Compound Interest

Percentage Accurate: 28.3% → 98.4%

Time: 26.1s

Alternatives: 15

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: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, t_0, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 INFINITY)
       (* n (/ (fma 100.0 t_0 -100.0) i))
       (* 100.0 (/ n (+ 1.0 (fma 0.08333333333333333 (* i i) (* 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 tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (fma(100.0, t_0, -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + fma(0.08333333333333333, (i * i), (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))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(fma(100.0, t_0, -100.0) / i));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + fma(0.08333333333333333, Float64(i * i), 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]}, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(100.0 * t$95$0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(0.08333333333333333 * N[(i * i), $MachinePrecision] + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.1%

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

      1. *-un-lft-identity 26.1%

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

      2. pow-to-exp 25.1%

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

      3. expm1-def 35.6%

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

      4. *-commutative 35.6%

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

      5. log1p-udef 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. *-lft-identity 98.6%

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

   5. Simplified 98.6%

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

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

   1. Initial program 99.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 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 +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 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 78.3%

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

   4. Simplified 78.3%

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

   5. Taylor expanded in i around 0 99.9%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. fma-def 99.9%

         \[\leadsto \frac{n}{1 + \color{blue}{\mathsf{fma}\left(0.08333333333333333, {i}^{2}, -0.5 \cdot i\right)}} \cdot 100 \]

      2. unpow2 99.9%

         \[\leadsto \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, \color{blue}{i \cdot i}, -0.5 \cdot i\right)} \cdot 100 \]

      3. *-commutative 99.9%

         \[\leadsto \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, \color{blue}{i \cdot -0.5}\right)} \cdot 100 \]

   7. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_0 INFINITY)
       (* t_0 100.0)
       (* 100.0 (/ n (+ 1.0 (fma 0.08333333333333333 (* i i) (* i -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * (n / (1.0 + fma(0.08333333333333333, (i * i), (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))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + fma(0.08333333333333333, Float64(i * i), 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]}, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(0.08333333333333333 * N[(i * i), $MachinePrecision] + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.1%

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

      1. *-un-lft-identity 26.1%

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

      2. pow-to-exp 25.1%

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

      3. expm1-def 35.6%

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

      4. *-commutative 35.6%

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

      5. log1p-udef 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. *-lft-identity 98.6%

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

   5. Simplified 98.6%

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

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

   1. Initial program 99.9%

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

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

   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 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 78.3%

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

   4. Simplified 78.3%

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

   5. Taylor expanded in i around 0 99.9%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. fma-def 99.9%

         \[\leadsto \frac{n}{1 + \color{blue}{\mathsf{fma}\left(0.08333333333333333, {i}^{2}, -0.5 \cdot i\right)}} \cdot 100 \]

      2. unpow2 99.9%

         \[\leadsto \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, \color{blue}{i \cdot i}, -0.5 \cdot i\right)} \cdot 100 \]

      3. *-commutative 99.9%

         \[\leadsto \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, \color{blue}{i \cdot -0.5}\right)} \cdot 100 \]

   7. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}\\ \end{array} \]

