ 46.4%

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

      4. metadata-eval 46.4%

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

   4. Simplified 46.4%

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

      1. flip-+ 43.9%

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

      2. *-commutative 43.9%

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

      3. *-commutative 43.9%

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

      4. *-commutative 43.9%

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

   6. Applied egg-rr 43.9%

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

   7. Taylor expanded in i around 0 74.4%

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

      1. unpow2 74.4%

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

   9. Simplified 74.4%

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

   if 8200 < n < 3.6000000000000002e186

   1. Initial program 49.5%

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

   2. Taylor expanded in n around inf 58.9%

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

      1. expm1-def 84.6%

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

   4. Simplified 84.6%

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

   if 3.6000000000000002e186 < n

   1. Initial program 16.0%

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

      1. associate-/r/ 16.7%

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

      2. associate-*r* 16.7%

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

      3. *-commutative 16.7%

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

      4. associate-*r/ 16.8%

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

      5. sub-neg 16.8%

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

      6. distribute-lft-in 16.8%

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

      7. fma-def 16.8%

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

      8. metadata-eval 16.8%

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

      9. metadata-eval 16.8%

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

   3. Simplified 16.8%

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

   4. Taylor expanded in i around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. distribute-lft-out 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in n around inf 76.6%

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

      1. distribute-lft-in 76.6%

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

      2. associate-+r+ 76.6%

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

      3. associate-*r* 76.6%

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

      4. metadata-eval 76.6%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 76.6%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 76.6%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 76.6%

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

   9. Simplified 76.6%

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

   10. Taylor expanded in i around 0 78.8%

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

       1. *-commutative 78.8%

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

   12. Simplified 78.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+153}:\\ \;\;\;\;n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 8200:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 4: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -135000000000 \lor \neg \left(n \leq 5.5 \cdot 10^{-10}\right):\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -135000000000.0) (not (<= n 5.5e-10)))
   (* n (* (/ 100.0 i) (expm1 i)))
   (* 100.0 (/ (* n n) (+ n (* (* i n) (- (/ 0.5 n) 0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -135000000000.0) || !(n <= 5.5e-10)) {
		tmp = n * ((100.0 / i) * expm1(i));
	} else {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -135000000000.0) || !(n <= 5.5e-10)) {
		tmp = n * ((100.0 / i) * Math.expm1(i));
	} else {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -135000000000.0) or not (n <= 5.5e-10):
		tmp = n * ((100.0 / i) * math.expm1(i))
	else:
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -135000000000.0) || !(n <= 5.5e-10))
		tmp = Float64(n * Float64(Float64(100.0 / i) * expm1(i)));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / Float64(n + Float64(Float64(i * n) * Float64(Float64(0.5 / n) - 0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -135000000000.0], N[Not[LessEqual[n, 5.5e-10]], $MachinePrecision]], N[(n * N[(N[(100.0 / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / N[(n + N[(N[(i * n), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.35e11 or 5.4999999999999996e-10 < n

   1. Initial program 28.4%

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

      1. associate-/r/ 28.8%

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

      2. associate-*r* 28.8%

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

      3. *-commutative 28.8%

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

      4. associate-*r/ 28.9%

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

      5. sub-neg 28.9%

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

      6. distribute-lft-in 28.9%

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

      7. fma-def 28.9%

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

      8. metadata-eval 28.9%

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

      9. metadata-eval 28.9%

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

   3. Simplified 28.9%

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

      1. expm1-log1p-u 25.0%

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

      2. expm1-udef 15.1%

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

   5. Applied egg-rr 52.1%

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

      1. expm1-def 67.7%

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

      2. expm1-log1p 68.2%

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

      3. *-rgt-identity 68.2%

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

      4. associate-*r/ 68.1%

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

      5. associate-*l* 68.1%

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

      6. associate-*r/ 68.2%

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

      7. metadata-eval 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in n around inf 90.9%

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

   if -1.35e11 < n < 5.4999999999999996e-10

   1. Initial program 30.1%

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

   2. Taylor expanded in i around 0 36.3%

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

      1. associate-*r* 36.7%

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

      2. *-commutative 36.7%

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

      3. associate-*r/ 36.7%

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

      4. metadata-eval 36.7%

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

   4. Simplified 36.7%

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

      1. flip-+ 33.4%

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

      2. *-commutative 33.4%

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

      3. *-commutative 33.4%

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

      4. *-commutative 33.4%

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

   6. Applied egg-rr 33.4%

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

   7. Taylor expanded in i around 0 74.7%

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

      1. unpow2 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -135000000000 \lor \neg \left(n \leq 5.5 \cdot 10^{-10}\right):\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \end{array} \]

