Compound Interest

Percentage Accurate: 27.9% → 97.6%

Time: 19.7s

Alternatives: 16

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: 27.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := t_0 \cdot 100\\ \mathbf{if}\;t_0 \leq -200000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n \cdot 100}{i}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))) (t_1 (* t_0 100.0)))
   (if (<= t_0 -200000000.0)
     t_1
     (if (<= t_0 0.0)
       (* (expm1 (* n (log1p (/ i n)))) (/ (* n 100.0) i))
       (if (<= t_0 INFINITY)
         t_1
         (/ n (+ 0.01 (* 0.01 (* i (+ (/ 0.5 n) -0.5))))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = expm1((n * log1p((i / n)))) * ((n * 100.0) / i);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) * ((n * 100.0) / i);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	t_1 = t_0 * 100.0
	tmp = 0
	if t_0 <= -200000000.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = math.expm1((n * math.log1p((i / n)))) * ((n * 100.0) / i)
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	t_1 = Float64(t_0 * 100.0)
	tmp = 0.0
	if (t_0 <= -200000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(Float64(n * 100.0) / i));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(Float64(0.5 / n) + -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0), $MachinePrecision]}, If[LessEqual[t$95$0, -200000000.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(n / N[(0.01 + N[(0.01 * N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -2e8 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.9%

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

   if -2e8 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 30.2%

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

      1. associate-*r/ 30.2%

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

      2. sub-neg 30.2%

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

      3. distribute-lft-in 30.2%

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

      4. metadata-eval 30.2%

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

      5. metadata-eval 30.2%

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

      6. fma-udef 30.2%

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

      7. associate-/r/ 29.9%

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

      8. *-commutative 29.9%

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

      9. expm1-log1p-u 29.9%

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

      10. expm1-udef 26.5%

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

   3. Applied egg-rr 47.8%

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

      1. expm1-def 79.9%

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

      2. expm1-log1p 99.4%

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

      3. associate-*l/ 99.5%

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

   5. Simplified 99.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. associate-/r/ 1.9%

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

      8. *-commutative 1.9%

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

      9. clear-num 1.9%

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

      10. un-div-inv 1.9%

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

      11. fma-udef 1.9%

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

      12. metadata-eval 1.9%

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

      13. metadata-eval 1.9%

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

      14. distribute-lft-in 1.9%

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

      15. sub-neg 1.9%

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

      16. *-commutative 1.9%

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

   3. Applied egg-rr 1.9%

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

   4. Taylor expanded in i around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -200000000:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n \cdot 100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \end{array} \]

Alternative 2: 98.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(n / N[(0.01 + N[(0.01 * N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 99.9%

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

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

   1. Initial program 30.6%

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

      1. associate-/r/ 30.4%

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

      2. pow-to-exp 30.4%

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

      3. expm1-def 39.8%

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

      4. add-log-exp 30.4%

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

      5. pow-to-exp 30.4%

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

      6. log-pow 39.8%

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

      7. log1p-udef 99.3%

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

   3. Applied egg-rr 99.3%

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

   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. associate-/r/ 1.9%

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

      8. *-commutative 1.9%

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

      9. clear-num 1.9%

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

      10. un-div-inv 1.9%

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

      11. fma-udef 1.9%

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

      12. metadata-eval 1.9%

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

      13. metadata-eval 1.9%

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

      14. distribute-lft-in 1.9%

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

      15. sub-neg 1.9%

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

      16. *-commutative 1.9%

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

   3. Applied egg-rr 1.9%

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

   4. Taylor expanded in i around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.5%

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

Alternative 3: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -6 \cdot 10^{+52} \lor \neg \left(i \leq 14500000\right) \land i \leq 7.8 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -6e+52) (and (not (<= i 14500000.0)) (<= i 7.8e+262)))
   (* (expm1 i) (* 100.0 (/ n i)))
   (/ n (+ 0.01 (* 0.01 (* i (+ (/ 0.5 n) -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -6e+52) || (!(i <= 14500000.0) && (i <= 7.8e+262))) {
		tmp = expm1(i) * (100.0 * (n / i));
	} else {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -6e+52) || (!(i <= 14500000.0) && (i <= 7.8e+262))) {
		tmp = Math.expm1(i) * (100.0 * (n / i));
	} else {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -6e+52) or (not (i <= 14500000.0) and (i <= 7.8e+262)):
		tmp = math.expm1(i) * (100.0 * (n / i))
	else:
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -6e+52) || (!(i <= 14500000.0) && (i <= 7.8e+262)))
		tmp = Float64(expm1(i) * Float64(100.0 * Float64(n / i)));
	else
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(Float64(0.5 / n) + -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -6e+52], And[N[Not[LessEqual[i, 14500000.0]], $MachinePrecision], LessEqual[i, 7.8e+262]]], N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(0.01 * N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -6e52 or 1.45e7 < i < 7.79999999999999971e262