Alternative 3: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{-42}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -1.7 \cdot 10^{-58}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot n\right)}{\frac{i}{\log \left(\frac{i}{n}\right)}}\\ \mathbf{elif}\;n \leq -1.05 \cdot 10^{-202}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.9e-42)
   (* 100.0 (/ n (/ i (expm1 i))))
   (if (<= n -1.7e-58)
     (/ (* 100.0 (* n n)) (/ i (log (/ i n))))
     (if (<= n -1.05e-202)
       (* 100.0 (/ n (+ 1.0 (fma 0.08333333333333333 (* i i) (* i -0.5)))))
       (if (<= n 4.9e-31)
         (/
          (* n (* n 10000.0))
          (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))
         (* 100.0 (* n (/ (expm1 i) i))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.9e-42) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (n <= -1.7e-58) {
		tmp = (100.0 * (n * n)) / (i / log((i / n)));
	} else if (n <= -1.05e-202) {
		tmp = 100.0 * (n / (1.0 + fma(0.08333333333333333, (i * i), (i * -0.5))));
	} else if (n <= 4.9e-31) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = 100.0 * (n * (expm1(i) / i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.9e-42)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (n <= -1.7e-58)
		tmp = Float64(Float64(100.0 * Float64(n * n)) / Float64(i / log(Float64(i / n))));
	elseif (n <= -1.05e-202)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + fma(0.08333333333333333, Float64(i * i), Float64(i * -0.5)))));
	elseif (n <= 4.9e-31)
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	else
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.9e-42], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.7e-58], N[(N[(100.0 * N[(n * n), $MachinePrecision]), $MachinePrecision] / N[(i / N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.05e-202], N[(100.0 * N[(n / N[(1.0 + N[(0.08333333333333333 * N[(i * i), $MachinePrecision] + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.9e-31], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -4.9e-42

   1. Initial program 34.0%

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

      1. *-un-lft-identity 34.0%

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

      2. pow-to-exp 17.8%

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

      3. expm1-def 18.6%

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

      4. *-commutative 18.6%

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

      5. log1p-udef 66.0%

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

   3. Applied egg-rr 66.0%

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

      1. *-lft-identity 66.0%

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

   5. Simplified 66.0%

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

      1. *-commutative 66.0%

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

      2. div-inv 64.6%

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

      3. clear-num 64.7%

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

      4. associate-*r* 64.7%

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

      5. associate-/r/ 64.6%

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

      6. clear-num 64.5%

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

      7. un-div-inv 64.8%

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

      8. div-inv 64.8%

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

      9. metadata-eval 64.8%

         \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot \color{blue}{0.01}}{n}} \]

   7. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot 0.01}{n}}} \]
   8. Step-by-step derivation

      1. associate-/l* 64.9%

         \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{i}{\frac{n}{0.01}}}} \]

   9. Simplified 64.9%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{\frac{n}{0.01}}}} \]

   10. Taylor expanded in n around inf 41.7%

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

       1. associate-/l* 41.7%

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

       2. expm1-def 83.9%

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

   12. Simplified 83.9%

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

   if -4.9e-42 < n < -1.69999999999999987e-58

   1. Initial program 25.0%

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

      1. *-un-lft-identity 25.0%

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

      2. pow-to-exp 25.0%

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

      3. expm1-def 99.1%

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

      4. *-commutative 99.1%

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

      5. log1p-udef 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. *-lft-identity 99.1%

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

   5. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. div-inv 99.1%

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

      3. clear-num 99.1%

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

      4. associate-*r* 99.1%

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

      5. associate-/r/ 99.4%

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

      6. clear-num 99.4%

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

      7. un-div-inv 99.4%

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

      8. div-inv 99.4%

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

      9. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot \color{blue}{0.01}}{n}} \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot 0.01}{n}}} \]
   8. Step-by-step derivation

      1. associate-/l* 99.7%

         \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{i}{\frac{n}{0.01}}}} \]

   9. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{\frac{n}{0.01}}}} \]

   10. Taylor expanded in n around 0 0.0%

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

       1. associate-/l* 0.0%

          \[\leadsto 100 \cdot \color{blue}{\frac{{n}^{2}}{\frac{i}{-1 \cdot \log n + \log i}}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \color{blue}{\frac{100 \cdot {n}^{2}}{\frac{i}{-1 \cdot \log n + \log i}}} \]

