Compound Interest

Percentage Accurate: 28.3% → 98.5%

Time: 22.9s

Alternatives: 20

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ t_2 := n \cdot \frac{\mathsf{fma}\left(100, t_0, -100\right)}{i}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

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 (/ (fma 100.0 t_0 -100.0) i))))
   (if (<= t_1 -4e-16)
     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 * (fma(100.0, t_0, -100.0) / i);
	double tmp;
	if (t_1 <= -4e-16) {
		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;
}

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(fma(100.0, t_0, -100.0) / i))
	tmp = 0.0
	if (t_1 <= -4e-16)
		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[(100.0 * t$95$0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-16], 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)) < -3.9999999999999999e-16 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.6%

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

      1. associate-/r/ 99.8%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. fma-def 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   if -3.9999999999999999e-16 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-/r/ 26.5%

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

      3. sub-neg 26.5%

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

      4. metadata-eval 26.5%

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

      5. associate-*r* 26.5%

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

      6. metadata-eval 26.5%

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

      7. sub-neg 26.5%

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

      8. associate-*l/ 26.5%

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

      9. associate-/l* 27.1%

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

      10. add-exp-log 27.1%

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

      11. expm1-def 27.1%

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

      12. log-pow 35.1%

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

      13. log1p-udef 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.2%

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

   5. Simplified 98.2%

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

   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. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-16}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 2: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := t_0 \cdot 100\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))) (t_1 (* t_0 100.0)))
   (if (<= t_0 -1e-218)
     t_1
     (if (<= t_0 0.0)
       (* (* n 100.0) (/ (expm1 i) i))
       (if (<= t_0 INFINITY) t_1 (* 100.0 (/ n (+ 1.0 (* i -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 <= -1e-218) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} 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) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -1e-218) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} 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) + -1.0) / (i / n)
	t_1 = t_0 * 100.0
	tmp = 0
	if t_0 <= -1e-218:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = 100.0 * (n / (1.0 + (i * -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 <= -1e-218)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (t_0 <= Inf)
		tmp = t_1;
	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[(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, -1e-218], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, 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)) < -1e-218 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.0%

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

   if -1e-218 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 21.9%

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

      1. *-commutative 21.9%

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

      2. associate-/r/ 21.2%

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

      3. associate-*l* 21.2%

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

      4. sub-neg 21.2%

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

      5. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in n around inf 36.2%

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

      1. expm1-def 81.6%

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

   6. Simplified 81.6%

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

   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. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 3: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ t_2 := \left(n \cdot 100\right) \cdot \frac{t_0}{i}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* (* n 100.0) (/ t_0 i))))
   (if (<= t_1 -1e-169)
     t_2
     (if (<= t_1 0.0)
       (* 100.0 (/ n (/ i (expm1 i))))
       (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) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = (n * 100.0) * (t_0 / i);
	double tmp;
	if (t_1 <= -1e-169) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} 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) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = (n * 100.0) * (t_0 / i);
	double tmp;
	if (t_1 <= -1e-169) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} 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) + -1.0
	t_1 = t_0 / (i / n)
	t_2 = (n * 100.0) * (t_0 / i)
	tmp = 0
	if t_1 <= -1e-169:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	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((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(Float64(n * 100.0) * Float64(t_0 / i))
	tmp = 0.0
	if (t_1 <= -1e-169)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	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[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-169], t$95$2, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $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)) < -1.00000000000000002e-169 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/r/ 99.1%

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

      3. associate-*l* 99.2%

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

      4. sub-neg 99.2%

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

      5. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   if -1.00000000000000002e-169 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 23.8%

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

   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. associate-/l* 37.7%

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

      3. expm1-def 82.1%

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

   4. Simplified 82.1%

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

   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. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-169}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 4: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-169}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(\frac{t_0}{i} + \frac{-1}{i}\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{t_1}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -1e-169)
     (* (* n 100.0) (+ (/ t_0 i) (/ -1.0 i)))
     (if (<= t_2 0.0)
       (* 100.0 (/ n (/ i (expm1 i))))
       (if (<= t_2 INFINITY)
         (* (* n 100.0) (/ t_1 i))
         (* 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;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -1e-169) {
		tmp = (n * 100.0) * ((t_0 / i) + (-1.0 / i));
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (n * 100.0) * (t_1 / i);
	} 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;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -1e-169) {
		tmp = (n * 100.0) * ((t_0 / i) + (-1.0 / i));
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (n * 100.0) * (t_1 / i);
	} 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
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -1e-169:
		tmp = (n * 100.0) * ((t_0 / i) + (-1.0 / i))
	elif t_2 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif t_2 <= math.inf:
		tmp = (n * 100.0) * (t_1 / i)
	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(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -1e-169)
		tmp = Float64(Float64(n * 100.0) * Float64(Float64(t_0 / i) + Float64(-1.0 / i)));
	elseif (t_2 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(n * 100.0) * Float64(t_1 / i));
	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[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-169], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(t$95$0 / i), $MachinePrecision] + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -1.00000000000000002e-169

