Compound Interest

Percentage Accurate: 28.8% → 98.5%

Time: 29.3s

Alternatives: 10

Speedup: 8.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ t_2 := n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (* n (/ (+ (* t_0 100.0) -100.0) i))))
   (if (<= t_1 -5e-43)
     t_2
     (if (<= t_1 0.0)
       (* (/ (expm1 (* n (log1p (/ i n)))) i) (* n 100.0))
       (if (<= t_1 INFINITY) t_2 (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = n * (((t_0 * 100.0) + -100.0) / i);
	double tmp;
	if (t_1 <= -5e-43) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = n * (((t_0 * 100.0) + -100.0) / i);
	double tmp;
	if (t_1 <= -5e-43) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	t_2 = n * (((t_0 * 100.0) + -100.0) / i)
	tmp = 0
	if t_1 <= -5e-43:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) / i) * (n * 100.0)
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i))
	tmp = 0.0
	if (t_1 <= -5e-43)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i) * Float64(n * 100.0));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-43], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $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)) < -5.00000000000000019e-43 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.2%

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

      1. associate-/r/ 99.2%

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

      2. associate-*r* 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*r/ 99.3%

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

      5. sub-neg 99.3%

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

      6. distribute-lft-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. fma-def 99.3%

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

      11. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. fma-udef 99.5%

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

      2. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   if -5.00000000000000019e-43 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 27.8%

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

      1. *-commutative 27.8%

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

      2. associate-/r/ 27.3%

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

      3. associate-*l* 27.3%

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

      4. sub-neg 27.3%

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

      5. metadata-eval 27.3%

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

   3. Simplified 27.3%

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

      1. *-un-lft-identity 27.3%

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

      2. metadata-eval 27.3%

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

      3. sub-neg 27.3%

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

      4. add-exp-log 27.3%

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

      5. expm1-def 27.3%

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

      6. log-pow 37.4%

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

      7. log1p-udef 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. *-lft-identity 98.2%

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

   8. Simplified 98.2%

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

   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/ 1.9%

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

      2. sub-neg 1.9%

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

      3. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in n around inf 1.9%

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

      1. associate-/l* 1.9%

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

      2. expm1-def 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in i around 0 100.0%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \lor \neg \left(n \leq 1.3 \cdot 10^{+15}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.8) (not (<= n 1.3e+15)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/ (* n 100.0) (+ 1.0 (* i (+ -0.5 (/ 0.5 n)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.8) || !(n <= 1.3e+15)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * (-0.5 + (0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.8) || !(n <= 1.3e+15)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * (-0.5 + (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.8) or not (n <= 1.3e+15):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = (n * 100.0) / (1.0 + (i * (-0.5 + (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.8) || !(n <= 1.3e+15))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(-0.5 + Float64(0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.8], N[Not[LessEqual[n, 1.3e+15]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.7999999999999998 or 1.3e15 < n

   1. Initial program 32.5%

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

      1. associate-/r/ 33.0%

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

      2. sub-neg 33.0%

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

      3. metadata-eval 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in n around inf 49.3%