Alternative 5: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -135000000000 \lor \neg \left(n \leq 5.5 \cdot 10^{-10}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -135000000000.0) (not (<= n 5.5e-10)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (* 100.0 (/ (* n n) (+ n (* (* i n) (- (/ 0.5 n) 0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -135000000000.0) || !(n <= 5.5e-10)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -135000000000.0) || !(n <= 5.5e-10)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -135000000000.0) or not (n <= 5.5e-10):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -135000000000.0) || !(n <= 5.5e-10))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / Float64(n + Float64(Float64(i * n) * Float64(Float64(0.5 / n) - 0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -135000000000.0], N[Not[LessEqual[n, 5.5e-10]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / N[(n + N[(N[(i * n), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.35e11 or 5.4999999999999996e-10 < n

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/l* 41.7%

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

      3. expm1-def 90.9%

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

   4. Simplified 90.9%

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

   if -1.35e11 < n < 5.4999999999999996e-10

   1. Initial program 30.1%

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

   2. Taylor expanded in i around 0 36.3%

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

      1. associate-*r* 36.7%

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

      2. *-commutative 36.7%

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

      3. associate-*r/ 36.7%

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

      4. metadata-eval 36.7%

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

   4. Simplified 36.7%

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

      1. flip-+ 33.4%

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

      2. *-commutative 33.4%

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

      3. *-commutative 33.4%

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

      4. *-commutative 33.4%

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

   6. Applied egg-rr 33.4%

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

   7. Taylor expanded in i around 0 74.7%

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

      1. unpow2 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -135000000000 \lor \neg \left(n \leq 5.5 \cdot 10^{-10}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \end{array} \]

Alternative 6: 69.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{+153}:\\ \;\;\;\;n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot 0.16666666666666666 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.2e+153)
   (* n (+ (+ 100.0 (* 16.666666666666668 (* i i))) (* i 50.0)))
   (if (<= n 5.5e-10)
     (* 100.0 (/ (* n n) (+ n (* (* i n) (- (/ 0.5 n) 0.5)))))
     (*
      100.0
      (+
       n
       (* n (+ (* (* i i) 0.16666666666666666) (* i (- 0.5 (/ 0.5 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.2e+153) {
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	} else if (n <= 5.5e-10) {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	} else {
		tmp = 100.0 * (n + (n * (((i * i) * 0.16666666666666666) + (i * (0.5 - (0.5 / n))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.2d+153)) then
        tmp = n * ((100.0d0 + (16.666666666666668d0 * (i * i))) + (i * 50.0d0))
    else if (n <= 5.5d-10) then
        tmp = 100.0d0 * ((n * n) / (n + ((i * n) * ((0.5d0 / n) - 0.5d0))))
    else
        tmp = 100.0d0 * (n + (n * (((i * i) * 0.16666666666666666d0) + (i * (0.5d0 - (0.5d0 / n))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.2e+153) {
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	} else if (n <= 5.5e-10) {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	} else {
		tmp = 100.0 * (n + (n * (((i * i) * 0.16666666666666666) + (i * (0.5 - (0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.2e+153:
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0))
	elif n <= 5.5e-10:
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))))
	else:
		tmp = 100.0 * (n + (n * (((i * i) * 0.16666666666666666) + (i * (0.5 - (0.5 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.2e+153)
		tmp = Float64(n * Float64(Float64(100.0 + Float64(16.666666666666668 * Float64(i * i))) + Float64(i * 50.0)));
	elseif (n <= 5.5e-10)
		tmp = Float64(100.0 * Float64(Float64(n * n) / Float64(n + Float64(Float64(i * n) * Float64(Float64(0.5 / n) - 0.5)))));
	else
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(Float64(i * i) * 0.16666666666666666) + Float64(i * Float64(0.5 - Float64(0.5 / n)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -5.2e+153)
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	elseif (n <= 5.5e-10)
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	else
		tmp = 100.0 * (n + (n * (((i * i) * 0.16666666666666666) + (i * (0.5 - (0.5 / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.2e+153], N[(n * N[(N[(100.0 + N[(16.666666666666668 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e-10], N[(100.0 * N[(N[(n * n), $MachinePrecision] / N[(n + N[(N[(i * n), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(n * N[(N[(N[(i * i), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e153