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-/r/ 98.2%

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

      3. associate-*l* 98.2%

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

      4. sub-neg 98.2%

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

      5. metadata-eval 98.2%

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

   3. Simplified 98.2%

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

      1. metadata-eval 98.2%

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

      2. sub-neg 98.2%

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

      3. div-sub 99.0%

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

   5. Applied egg-rr 99.0%

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

   if -1.00000000000000002e-169 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 23.8%

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

   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. associate-/l* 37.7%

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

      3. expm1-def 82.1%

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

   4. Simplified 82.1%

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

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

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   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. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-169}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 5: 97.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = expm1((n * log1p((i / n)))) * (n / (i / 100.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (n * 100.0) * (t_0 / i);
	} 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) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) * (n / (i / 100.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (n * 100.0) * (t_0 / i);
	} 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) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = math.expm1((n * math.log1p((i / n)))) * (n / (i / 100.0))
	elif t_1 <= math.inf:
		tmp = (n * 100.0) * (t_0 / i)
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(n / Float64(i / 100.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(n * 100.0) * Float64(t_0 / i));
	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[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(n / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 28.3%

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

      1. *-commutative 28.3%

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

      2. associate-/r/ 27.8%

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

      3. sub-neg 27.8%

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

      4. metadata-eval 27.8%

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

      5. associate-*r* 27.7%

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

      6. metadata-eval 27.7%

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

      7. sub-neg 27.7%

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

      8. associate-*l/ 27.7%

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

      9. associate-/l* 28.3%

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

      10. add-exp-log 28.3%

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

      11. expm1-def 28.3%

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

      12. log-pow 36.1%

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

      13. log1p-udef 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.2%

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

   5. Simplified 98.2%

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

      1. expm1-log1p-u 77.6%

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

      2. expm1-udef 44.3%

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

      3. div-inv 44.3%

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

      4. associate-*l* 43.9%

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

      5. associate-*l/ 43.9%

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

      6. *-un-lft-identity 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. expm1-def 77.3%

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

      2. expm1-log1p 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-/l* 96.8%

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

   9. Simplified 96.8%

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

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

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   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. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.8%

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

Alternative 6: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \left(\left(\frac{n}{i} - \frac{n}{i}\right) + \left(n \cdot \frac{t_0}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{t_1}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -4e-16)
     (* 100.0 (+ (- (/ n i) (/ n i)) (- (* n (/ t_0 i)) (/ n i))))
     (if (<= t_2 0.0)
       (* (/ (expm1 (* n (log1p (/ i n)))) i) (* n 100.0))
       (if (<= t_2 INFINITY)
         (* (* n 100.0) (/ t_1 i))
         (* 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;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -4e-16) {
		tmp = 100.0 * (((n / i) - (n / i)) + ((n * (t_0 / i)) - (n / i)));
	} else if (t_2 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (n * 100.0) * (t_1 / i);
	} 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;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -4e-16) {
		tmp = 100.0 * (((n / i) - (n / i)) + ((n * (t_0 / i)) - (n / i)));
	} else if (t_2 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (n * 100.0) * (t_1 / i);
	} 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
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -4e-16:
		tmp = 100.0 * (((n / i) - (n / i)) + ((n * (t_0 / i)) - (n / i)))
	elif t_2 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) / i) * (n * 100.0)
	elif t_2 <= math.inf:
		tmp = (n * 100.0) * (t_1 / i)
	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(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -4e-16)
		tmp = Float64(100.0 * Float64(Float64(Float64(n / i) - Float64(n / i)) + Float64(Float64(n * Float64(t_0 / i)) - Float64(n / i))));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i) * Float64(n * 100.0));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(n * 100.0) * Float64(t_1 / i));
	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[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-16], N[(100.0 * N[(N[(N[(n / i), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 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$2, Infinity], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -3.9999999999999999e-16

   1. Initial program 99.7%

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

      1. div-sub 99.7%

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

      2. associate-/r/ 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. clear-num 99.8%

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

      5. prod-diff 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-udef 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. fma-udef 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. unsub-neg 99.8%

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

   5. Simplified 99.8%

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

   if -3.9999999999999999e-16 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-/r/ 26.5%