   1. Initial program 57.4%

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

   2. Taylor expanded in n around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. expm1-def 63.7%

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

   4. Simplified 63.7%

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

   5. Taylor expanded in n around 0 63.7%

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

      1. expm1-def 63.7%

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

      2. associate-*l/ 63.6%

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

      3. associate-*r* 63.6%

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

   7. Simplified 63.6%

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

   if -6e52 < i < 1.45e7 or 7.79999999999999971e262 < i

   1. Initial program 13.9%

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

      1. associate-*r/ 13.9%

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

      2. sub-neg 13.9%

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

      3. distribute-lft-in 13.8%

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

      4. metadata-eval 13.8%

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

      5. metadata-eval 13.8%

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

      6. fma-udef 13.9%

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

      7. associate-/r/ 14.5%

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

      8. *-commutative 14.5%

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

      9. clear-num 14.5%

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

      10. un-div-inv 14.5%

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

      11. fma-udef 14.5%

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

      12. metadata-eval 14.5%

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

      13. metadata-eval 14.5%

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

      14. distribute-lft-in 14.5%

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

      15. sub-neg 14.5%

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

      16. *-commutative 14.5%

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

   3. Applied egg-rr 65.8%

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

   4. Taylor expanded in i around 0 91.6%

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

      1. sub-neg 91.6%

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

      2. associate-*r/ 91.6%

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

      3. metadata-eval 91.6%

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

      4. metadata-eval 91.6%

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

   6. Simplified 91.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6 \cdot 10^{+52} \lor \neg \left(i \leq 14500000\right) \land i \leq 7.8 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \end{array} \]

Alternative 4: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-8} \lor \neg \left(n \leq 1.05 \cdot 10^{-7}\right):\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9.8e-8) (not (<= n 1.05e-7)))
   (* 100.0 (/ (* n (expm1 i)) i))
   (/ n (+ 0.01 (* 0.01 (* i (+ (/ 0.5 n) -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9.8e-8) || !(n <= 1.05e-7)) {
		tmp = 100.0 * ((n * expm1(i)) / i);
	} else {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9.8e-8) || !(n <= 1.05e-7)) {
		tmp = 100.0 * ((n * Math.expm1(i)) / i);
	} else {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9.8e-8) or not (n <= 1.05e-7):
		tmp = 100.0 * ((n * math.expm1(i)) / i)
	else:
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9.8e-8) || !(n <= 1.05e-7))
		tmp = Float64(100.0 * Float64(Float64(n * expm1(i)) / i));
	else
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(Float64(0.5 / n) + -0.5)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9.8e-8], N[Not[LessEqual[n, 1.05e-7]], $MachinePrecision]], N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(0.01 * N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.8000000000000004e-8 or 1.05e-7 < n

   1. Initial program 23.9%

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

   2. Taylor expanded in n around inf 37.7%

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

      1. *-commutative 37.7%

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

      2. expm1-def 90.2%

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

   4. Simplified 90.2%

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

   if -9.8000000000000004e-8 < n < 1.05e-7

   1. Initial program 41.1%

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

      1. associate-*r/ 41.1%

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

      2. sub-neg 41.1%

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

      3. distribute-lft-in 41.1%

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

      4. metadata-eval 41.1%

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

      5. metadata-eval 41.1%

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

      6. fma-udef 41.1%

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

      7. associate-/r/ 40.9%

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

      8. *-commutative 40.9%

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

      9. clear-num 40.9%

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

      10. un-div-inv 40.9%

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

      11. fma-udef 40.8%

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

      12. metadata-eval 40.8%

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

      13. metadata-eval 40.8%

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

      14. distribute-lft-in 40.9%

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

      15. sub-neg 40.9%

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

      16. *-commutative 40.9%

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

   3. Applied egg-rr 88.8%

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

   4. Taylor expanded in i around 0 85.9%

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

      1. sub-neg 85.9%

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

      2. associate-*r/ 85.9%

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

      3. metadata-eval 85.9%

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

      4. metadata-eval 85.9%

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

   6. Simplified 85.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-8} \lor \neg \left(n \leq 1.05 \cdot 10^{-7}\right):\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \end{array} \]