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

      1. associate-/l* 49.3%

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

      2. expm1-def 91.9%

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

   7. Simplified 91.9%

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

   if -2.7999999999999998 < n < 1.3e15

   1. Initial program 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-/r/ 31.0%

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

      3. sub-neg 31.0%

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

      4. metadata-eval 31.0%

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

      5. associate-*r* 31.0%

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

      6. *-commutative 31.0%

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

      7. clear-num 31.0%

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

      8. un-div-inv 31.0%

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

      9. metadata-eval 31.0%

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

      10. sub-neg 31.0%

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

      11. add-exp-log 31.0%

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

      12. expm1-def 31.0%

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

      13. log-pow 47.4%

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

      14. log1p-udef 88.0%

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in i around 0 84.7%

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

      1. sub-neg 84.7%

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

      2. associate-*r/ 84.7%

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

      3. metadata-eval 84.7%

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

      4. metadata-eval 84.7%

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

   7. Simplified 84.7%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \lor \neg \left(n \leq 1.3 \cdot 10^{+15}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.05 \cdot 10^{+210}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -2.8 \cdot 10^{-196}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.05e+210)
   (/ (* 100.0 (* i n)) i)
   (if (<= n -2.8e-196)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 1.95e-159) 0.0 (+ (* n 100.0) (* (* i n) 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.05e+210) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -2.8e-196) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.95e-159) {
		tmp = 0.0;
	} 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 <= (-2.05d+210)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= (-2.8d-196)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1.95d-159) then
        tmp = 0.0d0
    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 <= -2.05e+210) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -2.8e-196) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.95e-159) {
		tmp = 0.0;
	} else {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.05e+210:
		tmp = (100.0 * (i * n)) / i
	elif n <= -2.8e-196:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1.95e-159:
		tmp = 0.0
	else:
		tmp = (n * 100.0) + ((i * n) * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.05e+210)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= -2.8e-196)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1.95e-159)
		tmp = 0.0;
	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 <= -2.05e+210)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= -2.8e-196)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1.95e-159)
		tmp = 0.0;
	else
		tmp = (n * 100.0) + ((i * n) * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.05e+210], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -2.8e-196], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e-159], 0.0, 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 < -2.05e210

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 56.7%

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

      1. associate-*r/ 56.6%

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

      2. expm1-def 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in i around 0 76.7%

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

      1. *-commutative 76.7%

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

   8. Simplified 76.7%

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

   if -2.05e210 < n < -2.7999999999999998e-196

   1. Initial program 37.4%

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

      1. associate-/r/ 37.5%

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

      2. sub-neg 37.5%

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

      3. metadata-eval 37.5%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in n around inf 36.7%

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

      1. associate-/l* 36.7%

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

      2. expm1-def 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in i around 0 55.5%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 55.5%

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

   10. Simplified 55.5%

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

   if -2.7999999999999998e-196 < n < 1.94999999999999988e-159

   1. Initial program 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-/r/ 54.6%

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

      3. associate-*l* 54.6%

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

      4. sub-neg 54.6%

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

      5. metadata-eval 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in i around 0 87.8%

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

   6. Taylor expanded in i around 0 87.8%

      \[\leadsto \color{blue}{0} \]

   if 1.94999999999999988e-159 < n

   1. Initial program 20.8%

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

      1. associate-/r/ 21.2%

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

      2. sub-neg 21.2%

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

      3. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in n around inf 32.7%

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

      1. associate-/l* 32.7%

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

      2. expm1-def 88.7%

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

   7. Simplified 88.7%

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

   8. Taylor expanded in i around 0 76.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.05 \cdot 10^{+210}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -2.8 \cdot 10^{-196}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+211}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.96:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+211)
   (/ (* 100.0 (* i n)) i)
   (if (<= n 1.96)
     (/ (* n 100.0) (+ 1.0 (* i (+ -0.5 (/ 0.5 n)))))
     (+ (* n 100.0) (* (* i n) 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+211) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 1.96) {
		tmp = (n * 100.0) / (1.0 + (i * (-0.5 + (0.5 / 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 <= (-2.8d+211)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= 1.96d0) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((-0.5d0) + (0.5d0 / 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 <= -2.8e+211) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 1.96) {
		tmp = (n * 100.0) / (1.0 + (i * (-0.5 + (0.5 / n))));
	} else {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.8e+211:
		tmp = (100.0 * (i * n)) / i
	elif n <= 1.96:
		tmp = (n * 100.0) / (1.0 + (i * (-0.5 + (0.5 / n))))
	else:
		tmp = (n * 100.0) + ((i * n) * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+211)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= 1.96)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(-0.5 + Float64(0.5 / 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 <= -2.8e+211)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= 1.96)
		tmp = (n * 100.0) / (1.0 + (i * (-0.5 + (0.5 / n))));
	else
		tmp = (n * 100.0) + ((i * n) * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+211], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.96], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.8e211

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 56.7%

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

      1. associate-*r/ 56.6%

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

      2. expm1-def 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in i around 0 76.7%

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

      1. *-commutative 76.7%

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

   8. Simplified 76.7%

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

   if -2.8e211 < n < 1.96

   1. Initial program 37.2%

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

      1. *-commutative 37.2%

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

      2. associate-/r/ 36.9%

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

      3. sub-neg 36.9%

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

      4. metadata-eval 36.9%

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

      5. associate-*r* 36.9%

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

      6. *-commutative 36.9%

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

      7. clear-num 36.9%

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

      8. un-div-inv 36.9%

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

      9. metadata-eval 36.9%

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

      10. sub-neg 36.9%

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

      11. add-exp-log 36.9%

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

      12. expm1-def 36.9%

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

      13. log-pow 44.5%

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

      14. log1p-udef 84.6%

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

   4. Applied egg-rr 84.6%

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

   5. Taylor expanded in i around 0 71.7%

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

      1. sub-neg 71.7%

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

      2. associate-*r/ 71.7%

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

      3. metadata-eval 71.7%

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

      4. metadata-eval 71.7%

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

   7. Simplified 71.7%

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

   if 1.96 < n

   1. Initial program 24.8%

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

      1. associate-/r/ 25.5%

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

      2. sub-neg 25.5%

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

      3. metadata-eval 25.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in n around inf 41.7%