       3. unpow2 0.0%

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

       4. mul-1-neg 0.0%

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

       5. log-rec 0.0%

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

       6. +-commutative 0.0%

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

       7. log-rec 0.0%

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

       8. unsub-neg 0.0%

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

       9. log-div 99.4%

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

   12. Simplified 99.4%

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

   if -1.69999999999999987e-58 < n < -1.04999999999999993e-202

   1. Initial program 21.3%

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

   2. Taylor expanded in n around inf 13.1%

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

      1. *-commutative 13.1%

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

      2. associate-/l* 13.1%

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

      3. expm1-def 78.0%

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

   4. Simplified 78.0%

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

   5. Taylor expanded in i around 0 86.2%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. fma-def 86.2%

         \[\leadsto \frac{n}{1 + \color{blue}{\mathsf{fma}\left(0.08333333333333333, {i}^{2}, -0.5 \cdot i\right)}} \cdot 100 \]

      2. unpow2 86.2%

         \[\leadsto \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, \color{blue}{i \cdot i}, -0.5 \cdot i\right)} \cdot 100 \]

      3. *-commutative 86.2%

         \[\leadsto \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, \color{blue}{i \cdot -0.5}\right)} \cdot 100 \]

   7. Simplified 86.2%

      \[\leadsto \frac{n}{\color{blue}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}} \cdot 100 \]

   if -1.04999999999999993e-202 < n < 4.90000000000000023e-31

   1. Initial program 42.0%

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

      1. associate-/r/ 42.2%

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

      2. associate-*r* 42.2%

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

      3. *-commutative 42.2%

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

      4. associate-*r/ 42.2%

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

      5. sub-neg 42.2%

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

      6. distribute-lft-in 42.2%

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

      7. fma-def 42.2%

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

      8. metadata-eval 42.2%

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

      9. metadata-eval 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in i around 0 27.4%

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

      1. associate-*r* 27.4%

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

      2. *-commutative 27.4%

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

      3. associate-*r/ 27.4%

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

      4. metadata-eval 27.4%

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

   6. Simplified 27.4%

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

      1. distribute-lft-in 27.4%

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

      2. flip-+ 23.4%

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

   8. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)}} \]

   9. Taylor expanded in i around 0 76.1%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 76.1%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       2. unpow2 76.1%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       3. associate-*l* 76.1%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   11. Simplified 76.1%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   if 4.90000000000000023e-31 < n

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/l* 40.6%

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

      3. expm1-def 98.7%

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

   4. Simplified 98.7%

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

      1. div-inv 98.7%

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

      2. clear-num 98.8%

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

   6. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \cdot 100 \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{-42}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -1.7 \cdot 10^{-58}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot n\right)}{\frac{i}{\log \left(\frac{i}{n}\right)}}\\ \mathbf{elif}\;n \leq -1.05 \cdot 10^{-202}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \mathsf{fma}\left(0.08333333333333333, i \cdot i, i \cdot -0.5\right)}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 4: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-6} \lor \neg \left(n \leq 4.9 \cdot 10^{-31}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -8e-6) (not (<= n 4.9e-31)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/
    (* n (* n 10000.0))
    (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -8e-6) || !(n <= 4.9e-31)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -8e-6) || !(n <= 4.9e-31)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -8e-6) or not (n <= 4.9e-31):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -8e-6) || !(n <= 4.9e-31))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -8e-6], N[Not[LessEqual[n, 4.9e-31]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7.99999999999999964e-6 or 4.90000000000000023e-31 < n

   1. Initial program 26.5%

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

      1. *-un-lft-identity 26.5%

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

      2. pow-to-exp 18.1%

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

      3. expm1-def 18.1%

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

      4. *-commutative 18.1%

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

      5. log1p-udef 70.6%

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

   3. Applied egg-rr 70.6%

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

      1. *-lft-identity 70.6%

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

   5. Simplified 70.6%

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

      1. *-commutative 70.6%

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

      2. div-inv 68.0%

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

      3. clear-num 68.1%

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

      4. associate-*r* 68.1%

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

      5. associate-/r/ 68.1%

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

      6. clear-num 68.0%

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

      7. un-div-inv 69.8%

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

      8. div-inv 69.8%

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

      9. metadata-eval 69.8%

         \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot \color{blue}{0.01}}{n}} \]

   7. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot 0.01}{n}}} \]
   8. Step-by-step derivation