   1. Initial program 16.2%

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

      1. associate-/r/ 16.9%

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

      2. associate-*r* 16.9%

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

      3. *-commutative 16.9%

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

      4. associate-*r/ 17.0%

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

      5. sub-neg 17.0%

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

      6. distribute-lft-in 17.0%

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

      7. fma-def 17.0%

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

      8. metadata-eval 17.0%

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

      9. metadata-eval 17.0%

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

   3. Simplified 17.0%

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

   4. Taylor expanded in i around 0 68.0%

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

      1. +-commutative 68.0%

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

      2. distribute-lft-out 68.0%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in n around inf 68.0%

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

      1. distribute-lft-in 68.0%

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

      2. associate-+r+ 68.0%

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

      3. associate-*r* 68.0%

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

      4. metadata-eval 68.0%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 68.0%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 68.0%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 68.0%

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

   9. Simplified 68.0%

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

   if -5.1999999999999998e153 < n < 5.4999999999999996e-10

   1. Initial program 30.3%

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

   2. Taylor expanded in i around 0 44.5%

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

      1. associate-*r* 44.8%

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

      2. *-commutative 44.8%

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

      3. associate-*r/ 44.8%

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

      4. metadata-eval 44.8%

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

   4. Simplified 44.8%

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

      1. flip-+ 42.3%

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

      2. *-commutative 42.3%

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

      3. *-commutative 42.3%

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

      4. *-commutative 42.3%

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in i around 0 73.6%

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

      1. unpow2 73.6%

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

   9. Simplified 73.6%

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

   if 5.4999999999999996e-10 < n

   1. Initial program 33.2%

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

   2. Taylor expanded in i around 0 72.8%

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

      1. distribute-lft-out 73.1%

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

      2. unpow2 73.1%

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

      3. associate--l+ 73.1%

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

      4. associate-*r/ 73.1%

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

      5. metadata-eval 73.1%

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

      6. unpow2 73.1%

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

      7. associate-*r/ 73.1%

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

      8. metadata-eval 73.1%

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

      9. associate-*r/ 73.1%

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

      10. metadata-eval 73.1%

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

   4. Simplified 73.1%

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

   5. Taylor expanded in n around inf 73.1%

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

      1. *-commutative 73.1%

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

      2. unpow2 73.1%

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

   7. Simplified 73.1%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{+153}:\\ \;\;\;\;n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot 0.16666666666666666 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 7: 69.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+153} \lor \neg \left(n \leq 5.5 \cdot 10^{-10}\right):\\ \;\;\;\;n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3.2e+153) (not (<= n 5.5e-10)))
   (* n (+ (+ 100.0 (* 16.666666666666668 (* i i))) (* i 50.0)))
   (* 100.0 (/ (* n n) (+ n (* (* i n) (- (/ 0.5 n) 0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3.2e+153) || !(n <= 5.5e-10)) {
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	} else {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-3.2d+153)) .or. (.not. (n <= 5.5d-10))) then
        tmp = n * ((100.0d0 + (16.666666666666668d0 * (i * i))) + (i * 50.0d0))
    else
        tmp = 100.0d0 * ((n * n) / (n + ((i * n) * ((0.5d0 / n) - 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -3.2e+153) || !(n <= 5.5e-10)) {
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	} else {
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -3.2e+153) or not (n <= 5.5e-10):
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0))
	else:
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3.2e+153) || !(n <= 5.5e-10))
		tmp = Float64(n * Float64(Float64(100.0 + Float64(16.666666666666668 * Float64(i * i))) + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / Float64(n + Float64(Float64(i * n) * Float64(Float64(0.5 / n) - 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -3.2e+153) || ~((n <= 5.5e-10)))
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	else
		tmp = 100.0 * ((n * n) / (n + ((i * n) * ((0.5 / n) - 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -3.2e+153], N[Not[LessEqual[n, 5.5e-10]], $MachinePrecision]], N[(n * N[(N[(100.0 + N[(16.666666666666668 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / N[(n + N[(N[(i * n), $MachinePrecision] * N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.2000000000000001e153 or 5.4999999999999996e-10 < n