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

      1. associate-/l* 41.7%

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

      2. expm1-def 93.7%

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

   7. Simplified 93.7%

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

   8. Taylor expanded in i around 0 77.2%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+211}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.96:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -5.8 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2.45 \cdot 10^{-185}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -5.8e+102)
     t_0
     (if (<= n -2.45e-185)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 4.2e-159) 0.0 t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -5.8e+102) {
		tmp = t_0;
	} else if (n <= -2.45e-185) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 4.2e-159) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-5.8d+102)) then
        tmp = t_0
    else if (n <= (-2.45d-185)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 4.2d-159) then
        tmp = 0.0d0
    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 tmp;
	if (n <= -5.8e+102) {
		tmp = t_0;
	} else if (n <= -2.45e-185) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 4.2e-159) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -5.8e+102:
		tmp = t_0
	elif n <= -2.45e-185:
		tmp = 100.0 * (i / (i / n))
	elif n <= 4.2e-159:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -5.8e+102)
		tmp = t_0;
	elseif (n <= -2.45e-185)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 4.2e-159)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -5.8e+102)
		tmp = t_0;
	elseif (n <= -2.45e-185)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 4.2e-159)
		tmp = 0.0;
	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]}, If[LessEqual[n, -5.8e+102], t$95$0, If[LessEqual[n, -2.45e-185], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.2e-159], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.8000000000000005e102 or 4.1999999999999998e-159 < n

   1. Initial program 26.5%

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

      1. associate-/r/ 27.0%

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

      2. associate-*r* 27.0%

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

      3. *-commutative 27.0%

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

      4. associate-*r/ 27.0%

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

      5. sub-neg 27.0%

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

      6. distribute-lft-in 27.0%

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

      7. metadata-eval 27.0%

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

      8. metadata-eval 27.0%

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

      9. metadata-eval 27.0%

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

      10. fma-def 27.0%

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

      11. metadata-eval 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in n around inf 41.7%

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

      1. *-commutative 41.7%

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

      2. fma-neg 41.7%

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

      3. metadata-eval 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in i around 0 67.5%

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

      1. *-commutative 67.5%

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

   10. Simplified 67.5%

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

   if -5.8000000000000005e102 < n < -2.4500000000000001e-185

   1. Initial program 30.7%

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

   3. Taylor expanded in i around 0 10.6%

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

      1. +-commutative 10.6%

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

   5. Simplified 10.6%

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

   6. Taylor expanded in i around 0 61.4%

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

   if -2.4500000000000001e-185 < n < 4.1999999999999998e-159

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/r/ 54.5%

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

      3. associate-*l* 54.5%

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

      4. sub-neg 54.5%

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

      5. metadata-eval 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in i around 0 86.0%

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

   6. Taylor expanded in i around 0 86.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{+102}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -2.45 \cdot 10^{-185}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+102}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -2.45 \cdot 10^{-185}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e+102)
   (* (* i n) (/ 100.0 i))
   (if (<= n -2.45e-185)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 3e-159) 0.0 (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e+102) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= -2.45e-185) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3e-159) {
		tmp = 0.0;
	} 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 <= (-5d+102)) then
        tmp = (i * n) * (100.0d0 / i)
    else if (n <= (-2.45d-185)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 3d-159) then
        tmp = 0.0d0
    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 <= -5e+102) {
		tmp = (i * n) * (100.0 / i);
	} else if (n <= -2.45e-185) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3e-159) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5e+102:
		tmp = (i * n) * (100.0 / i)
	elif n <= -2.45e-185:
		tmp = 100.0 * (i / (i / n))
	elif n <= 3e-159:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e+102)
		tmp = Float64(Float64(i * n) * Float64(100.0 / i));
	elseif (n <= -2.45e-185)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 3e-159)
		tmp = 0.0;
	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 <= -5e+102)
		tmp = (i * n) * (100.0 / i);
	elseif (n <= -2.45e-185)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 3e-159)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e+102], N[(N[(i * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.45e-185], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3e-159], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5e102