      1. associate-/l* 69.8%

         \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{i}{\frac{n}{0.01}}}} \]

   9. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{\frac{n}{0.01}}}} \]

   10. Taylor expanded in n around inf 42.0%

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

       1. associate-/l* 42.0%

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

       2. expm1-def 92.2%

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

   12. Simplified 92.2%

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

   if -7.99999999999999964e-6 < n < 4.90000000000000023e-31

   1. Initial program 33.1%

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

      1. associate-/r/ 33.2%

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

      2. associate-*r* 33.2%

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

      3. *-commutative 33.2%

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

      4. associate-*r/ 33.2%

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

      5. sub-neg 33.2%

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

      6. distribute-lft-in 33.2%

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

      7. fma-def 33.2%

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

      8. metadata-eval 33.2%

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

      9. metadata-eval 33.2%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in i around 0 41.9%

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

      1. associate-*r* 41.9%

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

      2. *-commutative 41.9%

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

      3. associate-*r/ 41.9%

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

      4. metadata-eval 41.9%

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

   6. Simplified 41.9%

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

      1. distribute-lft-in 41.9%

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

      2. flip-+ 38.0%

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

   8. Applied egg-rr 38.0%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)}} \]

   9. Taylor expanded in i around 0 75.9%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 75.9%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       2. unpow2 75.9%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       3. associate-*l* 76.3%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   11. Simplified 76.3%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-6} \lor \neg \left(n \leq 4.9 \cdot 10^{-31}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 5: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.00032:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -0.00032)
   (* 100.0 (/ n (/ i (expm1 i))))
   (if (<= n 4.9e-31)
     (/
      (* n (* n 10000.0))
      (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))
     (* 100.0 (* n (/ (expm1 i) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -0.00032) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (n <= 4.9e-31) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = 100.0 * (n * (expm1(i) / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -0.00032) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (n <= 4.9e-31) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -0.00032:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif n <= 4.9e-31:
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))))
	else:
		tmp = 100.0 * (n * (math.expm1(i) / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -0.00032)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (n <= 4.9e-31)
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	else
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -0.00032], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.9e-31], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.20000000000000026e-4

   1. Initial program 35.2%

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

      1. *-un-lft-identity 35.2%

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

      2. pow-to-exp 18.1%

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

      3. expm1-def 18.1%

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

      4. *-commutative 18.1%

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

      5. log1p-udef 64.0%

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

   3. Applied egg-rr 64.0%

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

      1. *-lft-identity 64.0%

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

   5. Simplified 64.0%

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

      1. *-commutative 64.0%

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

      2. div-inv 62.6%

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

      3. clear-num 62.7%

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

      4. associate-*r* 62.7%

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

      5. associate-/r/ 62.7%

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

      6. clear-num 62.6%

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

      7. un-div-inv 62.9%

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

      8. div-inv 62.9%

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

      9. metadata-eval 62.9%

         \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot \color{blue}{0.01}}{n}} \]

   7. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i \cdot 0.01}{n}}} \]
   8. Step-by-step derivation

      1. associate-/l* 62.9%

         \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{i}{\frac{n}{0.01}}}} \]

   9. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{\frac{n}{0.01}}}} \]

   10. Taylor expanded in n around inf 43.8%

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

       1. associate-/l* 43.8%

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

       2. expm1-def 84.3%

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

   12. Simplified 84.3%

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

   if -3.20000000000000026e-4 < n < 4.90000000000000023e-31

   1. Initial program 33.1%

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

      1. associate-/r/ 33.2%

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

      2. associate-*r* 33.2%

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

      3. *-commutative 33.2%

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

      4. associate-*r/ 33.2%

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

      5. sub-neg 33.2%

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

      6. distribute-lft-in 33.2%

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

      7. fma-def 33.2%

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

      8. metadata-eval 33.2%

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

      9. metadata-eval 33.2%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in i around 0 41.9%