Alternative 5: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1e-7)
   (* 100.0 (/ (* n (expm1 i)) i))
   (if (<= n 1.15e-7)
     (/ n (+ 0.01 (* 0.01 (* i (+ (/ 0.5 n) -0.5)))))
     (/ n (* 0.01 (/ i (expm1 i)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1e-7) {
		tmp = 100.0 * ((n * expm1(i)) / i);
	} else if (n <= 1.15e-7) {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	} else {
		tmp = n / (0.01 * (i / expm1(i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1e-7) {
		tmp = 100.0 * ((n * Math.expm1(i)) / i);
	} else if (n <= 1.15e-7) {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	} else {
		tmp = n / (0.01 * (i / Math.expm1(i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1e-7:
		tmp = 100.0 * ((n * math.expm1(i)) / i)
	elif n <= 1.15e-7:
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))))
	else:
		tmp = n / (0.01 * (i / math.expm1(i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1e-7)
		tmp = Float64(100.0 * Float64(Float64(n * expm1(i)) / i));
	elseif (n <= 1.15e-7)
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(n / Float64(0.01 * Float64(i / expm1(i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1e-7], N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-7], N[(n / N[(0.01 + N[(0.01 * N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 * N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.9999999999999995e-8

   1. Initial program 24.6%

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

   2. Taylor expanded in n around inf 34.5%

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

      1. *-commutative 34.5%

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

      2. expm1-def 86.3%

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

   4. Simplified 86.3%

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

   if -9.9999999999999995e-8 < n < 1.14999999999999997e-7

   1. Initial program 41.1%

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

      1. associate-*r/ 41.1%

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

      2. sub-neg 41.1%

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

      3. distribute-lft-in 41.1%

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

      4. metadata-eval 41.1%

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

      5. metadata-eval 41.1%

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

      6. fma-udef 41.1%

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

      7. associate-/r/ 40.9%

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

      8. *-commutative 40.9%

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

      9. clear-num 40.9%

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

      10. un-div-inv 40.9%

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

      11. fma-udef 40.8%

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

      12. metadata-eval 40.8%

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

      13. metadata-eval 40.8%

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

      14. distribute-lft-in 40.9%

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

      15. sub-neg 40.9%

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

      16. *-commutative 40.9%

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

   3. Applied egg-rr 88.8%

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

   4. Taylor expanded in i around 0 85.9%

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

      1. sub-neg 85.9%

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

      2. associate-*r/ 85.9%

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

      3. metadata-eval 85.9%

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

      4. metadata-eval 85.9%

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

   6. Simplified 85.9%

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

   if 1.14999999999999997e-7 < n

   1. Initial program 23.3%

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

      1. associate-*r/ 23.3%

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

      2. sub-neg 23.3%

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

      3. distribute-lft-in 23.3%

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

      4. metadata-eval 23.3%

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

      5. metadata-eval 23.3%

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

      6. fma-udef 23.3%

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

      7. associate-/r/ 23.9%

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

      8. *-commutative 23.9%

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

      9. clear-num 23.9%

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

      10. un-div-inv 24.0%

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

      11. fma-udef 24.0%

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

      12. metadata-eval 24.0%

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

      13. metadata-eval 24.0%

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

      14. distribute-lft-in 24.0%

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

      15. sub-neg 24.0%

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

      16. *-commutative 24.0%

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

   3. Applied egg-rr 63.4%

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

   4. Taylor expanded in n around inf 40.7%

      \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \frac{i}{e^{i} - 1}}} \]
   5. Step-by-step derivation