   1. Initial program 38.6%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. associate-*r/ 60.4%

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

      2. expm1-def 89.8%

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

   5. Simplified 89.8%

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

   6. Taylor expanded in i around 0 50.1%

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

      1. *-commutative 50.1%

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

   8. Simplified 50.1%

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

      1. associate-/l* 50.1%

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

      2. associate-/r/ 50.0%

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

   10. Applied egg-rr 50.0%

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

   if -5e102 < n < -2.4500000000000001e-185

   1. Initial program 30.7%

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

   3. Taylor expanded in i around 0 10.6%

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

      1. +-commutative 10.6%

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

   5. Simplified 10.6%

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

   6. Taylor expanded in i around 0 61.4%

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

   if -2.4500000000000001e-185 < n < 3.00000000000000009e-159

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/r/ 54.5%

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

      3. associate-*l* 54.5%

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

      4. sub-neg 54.5%

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

      5. metadata-eval 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in i around 0 86.0%

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

   6. Taylor expanded in i around 0 86.0%

      \[\leadsto \color{blue}{0} \]

   if 3.00000000000000009e-159 < n

   1. Initial program 20.8%

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

      1. associate-/r/ 21.2%

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

      2. associate-*r* 21.2%

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

      3. *-commutative 21.2%

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

      4. associate-*r/ 21.2%

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

      5. sub-neg 21.2%

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

      6. distribute-lft-in 21.3%

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

      7. metadata-eval 21.3%

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

      8. metadata-eval 21.3%

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

      9. metadata-eval 21.3%

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

      10. fma-def 21.2%

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

      11. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in n around inf 32.7%

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

      1. *-commutative 32.7%

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

      2. fma-neg 32.7%

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

      3. metadata-eval 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in i around 0 76.2%

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

      1. *-commutative 76.2%

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

   10. Simplified 76.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+102}:\\ \;\;\;\;\left(i \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq -2.45 \cdot 10^{-185}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+102}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{-184}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{-160}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e+102)
   (/ (* 100.0 (* i n)) i)
   (if (<= n -3.7e-184)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 5.1e-160) 0.0 (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e+102) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -3.7e-184) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.1e-160) {
		tmp = 0.0;
	} 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 <= (-5d+102)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= (-3.7d-184)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 5.1d-160) then
        tmp = 0.0d0
    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 <= -5e+102) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -3.7e-184) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.1e-160) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5e+102:
		tmp = (100.0 * (i * n)) / i
	elif n <= -3.7e-184:
		tmp = 100.0 * (i / (i / n))
	elif n <= 5.1e-160:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e+102)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= -3.7e-184)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 5.1e-160)
		tmp = 0.0;
	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 <= -5e+102)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= -3.7e-184)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 5.1e-160)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e+102], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -3.7e-184], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.1e-160], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5e102

   1. Initial program 38.6%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. associate-*r/ 60.4%

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

      2. expm1-def 89.8%

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

   5. Simplified 89.8%

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

   6. Taylor expanded in i around 0 50.1%

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

      1. *-commutative 50.1%

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

   8. Simplified 50.1%

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

   if -5e102 < n < -3.6999999999999999e-184

   1. Initial program 30.7%

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

   3. Taylor expanded in i around 0 10.6%

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

      1. +-commutative 10.6%

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

   5. Simplified 10.6%

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

   6. Taylor expanded in i around 0 61.4%

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

   if -3.6999999999999999e-184 < n < 5.1e-160

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/r/ 54.5%

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

      3. associate-*l* 54.5%

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

      4. sub-neg 54.5%

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

      5. metadata-eval 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in i around 0 86.0%

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

   6. Taylor expanded in i around 0 86.0%

      \[\leadsto \color{blue}{0} \]

   if 5.1e-160 < n

   1. Initial program 20.8%

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

      1. associate-/r/ 21.2%

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

      2. associate-*r* 21.2%

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

      3. *-commutative 21.2%

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

      4. associate-*r/ 21.2%

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

      5. sub-neg 21.2%

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

      6. distribute-lft-in 21.3%

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

      7. metadata-eval 21.3%

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

      8. metadata-eval 21.3%

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

      9. metadata-eval 21.3%

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

      10. fma-def 21.2%

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

      11. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in n around inf 32.7%

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

      1. *-commutative 32.7%

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

      2. fma-neg 32.7%

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

      3. metadata-eval 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in i around 0 76.2%