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

      1. associate-*r* 41.9%

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

      2. *-commutative 41.9%

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

      3. associate-*r/ 41.9%

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

      4. metadata-eval 41.9%

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

   6. Simplified 41.9%

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

      1. distribute-lft-in 41.9%

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

      2. flip-+ 38.0%

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

   8. Applied egg-rr 38.0%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)}} \]

   9. Taylor expanded in i around 0 75.9%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 75.9%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       2. unpow2 75.9%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       3. associate-*l* 76.3%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   11. Simplified 76.3%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   if 4.90000000000000023e-31 < n

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/l* 40.6%

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

      3. expm1-def 98.7%

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

   4. Simplified 98.7%

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

      1. div-inv 98.7%

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

      2. clear-num 98.8%

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

   6. Applied egg-rr 98.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.00032:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 6: 70.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+129} \lor \neg \left(n \leq 4.9 \cdot 10^{-31}\right):\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -8e+129) (not (<= n 4.9e-31)))
   (*
    100.0
    (+
     n
     (*
      n
      (+
       (*
        (* i i)
        (+ (/ 0.3333333333333333 (* n n)) (- 0.16666666666666666 (/ 0.5 n))))
       (* i (- 0.5 (/ 0.5 n)))))))
   (/
    (* n (* n 10000.0))
    (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -8e+129) || !(n <= 4.9e-31)) {
		tmp = 100.0 * (n + (n * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	} else {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8d+129)) .or. (.not. (n <= 4.9d-31))) then
        tmp = 100.0d0 * (n + (n * (((i * i) * ((0.3333333333333333d0 / (n * n)) + (0.16666666666666666d0 - (0.5d0 / n)))) + (i * (0.5d0 - (0.5d0 / n))))))
    else
        tmp = (n * (n * 10000.0d0)) / ((n * 100.0d0) - (n * (i * (50.0d0 + (100.0d0 * ((-0.5d0) / n))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -8e+129) || !(n <= 4.9e-31)) {
		tmp = 100.0 * (n + (n * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	} else {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -8e+129) or not (n <= 4.9e-31):
		tmp = 100.0 * (n + (n * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))) + (i * (0.5 - (0.5 / n))))))
	else:
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -8e+129) || !(n <= 4.9e-31))
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(Float64(i * i) * Float64(Float64(0.3333333333333333 / Float64(n * n)) + Float64(0.16666666666666666 - Float64(0.5 / n)))) + Float64(i * Float64(0.5 - Float64(0.5 / n)))))));
	else
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -8e+129) || ~((n <= 4.9e-31)))
		tmp = 100.0 * (n + (n * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	else
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -8e+129], N[Not[LessEqual[n, 4.9e-31]], $MachinePrecision]], N[(100.0 * N[(n + N[(n * N[(N[(N[(i * i), $MachinePrecision] * N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8e129 or 4.90000000000000023e-31 < n

   1. Initial program 20.1%

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

   2. Taylor expanded in i around 0 75.9%

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

      1. distribute-lft-out 76.1%

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

      2. unpow2 76.1%

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

      3. associate--l+ 76.1%

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

      4. associate-*r/ 76.1%

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

      5. metadata-eval 76.1%

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

      6. unpow2 76.1%

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

      7. associate-*r/ 76.1%

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

      8. metadata-eval 76.1%

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

      9. associate-*r/ 76.1%

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

      10. metadata-eval 76.1%

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

   4. Simplified 76.1%

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

   if -8e129 < n < 4.90000000000000023e-31

   1. Initial program 37.4%

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

      1. associate-/r/ 37.5%

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

      2. associate-*r* 37.5%

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

      3. *-commutative 37.5%

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

      4. associate-*r/ 37.5%

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

      5. sub-neg 37.5%

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

      6. distribute-lft-in 37.5%

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

      7. fma-def 37.5%

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

      8. metadata-eval 37.5%

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

      9. metadata-eval 37.5%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in i around 0 42.5%