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

      3. sub-neg 26.5%

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

      4. metadata-eval 26.5%

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

      5. associate-*r* 26.5%

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

      6. metadata-eval 26.5%

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

      7. sub-neg 26.5%

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

      8. associate-*l/ 26.5%

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

      9. associate-/l* 27.1%

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

      10. add-exp-log 27.1%

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

      11. expm1-def 27.1%

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

      12. log-pow 35.1%

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

      13. log1p-udef 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.2%

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

   5. Simplified 98.2%

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

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

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   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. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \left(\left(\frac{n}{i} - \frac{n}{i}\right) + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 7: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100}{i} \cdot \left(n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\\ t_1 := i \cdot \left(0.5 - \frac{0.5}{n}\right)\\ \mathbf{if}\;i \leq -1.15 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 8.4 \cdot 10^{-125}:\\ \;\;\;\;n \cdot \frac{1}{\frac{100 + -100 \cdot t_1}{10000 - {\left(100 \cdot t_1\right)}^{2}}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+238}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{n}{i} - \frac{n}{i}\right) + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ 100.0 i) (* n (expm1 (* n (log1p (/ i n)))))))
        (t_1 (* i (- 0.5 (/ 0.5 n)))))
   (if (<= i -1.15e-122)
     t_0
     (if (<= i 8.4e-125)
       (*
        n
        (/
         1.0
         (/ (+ 100.0 (* -100.0 t_1)) (- 10000.0 (pow (* 100.0 t_1) 2.0)))))
       (if (<= i 7e+238)
         t_0
         (*
          100.0
          (+
           (- (/ n i) (/ n i))
           (- (* n (/ (pow (+ 1.0 (/ i n)) n) i)) (/ n i)))))))))

C

double code(double i, double n) {
	double t_0 = (100.0 / i) * (n * expm1((n * log1p((i / n)))));
	double t_1 = i * (0.5 - (0.5 / n));
	double tmp;
	if (i <= -1.15e-122) {
		tmp = t_0;
	} else if (i <= 8.4e-125) {
		tmp = n * (1.0 / ((100.0 + (-100.0 * t_1)) / (10000.0 - pow((100.0 * t_1), 2.0))));
	} else if (i <= 7e+238) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (((n / i) - (n / i)) + ((n * (pow((1.0 + (i / n)), n) / i)) - (n / i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (100.0 / i) * (n * Math.expm1((n * Math.log1p((i / n)))));
	double t_1 = i * (0.5 - (0.5 / n));
	double tmp;
	if (i <= -1.15e-122) {
		tmp = t_0;
	} else if (i <= 8.4e-125) {
		tmp = n * (1.0 / ((100.0 + (-100.0 * t_1)) / (10000.0 - Math.pow((100.0 * t_1), 2.0))));
	} else if (i <= 7e+238) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (((n / i) - (n / i)) + ((n * (Math.pow((1.0 + (i / n)), n) / i)) - (n / i)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 / i) * (n * math.expm1((n * math.log1p((i / n)))))
	t_1 = i * (0.5 - (0.5 / n))
	tmp = 0
	if i <= -1.15e-122:
		tmp = t_0
	elif i <= 8.4e-125:
		tmp = n * (1.0 / ((100.0 + (-100.0 * t_1)) / (10000.0 - math.pow((100.0 * t_1), 2.0))))
	elif i <= 7e+238:
		tmp = t_0
	else:
		tmp = 100.0 * (((n / i) - (n / i)) + ((n * (math.pow((1.0 + (i / n)), n) / i)) - (n / i)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 / i) * Float64(n * expm1(Float64(n * log1p(Float64(i / n))))))
	t_1 = Float64(i * Float64(0.5 - Float64(0.5 / n)))
	tmp = 0.0
	if (i <= -1.15e-122)
		tmp = t_0;
	elseif (i <= 8.4e-125)
		tmp = Float64(n * Float64(1.0 / Float64(Float64(100.0 + Float64(-100.0 * t_1)) / Float64(10000.0 - (Float64(100.0 * t_1) ^ 2.0)))));
	elseif (i <= 7e+238)
		tmp = t_0;
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(n / i) - Float64(n / i)) + Float64(Float64(n * Float64((Float64(1.0 + Float64(i / n)) ^ n) / i)) - Float64(n / i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 / i), $MachinePrecision] * N[(n * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.15e-122], t$95$0, If[LessEqual[i, 8.4e-125], N[(n * N[(1.0 / N[(N[(100.0 + N[(-100.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(10000.0 - N[Power[N[(100.0 * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7e+238], t$95$0, N[(100.0 * N[(N[(N[(n / i), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.15000000000000003e-122 or 8.3999999999999999e-125 < i < 7.00000000000000005e238