      1. expm1-def 93.7%

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

   6. Simplified 93.7%

      \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 6: 72.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(\left(0.5 - \frac{0.5}{n}\right) + i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 1.15e-7)
   (/ n (+ 0.01 (* 0.01 (* i (+ (/ 0.5 n) -0.5)))))
   (*
    100.0
    (+ n (* n (* i (+ (- 0.5 (/ 0.5 n)) (* i 0.16666666666666666))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 1.15e-7) {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	} else {
		tmp = 100.0 * (n + (n * (i * ((0.5 - (0.5 / n)) + (i * 0.16666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1.15d-7) then
        tmp = n / (0.01d0 + (0.01d0 * (i * ((0.5d0 / n) + (-0.5d0)))))
    else
        tmp = 100.0d0 * (n + (n * (i * ((0.5d0 - (0.5d0 / n)) + (i * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 1.15e-7) {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	} else {
		tmp = 100.0 * (n + (n * (i * ((0.5 - (0.5 / n)) + (i * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 1.15e-7:
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))))
	else:
		tmp = 100.0 * (n + (n * (i * ((0.5 - (0.5 / n)) + (i * 0.16666666666666666)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 1.15e-7)
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(i * Float64(Float64(0.5 - Float64(0.5 / n)) + Float64(i * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 1.15e-7)
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	else
		tmp = 100.0 * (n + (n * (i * ((0.5 - (0.5 / n)) + (i * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 1.15e-7], N[(n / N[(0.01 + N[(0.01 * N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(n * N[(i * N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.14999999999999997e-7

   1. Initial program 34.3%

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

      1. associate-*r/ 34.3%

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

      2. sub-neg 34.3%

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

      3. distribute-lft-in 34.3%

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

      4. metadata-eval 34.3%

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

      5. metadata-eval 34.3%

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

      6. fma-udef 34.3%

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

      7. associate-/r/ 34.4%

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

      8. *-commutative 34.4%

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

      9. clear-num 34.5%

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

      10. un-div-inv 34.4%

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

      11. fma-udef 34.4%

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

      12. metadata-eval 34.4%

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

      13. metadata-eval 34.4%

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

      14. distribute-lft-in 34.4%

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

      15. sub-neg 34.4%

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

      16. *-commutative 34.4%

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

   3. Applied egg-rr 75.3%

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

   4. Taylor expanded in i around 0 75.7%

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

      1. sub-neg 75.7%

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

      2. associate-*r/ 75.7%

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

      3. metadata-eval 75.7%

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

      4. metadata-eval 75.7%

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

   6. Simplified 75.7%

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

   if 1.14999999999999997e-7 < n

   1. Initial program 23.3%

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

   2. Taylor expanded in i around 0 70.4%

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

      1. distribute-lft-out 70.7%

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

      2. unpow2 70.7%

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

      3. associate--l+ 70.7%

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

      4. associate-*r/ 70.7%

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

      5. metadata-eval 70.7%

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

      6. unpow2 70.7%

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

      7. associate-*r/ 70.7%

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

      8. metadata-eval 70.7%

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

      9. associate-*r/ 70.7%

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

      10. metadata-eval 70.7%

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

   4. Simplified 70.7%

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

   5. Taylor expanded in n around inf 70.7%

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

      1. unpow2 70.7%

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

      2. associate-*r* 70.7%

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

   7. Simplified 70.7%

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

      1. +-commutative 70.7%

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

      2. *-commutative 70.7%

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

      3. distribute-lft-out 70.7%

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

      4. *-commutative 70.7%

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

   9. Applied egg-rr 70.7%

      \[\leadsto 100 \cdot \left(n + n \cdot \color{blue}{\left(i \cdot \left(\left(0.5 - \frac{0.5}{n}\right) + i \cdot 0.16666666666666666\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(\left(0.5 - \frac{0.5}{n}\right) + i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]

Alternative 7: 64.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n + n \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-225}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (+ n (* n (* i (* i 0.16666666666666666)))))))
   (if (<= n -1.25e-81)
     t_0
     (if (<= n 3.8e-225)
       (* 100.0 (/ 0.0 (/ i n)))
       (if (<= n 5.8e-8) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * (i * (i * 0.16666666666666666))));
	double tmp;
	if (n <= -1.25e-81) {
		tmp = t_0;
	} else if (n <= 3.8e-225) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (n <= 5.8e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n + (n * (i * (i * 0.16666666666666666d0))))
    if (n <= (-1.25d-81)) then
        tmp = t_0
    else if (n <= 3.8d-225) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else if (n <= 5.8d-8) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * (i * (i * 0.16666666666666666))));
	double tmp;
	if (n <= -1.25e-81) {
		tmp = t_0;
	} else if (n <= 3.8e-225) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (n <= 5.8e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n + (n * (i * (i * 0.16666666666666666))))
	tmp = 0
	if n <= -1.25e-81:
		tmp = t_0
	elif n <= 3.8e-225:
		tmp = 100.0 * (0.0 / (i / n))
	elif n <= 5.8e-8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n + Float64(n * Float64(i * Float64(i * 0.16666666666666666)))))
	tmp = 0.0
	if (n <= -1.25e-81)
		tmp = t_0;
	elseif (n <= 3.8e-225)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	elseif (n <= 5.8e-8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n + (n * (i * (i * 0.16666666666666666))));
	tmp = 0.0;
	if (n <= -1.25e-81)
		tmp = t_0;
	elseif (n <= 3.8e-225)
		tmp = 100.0 * (0.0 / (i / n));
	elseif (n <= 5.8e-8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n + N[(n * N[(i * N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.25e-81], t$95$0, If[LessEqual[n, 3.8e-225], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.24999999999999995e-81 or 5.8000000000000003e-8 < n