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

      1. associate-*r* 42.5%

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

      2. *-commutative 42.5%

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

      3. associate-*r/ 42.5%

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

      4. metadata-eval 42.5%

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

   6. Simplified 42.5%

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

      1. distribute-lft-in 42.5%

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

      2. flip-+ 39.5%

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

   8. Applied egg-rr 39.5%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)}} \]

   9. Taylor expanded in i around 0 70.4%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 70.4%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       2. unpow2 70.4%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       3. associate-*l* 70.7%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   11. Simplified 70.7%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+129} \lor \neg \left(n \leq 4.9 \cdot 10^{-31}\right):\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 7: 69.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+151}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{100 - i \cdot 50}{10000 + \left(i \cdot i\right) \cdot -2500}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9e+151)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n 4.9e-31)
     (/
      (* n (* n 10000.0))
      (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))
     (/ n (/ (- 100.0 (* i 50.0)) (+ 10000.0 (* (* i i) -2500.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9e+151) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 4.9e-31) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = n / ((100.0 - (i * 50.0)) / (10000.0 + ((i * i) * -2500.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9d+151)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= 4.9d-31) then
        tmp = (n * (n * 10000.0d0)) / ((n * 100.0d0) - (n * (i * (50.0d0 + (100.0d0 * ((-0.5d0) / n))))))
    else
        tmp = n / ((100.0d0 - (i * 50.0d0)) / (10000.0d0 + ((i * i) * (-2500.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9e+151) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 4.9e-31) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = n / ((100.0 - (i * 50.0)) / (10000.0 + ((i * i) * -2500.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9e+151:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= 4.9e-31:
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))))
	else:
		tmp = n / ((100.0 - (i * 50.0)) / (10000.0 + ((i * i) * -2500.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9e+151)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= 4.9e-31)
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	else
		tmp = Float64(n / Float64(Float64(100.0 - Float64(i * 50.0)) / Float64(10000.0 + Float64(Float64(i * i) * -2500.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9e+151)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= 4.9e-31)
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	else
		tmp = n / ((100.0 - (i * 50.0)) / (10000.0 + ((i * i) * -2500.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9e+151], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.9e-31], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(100.0 - N[(i * 50.0), $MachinePrecision]), $MachinePrecision] / N[(10000.0 + N[(N[(i * i), $MachinePrecision] * -2500.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.9999999999999997e151

   1. Initial program 21.4%

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

      1. associate-/r/ 22.0%

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

      2. associate-*r* 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-*r/ 22.0%

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

      5. sub-neg 22.0%

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

      6. distribute-lft-in 22.0%

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

      7. fma-def 22.0%

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

      8. metadata-eval 22.0%

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

      9. metadata-eval 22.0%

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

   3. Simplified 22.0%

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

   4. Taylor expanded in i around 0 59.9%

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

      1. associate-*r* 59.9%

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

      2. *-commutative 59.9%

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

      3. associate-*r/ 59.9%

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

      4. metadata-eval 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in n around inf 59.9%

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

      1. *-commutative 59.9%

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

   9. Simplified 59.9%

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

   if -8.9999999999999997e151 < n < 4.90000000000000023e-31

   1. Initial program 37.0%

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

      1. associate-/r/ 37.2%

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

      2. associate-*r* 37.2%

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

      3. *-commutative 37.2%

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

      4. associate-*r/ 37.2%

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

      5. sub-neg 37.2%

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

      6. distribute-lft-in 37.2%

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

      7. fma-def 37.2%

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

      8. metadata-eval 37.2%

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

      9. metadata-eval 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in i around 0 42.3%

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

      1. associate-*r* 42.3%

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

      2. *-commutative 42.3%

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

      3. associate-*r/ 42.3%

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

      4. metadata-eval 42.3%

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

   6. Simplified 42.3%

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

      1. distribute-lft-in 42.3%

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

      2. flip-+ 40.0%

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

   8. Applied egg-rr 40.0%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)}} \]

   9. Taylor expanded in i around 0 69.4%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 69.4%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       2. unpow2 69.4%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