   1. Initial program 42.1%

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

      1. clear-num 42.1%

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

      2. un-div-inv 42.1%

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

      3. associate-/l/ 42.2%

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

      4. add-exp-log 42.2%

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

      5. expm1-def 42.2%

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

      6. log-pow 43.9%

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

      7. log1p-udef 89.2%

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

   3. Applied egg-rr 89.2%

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

      1. /-rgt-identity 89.2%

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

      2. associate-/r/ 89.2%

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

      3. associate-/r/ 89.3%

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

      4. associate-/r/ 89.3%

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

      5. /-rgt-identity 89.3%

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

      6. *-commutative 89.3%

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

   5. Simplified 89.3%

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

   if -1.15000000000000003e-122 < i < 8.3999999999999999e-125

   1. Initial program 4.9%

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

      1. associate-/r/ 5.8%

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

      2. associate-*r* 5.8%

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

      3. *-commutative 5.8%

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

      4. associate-*r/ 5.8%

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

      5. sub-neg 5.8%

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

      6. distribute-lft-in 5.8%

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

      7. fma-def 5.8%

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

      8. metadata-eval 5.8%

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

      9. metadata-eval 5.8%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in i around 0 95.3%

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

      1. associate-*r/ 95.3%

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

      2. metadata-eval 95.3%

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

   6. Simplified 95.3%

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

      1. flip-+ 95.3%

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

      2. clear-num 95.3%

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

      3. cancel-sign-sub-inv 95.3%

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

      4. metadata-eval 95.3%

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

      5. metadata-eval 95.3%

         \[\leadsto n \cdot \frac{1}{\frac{100 + -100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{\color{blue}{10000} - \left(100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right) \cdot \left(100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}} \]

      6. pow2 95.3%

         \[\leadsto n \cdot \frac{1}{\frac{100 + -100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{10000 - \color{blue}{{\left(100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}^{2}}}} \]

   8. Applied egg-rr 95.3%

      \[\leadsto n \cdot \color{blue}{\frac{1}{\frac{100 + -100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{10000 - {\left(100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}^{2}}}} \]

   if 7.00000000000000005e238 < i

   1. Initial program 72.0%

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

      1. div-sub 72.0%

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

      2. associate-/r/ 72.4%

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

      3. *-un-lft-identity 72.4%

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

      4. clear-num 72.6%

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

      5. prod-diff 72.6%

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

   3. Applied egg-rr 72.6%

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

      1. +-commutative 72.6%

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

      2. fma-udef 72.6%

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

      3. *-rgt-identity 72.6%

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

      4. distribute-neg-frac 72.6%

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

      5. *-rgt-identity 72.6%

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

      6. fma-udef 72.6%

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

      7. *-rgt-identity 72.6%

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

      8. unsub-neg 72.6%

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

   5. Simplified 72.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.15 \cdot 10^{-122}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 8.4 \cdot 10^{-125}:\\ \;\;\;\;n \cdot \frac{1}{\frac{100 + -100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{10000 - {\left(100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}^{2}}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+238}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{n}{i} - \frac{n}{i}\right) + \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{n}{i}\right)\right)\\ \end{array} \]

Alternative 8: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -4.4 \cdot 10^{-244}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-235}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -4.4e-244)
     t_0
     (if (<= n 4.5e-235)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 2.35e-59) (* 100.0 (/ n (+ 1.0 (* i -0.5)))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -4.4e-244) {
		tmp = t_0;
	} else if (n <= 4.5e-235) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.35e-59) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -4.4e-244) {
		tmp = t_0;
	} else if (n <= 4.5e-235) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.35e-59) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -4.4e-244:
		tmp = t_0
	elif n <= 4.5e-235:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.35e-59:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -4.4e-244)
		tmp = t_0;
	elseif (n <= 4.5e-235)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.35e-59)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.4e-244], t$95$0, If[LessEqual[n, 4.5e-235], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.35e-59], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.39999999999999969e-244 or 2.35e-59 < n

   1. Initial program 31.8%

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

   2. Taylor expanded in n around inf 37.6%

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

      1. *-commutative 37.6%

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

      2. associate-/l* 37.5%

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

      3. expm1-def 85.2%

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

   4. Simplified 85.2%

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

   if -4.39999999999999969e-244 < n < 4.4999999999999998e-235

   1. Initial program 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-/r/ 55.8%

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

      3. associate-*l* 55.8%

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

      4. sub-neg 55.8%

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

      5. metadata-eval 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in i around 0 78.9%

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

   if 4.4999999999999998e-235 < n < 2.35e-59

   1. Initial program 14.8%

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

   2. Taylor expanded in n around inf 4.0%

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

      1. *-commutative 4.0%

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

      2. associate-/l* 4.0%

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

      3. expm1-def 53.5%

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

   4. Simplified 53.5%

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

   5. Taylor expanded in i around 0 72.6%

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

      1. *-commutative 72.6%

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

   7. Simplified 72.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-244}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-235}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 9: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 7.4 \cdot 10^{-235}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.6e-248)
   (* 100.0 (/ n (/ i (expm1 i))))
   (if (<= n 7.4e-235)
     (* (* n 100.0) (/ 0.0 i))
     (if (<= n 2.35e-59)
       (* 100.0 (/ n (+ 1.0 (* i -0.5))))
       (* (* n 100.0) (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.6e-248) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (n <= 7.4e-235) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.35e-59) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = (n * 100.0) * (expm1(i) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.6e-248) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (n <= 7.4e-235) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 2.35e-59) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.6e-248:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif n <= 7.4e-235:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 2.35e-59:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.6e-248)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (n <= 7.4e-235)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 2.35e-59)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.6e-248], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.4e-235], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.35e-59], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.60000000000000007e-248