   1. Initial program 27.6%

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

      1. associate-/r/ 28.0%

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

      2. associate-*r* 28.0%

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

      3. *-commutative 28.0%

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

      4. associate-*r/ 28.1%

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

      5. sub-neg 28.1%

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

      6. distribute-lft-in 28.1%

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

      7. fma-def 28.1%

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

      8. metadata-eval 28.1%

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

      9. metadata-eval 28.1%

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

   3. Simplified 28.1%

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

   4. Taylor expanded in i around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. distribute-lft-out 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in n around inf 71.4%

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

      1. distribute-lft-in 71.4%

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

      2. associate-+r+ 71.4%

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

      3. associate-*r* 71.4%

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

      4. metadata-eval 71.4%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 71.4%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 71.4%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 71.4%

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

   9. Simplified 71.4%

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

   if -3.2000000000000001e153 < n < 5.4999999999999996e-10

   1. Initial program 30.3%

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

   2. Taylor expanded in i around 0 44.5%

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

      1. associate-*r* 44.8%

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

      2. *-commutative 44.8%

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

      3. associate-*r/ 44.8%

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

      4. metadata-eval 44.8%

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

   4. Simplified 44.8%

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

      1. flip-+ 42.3%

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

      2. *-commutative 42.3%

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

      3. *-commutative 42.3%

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

      4. *-commutative 42.3%

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in i around 0 73.6%

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

      1. unpow2 73.6%

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

   9. Simplified 73.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+153} \lor \neg \left(n \leq 5.5 \cdot 10^{-10}\right):\\ \;\;\;\;n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n + \left(i \cdot n\right) \cdot \left(\frac{0.5}{n} - 0.5\right)}\\ \end{array} \]

Alternative 8: 63.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-130} \lor \neg \left(n \leq 2 \cdot 10^{-79}\right):\\ \;\;\;\;n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.65e-130) (not (<= n 2e-79)))
   (* n (+ (+ 100.0 (* 16.666666666666668 (* i i))) (* i 50.0)))
   (* 100.0 (/ 0.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.65e-130) || !(n <= 2e-79)) {
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.65d-130)) .or. (.not. (n <= 2d-79))) then
        tmp = n * ((100.0d0 + (16.666666666666668d0 * (i * i))) + (i * 50.0d0))
    else
        tmp = 100.0d0 * (0.0d0 / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.65e-130) || !(n <= 2e-79)) {
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.65e-130) or not (n <= 2e-79):
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0))
	else:
		tmp = 100.0 * (0.0 / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.65e-130) || !(n <= 2e-79))
		tmp = Float64(n * Float64(Float64(100.0 + Float64(16.666666666666668 * Float64(i * i))) + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2.65e-130) || ~((n <= 2e-79)))
		tmp = n * ((100.0 + (16.666666666666668 * (i * i))) + (i * 50.0));
	else
		tmp = 100.0 * (0.0 / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.65e-130], N[Not[LessEqual[n, 2e-79]], $MachinePrecision]], N[(n * N[(N[(100.0 + N[(16.666666666666668 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.6500000000000002e-130 or 2e-79 < n

   1. Initial program 25.4%

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

      1. associate-/r/ 25.7%

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

      2. associate-*r* 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-*r/ 25.8%

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

      5. sub-neg 25.8%

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

      6. distribute-lft-in 25.7%

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

      7. fma-def 25.8%

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

      8. metadata-eval 25.8%

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

      9. metadata-eval 25.8%

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

   3. Simplified 25.8%

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

   4. Taylor expanded in i around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. distribute-lft-out 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in n around inf 70.4%

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

      1. distribute-lft-in 70.4%

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

      2. associate-+r+ 70.4%

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

      3. associate-*r* 70.4%

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

      4. metadata-eval 70.4%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 70.4%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 70.4%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 70.4%