   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 66.2%

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

      1. distribute-lft-out 66.5%

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

      2. unpow2 66.5%

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

      3. associate--l+ 66.5%

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

      4. associate-*r/ 66.5%

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

      5. metadata-eval 66.5%

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

      6. unpow2 66.5%

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

      7. associate-*r/ 66.5%

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

      8. metadata-eval 66.5%

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

      9. associate-*r/ 66.5%

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

      10. metadata-eval 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in n around inf 66.5%

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

      1. unpow2 66.5%

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

      2. associate-*r* 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in i around inf 66.0%

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

      1. unpow2 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*r* 66.0%

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

   10. Simplified 66.0%

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

   if -1.24999999999999995e-81 < n < 3.8000000000000003e-225

   1. Initial program 72.6%

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

   2. Taylor expanded in i around 0 76.6%

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

   if 3.8000000000000003e-225 < n < 5.8000000000000003e-8

   1. Initial program 12.0%

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

   2. Taylor expanded in i around 0 75.8%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-81}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-225}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]

Alternative 8: 62.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4.1 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 10^{-275}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -2.15e+79)
     t_0
     (if (<= n -4.1e-299)
       t_1
       (if (<= n 1e-275) (* (* i n) 50.0) (if (<= n 1.15e-7) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -2.15e+79) {
		tmp = t_0;
	} else if (n <= -4.1e-299) {
		tmp = t_1;
	} else if (n <= 1e-275) {
		tmp = (i * n) * 50.0;
	} else if (n <= 1.15e-7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    t_1 = 100.0d0 * (i / (i / n))
    if (n <= (-2.15d+79)) then
        tmp = t_0
    else if (n <= (-4.1d-299)) then
        tmp = t_1
    else if (n <= 1d-275) then
        tmp = (i * n) * 50.0d0
    else if (n <= 1.15d-7) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -2.15e+79) {
		tmp = t_0;
	} else if (n <= -4.1e-299) {
		tmp = t_1;
	} else if (n <= 1e-275) {
		tmp = (i * n) * 50.0;
	} else if (n <= 1.15e-7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -2.15e+79:
		tmp = t_0
	elif n <= -4.1e-299:
		tmp = t_1
	elif n <= 1e-275:
		tmp = (i * n) * 50.0
	elif n <= 1.15e-7:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -2.15e+79)
		tmp = t_0;
	elseif (n <= -4.1e-299)
		tmp = t_1;
	elseif (n <= 1e-275)
		tmp = Float64(Float64(i * n) * 50.0);
	elseif (n <= 1.15e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	t_1 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -2.15e+79)
		tmp = t_0;
	elseif (n <= -4.1e-299)
		tmp = t_1;
	elseif (n <= 1e-275)
		tmp = (i * n) * 50.0;
	elseif (n <= 1.15e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.15e+79], t$95$0, If[LessEqual[n, -4.1e-299], t$95$1, If[LessEqual[n, 1e-275], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision], If[LessEqual[n, 1.15e-7], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.1500000000000002e79 or 1.14999999999999997e-7 < n

   1. Initial program 20.3%

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

   2. Taylor expanded in n around inf 39.4%

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

      1. *-commutative 39.4%

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

      2. expm1-def 92.5%

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

   4. Simplified 92.5%

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

      1. associate-*l/ 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in i around 0 64.0%

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

      1. *-commutative 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.0%

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

      4. +-commutative 64.0%

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

      5. distribute-lft-in 64.0%

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

   9. Simplified 64.0%

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

   if -2.1500000000000002e79 < n < -4.1000000000000001e-299 or 9.99999999999999934e-276 < n < 1.14999999999999997e-7