   1. Initial program 38.0%

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

   2. Taylor expanded in n around inf 32.0%

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

      1. *-commutative 32.0%

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

      2. associate-/l* 32.0%

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

      3. expm1-def 79.2%

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

   4. Simplified 79.2%

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

   if -2.60000000000000007e-248 < n < 7.4000000000000002e-235

   1. Initial program 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-/r/ 55.8%

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

      3. associate-*l* 55.8%

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

      4. sub-neg 55.8%

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

      5. metadata-eval 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in i around 0 78.9%

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

   if 7.4000000000000002e-235 < n < 2.35e-59

   1. Initial program 14.8%

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

   2. Taylor expanded in n around inf 4.0%

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

      1. *-commutative 4.0%

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

      2. associate-/l* 4.0%

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

      3. expm1-def 53.5%

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

   4. Simplified 53.5%

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

   5. Taylor expanded in i around 0 72.6%

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

      1. *-commutative 72.6%

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

   7. Simplified 72.6%

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

   if 2.35e-59 < n

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

      2. associate-/r/ 22.8%

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

      3. associate-*l* 22.8%

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

      4. sub-neg 22.8%

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

      5. metadata-eval 22.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in n around inf 45.8%

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

      1. expm1-def 94.0%

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

   6. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(i\right)}{i}} \cdot \left(n \cdot 100\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 7.4 \cdot 10^{-235}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]

Alternative 10: 59.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.0146:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;i \leq 2.45 \cdot 10^{+188}:\\ \;\;\;\;n \cdot 100 + 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.0146)
   (* (* n 100.0) (/ 0.0 i))
   (if (<= i 2.45e+188)
     (+ (* n 100.0) (* 100.0 (* i (* n (+ 0.5 (* 0.5 (/ -1.0 n)))))))
     (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.0146) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (i <= 2.45e+188) {
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-0.0146d0)) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (i <= 2.45d+188) then
        tmp = (n * 100.0d0) + (100.0d0 * (i * (n * (0.5d0 + (0.5d0 * ((-1.0d0) / n))))))
    else
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.0146) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (i <= 2.45e+188) {
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.0146:
		tmp = (n * 100.0) * (0.0 / i)
	elif i <= 2.45e+188:
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))))
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.0146)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (i <= 2.45e+188)
		tmp = Float64(Float64(n * 100.0) + Float64(100.0 * Float64(i * Float64(n * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n)))))));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.0146)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (i <= 2.45e+188)
		tmp = (n * 100.0) + (100.0 * (i * (n * (0.5 + (0.5 * (-1.0 / n))))));
	else
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.0146], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.45e+188], N[(N[(n * 100.0), $MachinePrecision] + N[(100.0 * N[(i * N[(n * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 i < -0.0146000000000000001

   1. Initial program 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-/r/ 70.6%

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

      3. associate-*l* 70.5%

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

      4. sub-neg 70.5%

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

      5. metadata-eval 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in i around 0 36.3%

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

   if -0.0146000000000000001 < i < 2.45e188

   1. Initial program 14.1%

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

   2. Taylor expanded in i around 0 77.3%

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

   if 2.45e188 < i

   1. Initial program 66.4%

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

   2. Taylor expanded in n around inf 20.0%

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

      1. *-commutative 20.0%

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

      2. associate-/l* 20.0%

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

      3. expm1-def 20.0%

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

   4. Simplified 20.0%

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

   5. Taylor expanded in i around 0 38.6%

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

      1. *-commutative 38.6%

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

   7. Simplified 38.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0146:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;i \leq 2.45 \cdot 10^{+188}:\\ \;\;\;\;n \cdot 100 + 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 11: 59.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.0146:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;i \leq 2.45 \cdot 10^{+188}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.0146)
   (* (* n 100.0) (/ 0.0 i))
   (if (<= i 2.45e+188)
     (* n (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n))))))
     (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.0146) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (i <= 2.45e+188) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-0.0146d0)) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (i <= 2.45d+188) then
        tmp = n * (100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n)))))
    else
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.0146) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (i <= 2.45e+188) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.0146:
		tmp = (n * 100.0) * (0.0 / i)
	elif i <= 2.45e+188:
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.0146)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (i <= 2.45e+188)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.0146)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (i <= 2.45e+188)
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	else
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.0146], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.45e+188], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 i < -0.0146000000000000001

   1. Initial program 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-/r/ 70.6%

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

      3. associate-*l* 70.5%

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

      4. sub-neg 70.5%

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

      5. metadata-eval 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in i around 0 36.3%