       3. associate-*l* 69.7%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   11. Simplified 69.7%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   if 4.90000000000000023e-31 < n

   1. Initial program 19.3%

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

      1. associate-/r/ 19.7%

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

      2. associate-*r* 19.7%

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

      3. *-commutative 19.7%

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

      4. associate-*r/ 19.7%

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

      5. sub-neg 19.7%

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

      6. distribute-lft-in 19.7%

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

      7. fma-def 19.7%

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

      8. metadata-eval 19.7%

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

      9. metadata-eval 19.7%

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

   3. Simplified 19.7%

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

   4. Taylor expanded in i around 0 76.1%

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

      1. associate-*r* 76.1%

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

      2. *-commutative 76.1%

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

      3. associate-*r/ 76.1%

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

      4. metadata-eval 76.1%

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

   6. Simplified 76.1%

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

      1. distribute-lft-in 76.1%

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

      2. flip-+ 43.2%

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

   8. Applied egg-rr 43.2%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)}} \]

   9. Taylor expanded in n around inf 81.2%

      \[\leadsto \color{blue}{\frac{n \cdot \left(10000 - 2500 \cdot {i}^{2}\right)}{100 - 50 \cdot i}} \]
   10. Step-by-step derivation

       1. associate-/l* 79.2%

          \[\leadsto \color{blue}{\frac{n}{\frac{100 - 50 \cdot i}{10000 - 2500 \cdot {i}^{2}}}} \]

       2. *-commutative 79.2%

          \[\leadsto \frac{n}{\frac{100 - \color{blue}{i \cdot 50}}{10000 - 2500 \cdot {i}^{2}}} \]

       3. cancel-sign-sub-inv 79.2%

          \[\leadsto \frac{n}{\frac{100 - i \cdot 50}{\color{blue}{10000 + \left(-2500\right) \cdot {i}^{2}}}} \]

       4. metadata-eval 79.2%

          \[\leadsto \frac{n}{\frac{100 - i \cdot 50}{10000 + \color{blue}{-2500} \cdot {i}^{2}}} \]

       5. *-commutative 79.2%

          \[\leadsto \frac{n}{\frac{100 - i \cdot 50}{10000 + \color{blue}{{i}^{2} \cdot -2500}}} \]

       6. unpow2 79.2%

          \[\leadsto \frac{n}{\frac{100 - i \cdot 50}{10000 + \color{blue}{\left(i \cdot i\right)} \cdot -2500}} \]

   11. Simplified 79.2%

       \[\leadsto \color{blue}{\frac{n}{\frac{100 - i \cdot 50}{10000 + \left(i \cdot i\right) \cdot -2500}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+151}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{100 - i \cdot 50}{10000 + \left(i \cdot i\right) \cdot -2500}}\\ \end{array} \]

Alternative 8: 66.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+210}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -6.8 \cdot 10^{-204}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{100 - i \cdot 50}{10000 + \left(i \cdot i\right) \cdot -2500}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.4e+210)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n -6.8e-204)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 5e-123)
       (* (* n 100.0) (/ 0.0 i))
       (/ n (/ (- 100.0 (* i 50.0)) (+ 10000.0 (* (* i i) -2500.0))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.4e+210) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= -6.8e-204) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 5e-123) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = n / ((100.0 - (i * 50.0)) / (10000.0 + ((i * i) * -2500.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.4d+210)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= (-6.8d-204)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 5d-123) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = n / ((100.0d0 - (i * 50.0d0)) / (10000.0d0 + ((i * i) * (-2500.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.4e+210) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= -6.8e-204) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 5e-123) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = n / ((100.0 - (i * 50.0)) / (10000.0 + ((i * i) * -2500.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.4e+210:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= -6.8e-204:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 5e-123:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = n / ((100.0 - (i * 50.0)) / (10000.0 + ((i * i) * -2500.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.4e+210)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= -6.8e-204)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 5e-123)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = Float64(n / Float64(Float64(100.0 - Float64(i * 50.0)) / Float64(10000.0 + Float64(Float64(i * i) * -2500.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.4e+210)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= -6.8e-204)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 5e-123)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = n / ((100.0 - (i * 50.0)) / (10000.0 + ((i * i) * -2500.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.4e+210], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -6.8e-204], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5e-123], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(100.0 - N[(i * 50.0), $MachinePrecision]), $MachinePrecision] / N[(10000.0 + N[(N[(i * i), $MachinePrecision] * -2500.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.4000000000000001e210