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

   9. Simplified 70.4%

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

   if -2.6500000000000002e-130 < n < 2e-79

   1. Initial program 39.0%

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

   2. Taylor expanded in i around 0 68.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-130} \lor \neg \left(n \leq 2 \cdot 10^{-79}\right):\\ \;\;\;\;n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 9: 53.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-15}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+72}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{+189}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1e-15)
   (* n 100.0)
   (if (<= n 3.9e+72)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 5.5e+189) (* 16.666666666666668 (* n (* i i))) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1e-15) {
		tmp = n * 100.0;
	} else if (n <= 3.9e+72) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.5e+189) {
		tmp = 16.666666666666668 * (n * (i * i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1d-15)) then
        tmp = n * 100.0d0
    else if (n <= 3.9d+72) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 5.5d+189) then
        tmp = 16.666666666666668d0 * (n * (i * i))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1e-15) {
		tmp = n * 100.0;
	} else if (n <= 3.9e+72) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.5e+189) {
		tmp = 16.666666666666668 * (n * (i * i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1e-15:
		tmp = n * 100.0
	elif n <= 3.9e+72:
		tmp = 100.0 * (i / (i / n))
	elif n <= 5.5e+189:
		tmp = 16.666666666666668 * (n * (i * i))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1e-15)
		tmp = Float64(n * 100.0);
	elseif (n <= 3.9e+72)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 5.5e+189)
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1e-15)
		tmp = n * 100.0;
	elseif (n <= 3.9e+72)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 5.5e+189)
		tmp = 16.666666666666668 * (n * (i * i));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1e-15], N[(n * 100.0), $MachinePrecision], If[LessEqual[n, 3.9e+72], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e+189], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.0000000000000001e-15 or 5.5e189 < n

   1. Initial program 19.9%

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

   2. Taylor expanded in i around 0 61.7%

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

      1. *-commutative 61.7%

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

   4. Simplified 61.7%

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

   if -1.0000000000000001e-15 < n < 3.89999999999999992e72

   1. Initial program 34.6%

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

   2. Taylor expanded in i around 0 60.5%

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

   if 3.89999999999999992e72 < n < 5.5e189

   1. Initial program 46.9%

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

      1. associate-/r/ 47.0%

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

      2. associate-*r* 47.0%

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

      3. *-commutative 47.0%

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

      4. associate-*r/ 47.0%

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

      5. sub-neg 47.0%

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

      6. distribute-lft-in 47.0%

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

      7. fma-def 47.0%

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

      8. metadata-eval 47.0%

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

      9. metadata-eval 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in i around 0 68.6%

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

      1. +-commutative 68.6%

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

      2. distribute-lft-out 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in n around inf 68.6%

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

      1. distribute-lft-in 68.6%

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

      2. associate-+r+ 68.6%

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

      3. associate-*r* 68.6%

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

      4. metadata-eval 68.6%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 68.6%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 68.6%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 68.6%

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

   9. Simplified 68.6%

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

   10. Taylor expanded in i around inf 49.0%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot {i}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 49.0%

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

   12. Simplified 49.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-15}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+72}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{+189}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 10: 53.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-15}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{+73}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+189}:\\ \;\;\;\;i \cdot \left(n \cdot \left(i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1e-15)
   (* n 100.0)
   (if (<= n 1.25e+73)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 2.5e+189) (* i (* n (* i 16.666666666666668))) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1e-15) {
		tmp = n * 100.0;
	} else if (n <= 1.25e+73) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.5e+189) {
		tmp = i * (n * (i * 16.666666666666668));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1d-15)) then
        tmp = n * 100.0d0
    else if (n <= 1.25d+73) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 2.5d+189) then
        tmp = i * (n * (i * 16.666666666666668d0))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1e-15) {
		tmp = n * 100.0;
	} else if (n <= 1.25e+73) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.5e+189) {
		tmp = i * (n * (i * 16.666666666666668));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1e-15:
		tmp = n * 100.0
	elif n <= 1.25e+73:
		tmp = 100.0 * (i / (i / n))
	elif n <= 2.5e+189:
		tmp = i * (n * (i * 16.666666666666668))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1e-15)
		tmp = Float64(n * 100.0);
	elseif (n <= 1.25e+73)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.5e+189)
		tmp = Float64(i * Float64(n * Float64(i * 16.666666666666668)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1e-15)
		tmp = n * 100.0;
	elseif (n <= 1.25e+73)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 2.5e+189)
		tmp = i * (n * (i * 16.666666666666668));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1e-15], N[(n * 100.0), $MachinePrecision], If[LessEqual[n, 1.25e+73], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e+189], N[(i * N[(n * N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.0000000000000001e-15 or 2.5000000000000002e189 < n