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

   if -0.0146000000000000001 < i < 2.45e188

   1. Initial program 14.1%

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

      1. associate-/r/ 14.6%

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

      2. associate-*r* 14.6%

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

      3. *-commutative 14.6%

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

      4. associate-*r/ 14.6%

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

      5. sub-neg 14.6%

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

      6. distribute-lft-in 14.6%

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

      7. fma-def 14.6%

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

      8. metadata-eval 14.6%

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

      9. metadata-eval 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in i around 0 77.2%

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

      1. associate-*r/ 77.2%

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

      2. metadata-eval 77.2%

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

   6. Simplified 77.2%

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

   if 2.45e188 < i

   1. Initial program 66.4%

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

   2. Taylor expanded in n around inf 20.0%

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

      1. *-commutative 20.0%

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

      2. associate-/l* 20.0%

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

      3. expm1-def 20.0%

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

   4. Simplified 20.0%

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

   5. Taylor expanded in i around 0 38.6%

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

      1. *-commutative 38.6%

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

   7. Simplified 38.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0146:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;i \leq 2.45 \cdot 10^{+188}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 12: 60.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 8 \cdot 10^{-191} \lor \neg \left(i \leq 2.45 \cdot 10^{+188}\right):\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 \cdot \left(i \cdot n\right)}{-i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i 8e-191) (not (<= i 2.45e+188)))
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (/ (* -100.0 (* i n)) (- i))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= 8e-191) || !(i <= 2.45e+188)) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = (-100.0 * (i * n)) / -i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= 8d-191) .or. (.not. (i <= 2.45d+188))) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else
        tmp = ((-100.0d0) * (i * n)) / -i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= 8e-191) || !(i <= 2.45e+188)) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = (-100.0 * (i * n)) / -i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= 8e-191) or not (i <= 2.45e+188):
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = (-100.0 * (i * n)) / -i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= 8e-191) || !(i <= 2.45e+188))
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = Float64(Float64(-100.0 * Float64(i * n)) / Float64(-i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= 8e-191) || ~((i <= 2.45e+188)))
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	else
		tmp = (-100.0 * (i * n)) / -i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, 8e-191], N[Not[LessEqual[i, 2.45e+188]], $MachinePrecision]], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / (-i)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 8.0000000000000002e-191 or 2.45e188 < i

   1. Initial program 34.9%

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

   2. Taylor expanded in n around inf 30.8%

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

      1. *-commutative 30.8%

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

      2. associate-/l* 30.8%

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

      3. expm1-def 76.8%

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

   4. Simplified 76.8%

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

   5. Taylor expanded in i around 0 65.6%

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

      1. *-commutative 65.6%

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

   7. Simplified 65.6%

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

   if 8.0000000000000002e-191 < i < 2.45e188

   1. Initial program 23.7%

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

      1. *-commutative 23.7%

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

      2. associate-/r/ 23.9%

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

      3. sub-neg 23.9%

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

      4. metadata-eval 23.9%

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

      5. associate-*r* 23.8%

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

      6. *-commutative 23.8%

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

      7. frac-2neg 23.8%

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

      8. metadata-eval 23.8%

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

      9. sub-neg 23.8%

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

      10. associate-*r/ 23.8%

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

   3. Applied egg-rr 87.6%

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

   4. Taylor expanded in i around 0 61.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 8 \cdot 10^{-191} \lor \neg \left(i \leq 2.45 \cdot 10^{+188}\right):\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 \cdot \left(i \cdot n\right)}{-i}\\ \end{array} \]

Alternative 13: 59.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.0146:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;i \leq 2 \cdot 10^{+188}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.0146)
   (* (* n 100.0) (/ 0.0 i))
   (if (<= i 2e+188)
     (* 100.0 (* n (+ 1.0 (* i 0.5))))
     (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.0146) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (i <= 2e+188) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-0.0146d0)) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else if (i <= 2d+188) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * 0.5d0)))
    else
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.0146) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (i <= 2e+188) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.0146:
		tmp = (n * 100.0) * (0.0 / i)
	elif i <= 2e+188:
		tmp = 100.0 * (n * (1.0 + (i * 0.5)))
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.0146)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (i <= 2e+188)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * 0.5))));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.0146)
		tmp = (n * 100.0) * (0.0 / i);
	elseif (i <= 2e+188)
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	else
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.0146], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2e+188], N[(100.0 * N[(n * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 i < -0.0146000000000000001

   1. Initial program 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-/r/ 70.6%

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

      3. associate-*l* 70.5%

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

      4. sub-neg 70.5%

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

      5. metadata-eval 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in i around 0 36.3%