   1. Initial program 39.7%

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

   2. Taylor expanded in i around 0 61.6%

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

   if -4.1000000000000001e-299 < n < 9.99999999999999934e-276

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. *-commutative 83.6%

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

      2. expm1-def 83.6%

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

   4. Simplified 83.6%

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

   5. Taylor expanded in i around 0 9.2%

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

   6. Taylor expanded in i around inf 84.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -4.1 \cdot 10^{-299}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 10^{-275}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 9: 62.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -1 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -1e-14)
     t_0
     (if (<= n 1e-307)
       t_1
       (if (<= n 1.4e-275)
         t_0
         (if (<= n 1.1e-7) t_1 (* n (+ 100.0 (* i 50.0)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1e-14) {
		tmp = t_0;
	} else if (n <= 1e-307) {
		tmp = t_1;
	} else if (n <= 1.4e-275) {
		tmp = t_0;
	} else if (n <= 1.1e-7) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 100.0d0 * ((i * n) / i)
    t_1 = 100.0d0 * (i / (i / n))
    if (n <= (-1d-14)) then
        tmp = t_0
    else if (n <= 1d-307) then
        tmp = t_1
    else if (n <= 1.4d-275) then
        tmp = t_0
    else if (n <= 1.1d-7) then
        tmp = t_1
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1e-14) {
		tmp = t_0;
	} else if (n <= 1e-307) {
		tmp = t_1;
	} else if (n <= 1.4e-275) {
		tmp = t_0;
	} else if (n <= 1.1e-7) {
		tmp = t_1;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((i * n) / i)
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -1e-14:
		tmp = t_0
	elif n <= 1e-307:
		tmp = t_1
	elif n <= 1.4e-275:
		tmp = t_0
	elif n <= 1.1e-7:
		tmp = t_1
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(i * n) / i))
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -1e-14)
		tmp = t_0;
	elseif (n <= 1e-307)
		tmp = t_1;
	elseif (n <= 1.4e-275)
		tmp = t_0;
	elseif (n <= 1.1e-7)
		tmp = t_1;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * ((i * n) / i);
	t_1 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (n <= -1e-14)
		tmp = t_0;
	elseif (n <= 1e-307)
		tmp = t_1;
	elseif (n <= 1.4e-275)
		tmp = t_0;
	elseif (n <= 1.1e-7)
		tmp = t_1;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1e-14], t$95$0, If[LessEqual[n, 1e-307], t$95$1, If[LessEqual[n, 1.4e-275], t$95$0, If[LessEqual[n, 1.1e-7], t$95$1, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.99999999999999999e-15 or 9.99999999999999909e-308 < n < 1.39999999999999997e-275

   1. Initial program 29.7%

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

   2. Taylor expanded in n around inf 37.4%

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

      1. *-commutative 37.4%

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

      2. expm1-def 86.4%

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

   4. Simplified 86.4%

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

   5. Taylor expanded in i around 0 62.3%

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

   if -9.99999999999999999e-15 < n < 9.99999999999999909e-308 or 1.39999999999999997e-275 < n < 1.1000000000000001e-7

   1. Initial program 38.2%

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

   2. Taylor expanded in i around 0 62.6%

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

   if 1.1000000000000001e-7 < n

   1. Initial program 23.3%

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

   2. Taylor expanded in n around inf 40.7%

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

      1. *-commutative 40.7%

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

      2. expm1-def 93.7%

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

   4. Simplified 93.7%

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

      1. associate-*l/ 93.5%

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in i around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r* 65.5%

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

      4. +-commutative 65.5%

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

      5. distribute-lft-in 65.5%

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

   9. Simplified 65.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-14}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 10^{-307}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-275}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 10: 63.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-81}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-227}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.25e-81)
   (* 100.0 (/ (* i n) i))
   (if (<= n 6e-227)
     (* 100.0 (/ 0.0 (/ i n)))
     (if (<= n 5.8e-8)
       (* 100.0 (/ i (/ i n)))
       (+ (* n 100.0) (* (* i n) 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.25e-81) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 6e-227) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (n <= 5.8e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.25d-81)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 6d-227) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else if (n <= 5.8d-8) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) + ((i * n) * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.25e-81) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 6e-227) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (n <= 5.8e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.25e-81:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 6e-227:
		tmp = 100.0 * (0.0 / (i / n))
	elif n <= 5.8e-8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) + ((i * n) * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.25e-81)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 6e-227)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	elseif (n <= 5.8e-8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(i * n) * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.25e-81)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 6e-227)
		tmp = 100.0 * (0.0 / (i / n));
	elseif (n <= 5.8e-8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) + ((i * n) * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.25e-81], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6e-227], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.24999999999999995e-81