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

      1. *-commutative 76.2%

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

   10. Simplified 76.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+102}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{-184}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{-160}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{+211}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-196}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3e+211)
   (/ (* 100.0 (* i n)) i)
   (if (<= n -1.5e-196)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 7.5e-158) 0.0 (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3e+211) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -1.5e-196) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7.5e-158) {
		tmp = 0.0;
	} 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 <= (-3d+211)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= (-1.5d-196)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 7.5d-158) then
        tmp = 0.0d0
    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 <= -3e+211) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -1.5e-196) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7.5e-158) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3e+211:
		tmp = (100.0 * (i * n)) / i
	elif n <= -1.5e-196:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 7.5e-158:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3e+211)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= -1.5e-196)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 7.5e-158)
		tmp = 0.0;
	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 <= -3e+211)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= -1.5e-196)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 7.5e-158)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3e+211], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -1.5e-196], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.5e-158], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3e211

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 56.7%

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

      1. associate-*r/ 56.6%

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

      2. expm1-def 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in i around 0 76.7%

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

      1. *-commutative 76.7%

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

   8. Simplified 76.7%

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

   if -3e211 < n < -1.5e-196

   1. Initial program 37.4%

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

      1. associate-/r/ 37.5%

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

      2. sub-neg 37.5%

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

      3. metadata-eval 37.5%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in n around inf 36.7%

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

      1. associate-/l* 36.7%

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

      2. expm1-def 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in i around 0 55.5%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \]
   9. Step-by-step derivation

      1. *-commutative 55.5%

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

   10. Simplified 55.5%

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

   if -1.5e-196 < n < 7.5e-158

   1. Initial program 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-/r/ 54.6%

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

      3. associate-*l* 54.6%

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

      4. sub-neg 54.6%

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

      5. metadata-eval 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in i around 0 87.8%

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

   6. Taylor expanded in i around 0 87.8%

      \[\leadsto \color{blue}{0} \]

   if 7.5e-158 < n

   1. Initial program 20.8%

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

      1. associate-/r/ 21.2%

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

      2. associate-*r* 21.2%

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

      3. *-commutative 21.2%

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

      4. associate-*r/ 21.2%

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

      5. sub-neg 21.2%

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

      6. distribute-lft-in 21.3%

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

      7. metadata-eval 21.3%

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

      8. metadata-eval 21.3%

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

      9. metadata-eval 21.3%

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

      10. fma-def 21.2%

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

      11. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in n around inf 32.7%

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

      1. *-commutative 32.7%

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

      2. fma-neg 32.7%

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

      3. metadata-eval 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in i around 0 76.2%

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

      1. *-commutative 76.2%

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

   10. Simplified 76.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{+211}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-196}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-183} \lor \neg \left(n \leq 5.9 \cdot 10^{-158}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.6e-183) (not (<= n 5.9e-158))) (* n 100.0) 0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.6e-183) || !(n <= 5.9e-158)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.6d-183)) .or. (.not. (n <= 5.9d-158))) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.6e-183) || !(n <= 5.9e-158)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.6e-183) or not (n <= 5.9e-158):
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.6e-183) || !(n <= 5.9e-158))
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2.6e-183) || ~((n <= 5.9e-158)))
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.6e-183], N[Not[LessEqual[n, 5.9e-158]], $MachinePrecision]], N[(n * 100.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.5999999999999999e-183 or 5.9000000000000002e-158 < n

   1. Initial program 27.7%

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

   3. Taylor expanded in i around 0 52.5%

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

      1. *-commutative 52.5%

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

   5. Simplified 52.5%

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

   if -2.5999999999999999e-183 < n < 5.9000000000000002e-158

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-/r/ 54.5%

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

      3. associate-*l* 54.5%

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

      4. sub-neg 54.5%

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

      5. metadata-eval 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in i around 0 86.0%

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

   6. Taylor expanded in i around 0 86.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-183} \lor \neg \left(n \leq 5.9 \cdot 10^{-158}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 17.5% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 32.1%

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

   1. *-commutative 32.1%

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

   2. associate-/r/ 32.2%

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

   3. associate-*l* 32.2%

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

   4. sub-neg 32.2%

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

   5. metadata-eval 32.2%

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

3. Simplified 32.2%

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

5. Taylor expanded in i around 0 18.7%

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

6. Taylor expanded in i around 0 18.7%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 18.7%

   \[\leadsto 0 \]
8. Add Preprocessing

Developer target: 35.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

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

Reproduce

herbie shell --seed 2024018 
(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))))