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

   if -0.0146000000000000001 < i < 2e188

   1. Initial program 14.1%

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

      1. *-commutative 14.1%

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

      2. associate-/r/ 14.6%

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

      3. associate-*l* 14.5%

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

      4. sub-neg 14.5%

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

      5. metadata-eval 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in i around 0 77.2%

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

      1. associate-*r/ 77.2%

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

      2. metadata-eval 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in n around inf 77.2%

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

   if 2e188 < i

   1. Initial program 66.4%

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

   2. Taylor expanded in n around inf 20.0%

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

      1. *-commutative 20.0%

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

      2. associate-/l* 20.0%

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

      3. expm1-def 20.0%

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

   4. Simplified 20.0%

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

   5. Taylor expanded in i around 0 38.6%

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

      1. *-commutative 38.6%

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

   7. Simplified 38.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0146:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;i \leq 2 \cdot 10^{+188}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 14: 61.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 \cdot \left(i \cdot n\right)}{-i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e+16)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n 1.85e+27) (* 100.0 (/ i (/ i n))) (/ (* -100.0 (* i n)) (- i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e+16) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 1.85e+27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (-100.0 * (i * n)) / -i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.1d+16)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= 1.85d+27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = ((-100.0d0) * (i * n)) / -i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.1e+16) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 1.85e+27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (-100.0 * (i * n)) / -i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.1e+16:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= 1.85e+27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (-100.0 * (i * n)) / -i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e+16)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= 1.85e+27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(-100.0 * Float64(i * n)) / Float64(-i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.1e+16)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= 1.85e+27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (-100.0 * (i * n)) / -i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e+16], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e+27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / (-i)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1e16

   1. Initial program 32.8%

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

   2. Taylor expanded in n around inf 30.6%

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

      1. *-commutative 30.6%

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

      2. associate-/l* 30.6%

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

      3. expm1-def 84.7%

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

   4. Simplified 84.7%

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

   5. Taylor expanded in i around 0 61.6%

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

      1. associate-*r* 61.6%

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

      2. distribute-rgt-out 61.6%

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

   7. Simplified 61.6%

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

   if -1.1e16 < n < 1.85000000000000001e27

   1. Initial program 39.8%

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

   2. Taylor expanded in i around 0 65.1%

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

   if 1.85000000000000001e27 < n

   1. Initial program 18.7%

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

      1. *-commutative 18.7%

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

      2. associate-/r/ 19.2%

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

      3. sub-neg 19.2%

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

      4. metadata-eval 19.2%

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

      5. associate-*r* 19.2%

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

      6. *-commutative 19.2%

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

      7. frac-2neg 19.2%

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

      8. metadata-eval 19.2%

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

      9. sub-neg 19.2%

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

      10. associate-*r/ 19.2%

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

   3. Applied egg-rr 68.9%

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

   4. Taylor expanded in i around 0 64.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 \cdot \left(i \cdot n\right)}{-i}\\ \end{array} \]

Alternative 15: 61.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 \cdot \left(i \cdot n\right)}{-i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e+15)
   (* 100.0 (* n (+ 1.0 (* i 0.5))))
   (if (<= n 2.1e+27) (* 100.0 (/ i (/ i n))) (/ (* -100.0 (* i n)) (- i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+15) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else if (n <= 2.1e+27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (-100.0 * (i * n)) / -i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.6d+15)) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * 0.5d0)))
    else if (n <= 2.1d+27) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = ((-100.0d0) * (i * n)) / -i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+15) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else if (n <= 2.1e+27) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (-100.0 * (i * n)) / -i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.6e+15:
		tmp = 100.0 * (n * (1.0 + (i * 0.5)))
	elif n <= 2.1e+27:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (-100.0 * (i * n)) / -i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.6e+15)
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * 0.5))));
	elseif (n <= 2.1e+27)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(-100.0 * Float64(i * n)) / Float64(-i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.6e+15)
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	elseif (n <= 2.1e+27)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (-100.0 * (i * n)) / -i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.6e+15], N[(100.0 * N[(n * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e+27], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / (-i)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6e15

   1. Initial program 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/r/ 33.4%

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

      3. associate-*l* 33.3%

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

      4. sub-neg 33.3%

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

      5. metadata-eval 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in i around 0 61.6%

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

      1. associate-*r/ 61.6%

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

      2. metadata-eval 61.6%

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

   6. Simplified 61.6%

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

   7. Taylor expanded in n around inf 61.6%

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

   if -3.6e15 < n < 2.09999999999999995e27

   1. Initial program 39.8%

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

   2. Taylor expanded in i around 0 65.1%

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

   if 2.09999999999999995e27 < n

   1. Initial program 18.7%

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

      1. *-commutative 18.7%

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

      2. associate-/r/ 19.2%

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

      3. sub-neg 19.2%

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

      4. metadata-eval 19.2%

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

      5. associate-*r* 19.2%

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

      6. *-commutative 19.2%

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

      7. frac-2neg 19.2%

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

      8. metadata-eval 19.2%

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

      9. sub-neg 19.2%

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

      10. associate-*r/ 19.2%

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

   3. Applied egg-rr 68.9%

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

   4. Taylor expanded in i around 0 64.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+15}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-100 \cdot \left(i \cdot n\right)}{-i}\\ \end{array} \]