   1. Initial program 19.9%

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

   2. Taylor expanded in i around 0 61.7%

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

      1. *-commutative 61.7%

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

   4. Simplified 61.7%

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

   if -1.0000000000000001e-15 < n < 1.24999999999999994e73

   1. Initial program 34.6%

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

   2. Taylor expanded in i around 0 60.5%

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

   if 1.24999999999999994e73 < n < 2.5000000000000002e189

   1. Initial program 46.9%

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

      1. associate-/r/ 47.0%

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

      2. associate-*r* 47.0%

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

      3. *-commutative 47.0%

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

      4. associate-*r/ 47.0%

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

      5. sub-neg 47.0%

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

      6. distribute-lft-in 47.0%

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

      7. fma-def 47.0%

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

      8. metadata-eval 47.0%

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

      9. metadata-eval 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in i around 0 68.6%

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

      1. +-commutative 68.6%

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

      2. distribute-lft-out 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in n around inf 68.6%

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

      1. distribute-lft-in 68.6%

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

      2. associate-+r+ 68.6%

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

      3. associate-*r* 68.6%

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

      4. metadata-eval 68.6%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 68.6%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 68.6%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 68.6%

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

   9. Simplified 68.6%

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

   10. Taylor expanded in i around inf 49.0%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot {i}^{2}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 49.0%

          \[\leadsto \color{blue}{\left(n \cdot {i}^{2}\right) \cdot 16.666666666666668} \]

       2. associate-*l* 49.0%

          \[\leadsto \color{blue}{n \cdot \left({i}^{2} \cdot 16.666666666666668\right)} \]

       3. unpow2 49.0%

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

       4. associate-*l* 49.0%

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

   12. Simplified 49.0%

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

   13. Taylor expanded in n around 0 49.0%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot {i}^{2}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 49.0%

          \[\leadsto \color{blue}{\left(n \cdot {i}^{2}\right) \cdot 16.666666666666668} \]

       2. unpow2 49.0%

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

       3. associate-*r* 49.0%

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

       4. associate-*r* 49.0%

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

       5. *-commutative 49.0%

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

       6. associate-*l* 49.0%

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

   15. Simplified 49.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-15}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{+73}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+189}:\\ \;\;\;\;i \cdot \left(n \cdot \left(i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 11: 62.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{+15} \lor \neg \left(n \leq 2.6 \cdot 10^{-56}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.3e+15) (not (<= n 2.6e-56)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.3e+15) || !(n <= 2.6e-56)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-4.3d+15)) .or. (.not. (n <= 2.6d-56))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.3e+15) || !(n <= 2.6e-56)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.3e+15) or not (n <= 2.6e-56):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.3e+15) || !(n <= 2.6e-56))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -4.3e+15) || ~((n <= 2.6e-56)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.3e+15], N[Not[LessEqual[n, 2.6e-56]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.3e15 or 2.59999999999999997e-56 < n

   1. Initial program 27.5%

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

      1. associate-/r/ 28.0%

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

      2. associate-*r* 28.0%

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

      3. *-commutative 28.0%

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

      4. associate-*r/ 28.0%

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

      5. sub-neg 28.0%

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

      6. distribute-lft-in 28.0%

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

      7. fma-def 28.0%

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

      8. metadata-eval 28.0%

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

      9. metadata-eval 28.0%

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

   3. Simplified 28.0%

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

   4. Taylor expanded in i around 0 70.7%

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

      1. +-commutative 70.7%

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

      2. distribute-lft-out 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in n around inf 70.6%

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

      1. distribute-lft-in 70.6%

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

      2. associate-+r+ 70.7%

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

      3. associate-*r* 70.7%

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

      4. metadata-eval 70.7%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 70.7%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 70.7%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 70.7%