   1. Initial program 21.7%

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

   2. Taylor expanded in n around inf 29.4%

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

      1. *-commutative 29.4%

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

      2. expm1-def 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in i around 0 60.7%

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

   if -1.24999999999999995e-81 < n < 5.9999999999999999e-227

   1. Initial program 72.6%

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

   2. Taylor expanded in i around 0 76.6%

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

   if 5.9999999999999999e-227 < n < 5.8000000000000003e-8

   1. Initial program 12.0%

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

   2. Taylor expanded in i around 0 75.8%

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

   if 5.8000000000000003e-8 < n

   1. Initial program 23.3%

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

   2. Taylor expanded in n around inf 40.7%

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

      1. *-commutative 40.7%

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

      2. expm1-def 93.7%

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

   4. Simplified 93.7%

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

   5. Taylor expanded in i around 0 65.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-81}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-227}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 11: 71.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 1.15e-7)
   (/ n (+ 0.01 (* 0.01 (* i (+ (/ 0.5 n) -0.5)))))
   (* 100.0 (+ n (* n (* i (* i 0.16666666666666666)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 1.15e-7) {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	} else {
		tmp = 100.0 * (n + (n * (i * (i * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1.15d-7) then
        tmp = n / (0.01d0 + (0.01d0 * (i * ((0.5d0 / n) + (-0.5d0)))))
    else
        tmp = 100.0d0 * (n + (n * (i * (i * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 1.15e-7) {
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	} else {
		tmp = 100.0 * (n + (n * (i * (i * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 1.15e-7:
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))))
	else:
		tmp = 100.0 * (n + (n * (i * (i * 0.16666666666666666))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 1.15e-7)
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(Float64(0.5 / n) + -0.5)))));
	else
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(i * Float64(i * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 1.15e-7)
		tmp = n / (0.01 + (0.01 * (i * ((0.5 / n) + -0.5))));
	else
		tmp = 100.0 * (n + (n * (i * (i * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 1.15e-7], N[(n / N[(0.01 + N[(0.01 * N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(n * N[(i * N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.14999999999999997e-7

   1. Initial program 34.3%

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

      1. associate-*r/ 34.3%

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

      2. sub-neg 34.3%

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

      3. distribute-lft-in 34.3%

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

      4. metadata-eval 34.3%

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

      5. metadata-eval 34.3%

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

      6. fma-udef 34.3%

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

      7. associate-/r/ 34.4%

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

      8. *-commutative 34.4%

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

      9. clear-num 34.5%

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

      10. un-div-inv 34.4%

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

      11. fma-udef 34.4%

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

      12. metadata-eval 34.4%

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

      13. metadata-eval 34.4%

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

      14. distribute-lft-in 34.4%

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

      15. sub-neg 34.4%

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

      16. *-commutative 34.4%

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

   3. Applied egg-rr 75.3%

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

   4. Taylor expanded in i around 0 75.7%

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

      1. sub-neg 75.7%

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

      2. associate-*r/ 75.7%

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

      3. metadata-eval 75.7%

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

      4. metadata-eval 75.7%

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

   6. Simplified 75.7%

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

   if 1.14999999999999997e-7 < n

   1. Initial program 23.3%

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

   2. Taylor expanded in i around 0 70.4%

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

      1. distribute-lft-out 70.7%

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

      2. unpow2 70.7%

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

      3. associate--l+ 70.7%

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

      4. associate-*r/ 70.7%

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

      5. metadata-eval 70.7%

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

      6. unpow2 70.7%

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

      7. associate-*r/ 70.7%

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

      8. metadata-eval 70.7%

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

      9. associate-*r/ 70.7%

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

      10. metadata-eval 70.7%

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

   4. Simplified 70.7%

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

   5. Taylor expanded in n around inf 70.7%

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

      1. unpow2 70.7%

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

      2. associate-*r* 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in i around inf 70.0%

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

      1. unpow2 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-*r* 70.0%

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

   10. Simplified 70.0%

       \[\leadsto 100 \cdot \left(n + n \cdot \color{blue}{\left(i \cdot \left(i \cdot 0.16666666666666666\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]