Alternative 16: 62.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{+16} \lor \neg \left(n \leq 1.88 \cdot 10^{-59}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.65e+16) (not (<= n 1.88e-59)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.65e+16) || !(n <= 1.88e-59)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.65d+16)) .or. (.not. (n <= 1.88d-59))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.65e+16) || !(n <= 1.88e-59)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.65e+16) or not (n <= 1.88e-59):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.65e+16) || !(n <= 1.88e-59))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.65e+16) || ~((n <= 1.88e-59)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.65e+16], N[Not[LessEqual[n, 1.88e-59]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.65e16 or 1.88e-59 < n

   1. Initial program 27.3%

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

   2. Taylor expanded in n around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 38.5%

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

      3. expm1-def 89.6%

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

   4. Simplified 89.6%

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

   5. Taylor expanded in i around 0 62.3%

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

      1. associate-*r* 62.3%

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

      2. distribute-rgt-out 62.3%

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

   7. Simplified 62.3%

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

   if -1.65e16 < n < 1.88e-59

   1. Initial program 38.9%

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

   2. Taylor expanded in i around 0 66.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{+16} \lor \neg \left(n \leq 1.88 \cdot 10^{-59}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 17: 58.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+73}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2e+73)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 2.0) (* n 100.0) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2e+73) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 2.0) {
		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 <= (-2d+73)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 2.0d0) 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 <= -2e+73) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2e+73:
		tmp = 100.0 * (i / (i / n))
	elif i <= 2.0:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2e+73)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 2.0)
		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 <= -2e+73)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 2.0)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2e+73], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.0], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.99999999999999997e73

   1. Initial program 79.9%

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

   2. Taylor expanded in i around 0 33.1%

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

   if -1.99999999999999997e73 < i < 2

   1. Initial program 10.9%

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

   2. Taylor expanded in i around 0 81.2%

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

      1. *-commutative 81.2%

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

   4. Simplified 81.2%

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

   if 2 < i

   1. Initial program 47.9%

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

      1. *-commutative 47.9%

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

      2. associate-/r/ 48.2%

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

      3. associate-*l* 48.2%

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

      4. sub-neg 48.2%

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

      5. metadata-eval 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in n around inf 56.7%

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

      1. expm1-def 56.7%

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

   6. Simplified 56.7%

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

      1. div-inv 56.7%

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

   8. Applied egg-rr 56.7%

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

   9. Taylor expanded in i around 0 27.6%

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

       1. *-commutative 27.6%

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

   11. Simplified 27.6%

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

   12. Taylor expanded in i around inf 27.6%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+73}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 18: 54.7% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (if (<= i 2.0) (* n 100.0) (* (* i n) 50.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.0) {
		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 <= 2.0d0) 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 <= 2.0) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.0:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.0)
		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 <= 2.0)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.0], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2

   1. Initial program 26.9%

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

   2. Taylor expanded in i around 0 63.4%

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

      1. *-commutative 63.4%

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

   4. Simplified 63.4%

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

   if 2 < i

   1. Initial program 47.9%

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

      1. *-commutative 47.9%

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

      2. associate-/r/ 48.2%

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

      3. associate-*l* 48.2%

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

      4. sub-neg 48.2%

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

      5. metadata-eval 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in n around inf 56.7%

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

      1. expm1-def 56.7%

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

   6. Simplified 56.7%

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

      1. div-inv 56.7%

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

   8. Applied egg-rr 56.7%

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

   9. Taylor expanded in i around 0 27.6%

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

       1. *-commutative 27.6%

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

   11. Simplified 27.6%

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

   12. Taylor expanded in i around inf 27.6%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 19: 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 31.5%

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

   1. *-commutative 31.5%

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

   2. associate-/r/ 31.5%

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

   3. associate-*l* 31.5%

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

   4. sub-neg 31.5%

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

   5. metadata-eval 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in i around 0 54.7%

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

   1. associate-*r/ 54.7%

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

   2. metadata-eval 54.7%

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

6. Simplified 54.7%

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

7. Taylor expanded in n around 0 2.8%

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

9. Simplified 2.8%

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

10. Final simplification 2.8%

    \[\leadsto i \cdot -50 \]

Alternative 20: 49.3% 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 31.5%

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

2. Taylor expanded in i around 0 51.1%

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

   1. *-commutative 51.1%

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

4. Simplified 51.1%

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

5. Final simplification 51.1%

   \[\leadsto n \cdot 100 \]

Developer target: 33.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(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))))