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

   9. Simplified 70.7%

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

   10. Taylor expanded in i around 0 65.4%

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

       1. *-commutative 65.4%

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

   12. Simplified 65.4%

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

   if -4.3e15 < n < 2.59999999999999997e-56

   1. Initial program 31.6%

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

   2. Taylor expanded in i around 0 63.9%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{+15} \lor \neg \left(n \leq 2.6 \cdot 10^{-56}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 12: 62.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-57}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.05e+15)
   (* 100.0 (* n (+ 1.0 (* i 0.5))))
   (if (<= n 9.5e-57) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.05e+15) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else if (n <= 9.5e-57) {
		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 <= (-1.05d+15)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * 0.5d0)))
    else if (n <= 9.5d-57) 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 <= -1.05e+15) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else if (n <= 9.5e-57) {
		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 <= -1.05e+15:
		tmp = 100.0 * (n * (1.0 + (i * 0.5)))
	elif n <= 9.5e-57:
		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 <= -1.05e+15)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * 0.5))));
	elseif (n <= 9.5e-57)
		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 <= -1.05e+15)
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	elseif (n <= 9.5e-57)
		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, -1.05e+15], N[(100.0 * N[(n * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e-57], 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 < -1.05e15

   1. Initial program 22.7%

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

   2. Taylor expanded in i around 0 66.0%

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

      1. associate-*r* 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*r/ 66.0%

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

      4. metadata-eval 66.0%

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

   4. Simplified 66.0%

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

   5. Taylor expanded in n around inf 66.0%

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

   if -1.05e15 < n < 9.5000000000000005e-57

   1. Initial program 31.6%

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

   2. Taylor expanded in i around 0 63.9%

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

   if 9.5000000000000005e-57 < n

   1. Initial program 31.7%

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

      1. associate-/r/ 32.0%

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

      2. associate-*r* 32.0%

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

      3. *-commutative 32.0%

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

      4. associate-*r/ 32.1%

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

      5. sub-neg 32.1%

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

      6. distribute-lft-in 32.1%

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

      7. fma-def 32.1%

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

      8. metadata-eval 32.1%

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

      9. metadata-eval 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in i around 0 72.8%

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

      1. +-commutative 72.8%

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

      2. distribute-lft-out 72.8%

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

   6. Simplified 72.8%

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

   7. Taylor expanded in n around inf 72.8%

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

      1. distribute-lft-in 72.8%

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

      2. associate-+r+ 72.8%

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

      3. associate-*r* 72.8%

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

      4. metadata-eval 72.8%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 72.8%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 72.8%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 72.8%

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

   9. Simplified 72.8%

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

   10. Taylor expanded in i around 0 65.0%

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

       1. *-commutative 65.0%

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

   12. Simplified 65.0%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-57}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13: 61.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-79}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.65e-130)
   (* 100.0 (* n (+ 1.0 (* i 0.5))))
   (if (<= n 2.05e-79) (* 100.0 (/ 0.0 (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.65e-130) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else if (n <= 2.05e-79) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.65d-130)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * 0.5d0)))
    else if (n <= 2.05d-79) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.65e-130) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else if (n <= 2.05e-79) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.65e-130:
		tmp = 100.0 * (n * (1.0 + (i * 0.5)))
	elif n <= 2.05e-79:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.65e-130)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * 0.5))));
	elseif (n <= 2.05e-79)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.65e-130)
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	elseif (n <= 2.05e-79)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.65e-130], N[(100.0 * N[(n * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.05e-79], N[(100.0 * N[(0.0 / 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 < -2.6500000000000002e-130

   1. Initial program 20.4%

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

   2. Taylor expanded in i around 0 67.0%

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

      1. associate-*r* 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-*r/ 67.0%

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

      4. metadata-eval 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in n around inf 67.0%

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

   if -2.6500000000000002e-130 < n < 2.04999999999999997e-79

   1. Initial program 39.0%

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

   2. Taylor expanded in i around 0 68.0%

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

   if 2.04999999999999997e-79 < n

   1. Initial program 31.1%

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

      1. associate-/r/ 31.4%

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

      2. associate-*r* 31.4%

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

      3. *-commutative 31.4%

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

      4. associate-*r/ 31.4%

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

      5. sub-neg 31.4%

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

      6. distribute-lft-in 31.5%

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

      7. fma-def 31.4%

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

      8. metadata-eval 31.4%

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

      9. metadata-eval 31.4%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in i around 0 72.3%

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

      1. +-commutative 72.3%

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

      2. distribute-lft-out 72.3%

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

   6. Simplified 72.3%

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

   7. Taylor expanded in n around inf 72.3%

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

      1. distribute-lft-in 72.3%

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

      2. associate-+r+ 72.3%

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

      3. associate-*r* 72.3%

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

      4. metadata-eval 72.3%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 72.3%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 72.3%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 72.3%