Alternative 12: 63.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-81}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-230}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.25e-81)
   (* 100.0 (/ (* i n) i))
   (if (<= n 3.5e-230)
     (* 100.0 (/ 0.0 (/ i n)))
     (if (<= n 1.15e-7) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.25e-81) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 3.5e-230) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (n <= 1.15e-7) {
		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 <= (-1.25d-81)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 3.5d-230) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else if (n <= 1.15d-7) 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 <= -1.25e-81) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 3.5e-230) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (n <= 1.15e-7) {
		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 <= -1.25e-81:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 3.5e-230:
		tmp = 100.0 * (0.0 / (i / n))
	elif n <= 1.15e-7:
		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 <= -1.25e-81)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 3.5e-230)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	elseif (n <= 1.15e-7)
		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 <= -1.25e-81)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 3.5e-230)
		tmp = 100.0 * (0.0 / (i / n));
	elseif (n <= 1.15e-7)
		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, -1.25e-81], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-230], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-7], 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 4 regimes

2. if n < -1.24999999999999995e-81

   1. Initial program 21.7%

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

   2. Taylor expanded in n around inf 29.4%

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

      1. *-commutative 29.4%

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

      2. expm1-def 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in i around 0 60.7%

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

   if -1.24999999999999995e-81 < n < 3.49999999999999988e-230

   1. Initial program 72.6%

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

   2. Taylor expanded in i around 0 76.6%

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

   if 3.49999999999999988e-230 < n < 1.14999999999999997e-7

   1. Initial program 12.0%

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

   2. Taylor expanded in i around 0 75.8%

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

   if 1.14999999999999997e-7 < n

   1. Initial program 23.3%

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

   2. Taylor expanded in n around inf 40.7%

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

      1. *-commutative 40.7%

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

      2. expm1-def 93.7%

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

   4. Simplified 93.7%

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

      1. associate-*l/ 93.5%

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in i around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r* 65.5%

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

      4. +-commutative 65.5%

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

      5. distribute-lft-in 65.5%

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

   9. Simplified 65.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-81}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-230}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13: 56.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+55} \lor \neg \left(i \leq 1.3 \cdot 10^{-62}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2e+55) (not (<= i 1.3e-62)))
   (* 100.0 (/ i (/ i n)))
   (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2e+55) || !(i <= 1.3e-62)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-2d+55)) .or. (.not. (i <= 1.3d-62))) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2e+55) || !(i <= 1.3e-62)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2e+55) or not (i <= 1.3e-62):
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2e+55) || !(i <= 1.3e-62))
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -2e+55) || ~((i <= 1.3e-62)))
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2e+55], N[Not[LessEqual[i, 1.3e-62]], $MachinePrecision]], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2.00000000000000002e55 or 1.3e-62 < i

   1. Initial program 52.7%

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

   2. Taylor expanded in i around 0 29.1%

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

   if -2.00000000000000002e55 < i < 1.3e-62

   1. Initial program 13.6%

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

   2. Taylor expanded in i around 0 80.8%

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

      1. *-commutative 80.8%

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

   4. Simplified 80.8%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+55} \lor \neg \left(i \leq 1.3 \cdot 10^{-62}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 14: 55.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 350000000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 350000000000.0) (* n 100.0) (* (* i n) 50.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 350000000000.0) {
		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 <= 350000000000.0d0) 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 <= 350000000000.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 350000000000.0:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 350000000000.0)
		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 <= 350000000000.0)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 350000000000.0], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 3.5e11

   1. Initial program 26.3%

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

   2. Taylor expanded in i around 0 60.5%

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

      1. *-commutative 60.5%

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

   4. Simplified 60.5%

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

   if 3.5e11 < i

   1. Initial program 48.8%

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

   2. Taylor expanded in n around inf 48.8%

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

      1. *-commutative 48.8%

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

      2. expm1-def 48.8%

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

   4. Simplified 48.8%

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

   5. Taylor expanded in i around 0 20.5%

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

   6. Taylor expanded in i around inf 20.5%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 350000000000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 15: 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 30.9%

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

2. Taylor expanded in i around 0 51.9%

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

   1. associate-*r* 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*r/ 52.1%

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

   4. metadata-eval 52.1%

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

4. Simplified 52.1%

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

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 16: 49.7% 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 30.9%

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

2. Taylor expanded in i around 0 49.1%

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

   1. *-commutative 49.1%

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

4. Simplified 49.1%

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

5. Final simplification 49.1%

   \[\leadsto n \cdot 100 \]

Developer target: 34.0% 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 2023174 
(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))))