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

   9. Simplified 72.3%

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

   10. Taylor expanded in i around 0 64.6%

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

       1. *-commutative 64.6%

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

   12. Simplified 64.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-79}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 14: 57.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.8 \cdot 10^{+19}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.8e+19) (* n 100.0) (* 16.666666666666668 (* n (* i i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.8e+19) {
		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 <= 2.8d+19) 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 <= 2.8e+19) {
		tmp = n * 100.0;
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.8e+19:
		tmp = n * 100.0
	else:
		tmp = 16.666666666666668 * (n * (i * i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.8e+19)
		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 <= 2.8e+19)
		tmp = n * 100.0;
	else
		tmp = 16.666666666666668 * (n * (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.8e+19], 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 < 2.8e19

   1. Initial program 22.5%

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

   2. Taylor expanded in i around 0 63.1%

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

      1. *-commutative 63.1%

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

   4. Simplified 63.1%

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

   if 2.8e19 < i

   1. Initial program 46.8%

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

      1. associate-/r/ 47.4%

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

      2. associate-*r* 47.3%

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

      3. *-commutative 47.3%

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

      4. associate-*r/ 47.3%

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

      5. sub-neg 47.3%

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

      6. distribute-lft-in 47.3%

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

      7. fma-def 47.3%

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

      8. metadata-eval 47.3%

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

      9. metadata-eval 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in i around 0 38.6%

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

      1. +-commutative 38.6%

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

      2. distribute-lft-out 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in n around inf 39.4%

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

      1. distribute-lft-in 39.4%

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

      2. associate-+r+ 39.4%

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

      3. associate-*r* 39.4%

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

      4. metadata-eval 39.4%

         \[\leadsto n \cdot \left(\left(100 + \color{blue}{16.666666666666668} \cdot {i}^{2}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      5. unpow2 39.4%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)}\right) + 100 \cdot \left(0.5 \cdot i\right)\right) \]

      6. associate-*r* 39.4%

         \[\leadsto n \cdot \left(\left(100 + 16.666666666666668 \cdot \left(i \cdot i\right)\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right) \]

      7. metadata-eval 39.4%

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

   9. Simplified 39.4%

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

   10. Taylor expanded in i around inf 39.4%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot {i}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 39.4%

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

   12. Simplified 39.4%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.8 \cdot 10^{+19}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 15: 55.1% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5.1 \cdot 10^{+19}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 5.1e+19) (* n 100.0) (* (* i n) 50.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 5.1e+19) {
		tmp = n * 100.0;
	} else {
		tmp = (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 (i <= 5.1d+19) then
        tmp = n * 100.0d0
    else
        tmp = (i * n) * 50.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 5.1e+19) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 5.1e+19:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 5.1e+19)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 5.1e+19)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 5.1e+19], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.1e19

   1. Initial program 22.5%

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

   2. Taylor expanded in i around 0 63.1%

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

      1. *-commutative 63.1%

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

   4. Simplified 63.1%

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

   if 5.1e19 < i

   1. Initial program 46.8%

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

   2. Taylor expanded in i around 0 24.3%

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

      1. associate-*r* 24.4%

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

      2. *-commutative 24.4%

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

      3. associate-*r/ 24.4%

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

      4. metadata-eval 24.4%

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

   4. Simplified 24.4%

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

   5. Taylor expanded in n around inf 25.1%

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

   6. Taylor expanded in i around inf 25.1%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5.1 \cdot 10^{+19}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 16: 2.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

2. Taylor expanded in i around 0 53.7%

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

   1. associate-*r* 53.9%

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

   2. *-commutative 53.9%

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

   3. associate-*r/ 53.9%

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

   4. metadata-eval 53.9%

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

4. Simplified 53.9%

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

5. Taylor expanded in n around 0 2.7%

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

   1. *-commutative 2.7%

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

7. Simplified 2.7%

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

8. Final simplification 2.7%

   \[\leadsto i \cdot -50 \]

Alternative 17: 49.9% 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 29.1%

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

2. Taylor expanded in i around 0 47.5%

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

   1. *-commutative 47.5%

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

4. Simplified 47.5%

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

5. Final simplification 47.5%

   \[\leadsto n \cdot 100 \]

Developer target: 34.1% 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 2023171 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

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

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