   1. Initial program 15.7%

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

      1. associate-/r/ 16.4%

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

      2. associate-*r* 16.4%

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

      3. *-commutative 16.4%

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

      4. associate-*r/ 16.4%

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

      5. sub-neg 16.4%

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

      6. distribute-lft-in 16.4%

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

      7. fma-def 16.4%

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

      8. metadata-eval 16.4%

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

      9. metadata-eval 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in i around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-*r/ 70.8%

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

      4. metadata-eval 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in n around inf 70.8%

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

      1. *-commutative 70.8%

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

   9. Simplified 70.8%

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

   if -1.4000000000000001e210 < n < -6.8000000000000004e-204

   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 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/l* 28.5%

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

      3. expm1-def 77.4%

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

   4. Simplified 77.4%

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

   5. Taylor expanded in i around 0 60.1%

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

      1. *-commutative 60.1%

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

   7. Simplified 60.1%

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

   if -6.8000000000000004e-204 < n < 5.0000000000000003e-123

   1. Initial program 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-/r/ 52.1%

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

      3. associate-*l* 52.1%

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

      4. sub-neg 52.1%

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

      5. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in i around 0 75.4%

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

   if 5.0000000000000003e-123 < n

   1. Initial program 20.3%

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

      1. associate-/r/ 20.6%

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

      2. associate-*r* 20.6%

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

      3. *-commutative 20.6%

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

      4. associate-*r/ 20.6%

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

      5. sub-neg 20.6%

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

      6. distribute-lft-in 20.6%

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

      7. fma-def 20.6%

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

      8. metadata-eval 20.6%

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

      9. metadata-eval 20.6%

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

   3. Simplified 20.6%

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

   4. Taylor expanded in i around 0 69.7%

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

      1. associate-*r* 69.7%

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

      2. *-commutative 69.7%

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

      3. associate-*r/ 69.7%

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

      4. metadata-eval 69.7%

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

   6. Simplified 69.7%

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

      1. distribute-lft-in 69.7%

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

      2. flip-+ 43.3%

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

   8. Applied egg-rr 43.3%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)}} \]

   9. Taylor expanded in n around inf 73.9%

      \[\leadsto \color{blue}{\frac{n \cdot \left(10000 - 2500 \cdot {i}^{2}\right)}{100 - 50 \cdot i}} \]
   10. Step-by-step derivation

       1. associate-/l* 72.4%

          \[\leadsto \color{blue}{\frac{n}{\frac{100 - 50 \cdot i}{10000 - 2500 \cdot {i}^{2}}}} \]

       2. *-commutative 72.4%

          \[\leadsto \frac{n}{\frac{100 - \color{blue}{i \cdot 50}}{10000 - 2500 \cdot {i}^{2}}} \]

       3. cancel-sign-sub-inv 72.4%

          \[\leadsto \frac{n}{\frac{100 - i \cdot 50}{\color{blue}{10000 + \left(-2500\right) \cdot {i}^{2}}}} \]

       4. metadata-eval 72.4%

          \[\leadsto \frac{n}{\frac{100 - i \cdot 50}{10000 + \color{blue}{-2500} \cdot {i}^{2}}} \]

       5. *-commutative 72.4%

          \[\leadsto \frac{n}{\frac{100 - i \cdot 50}{10000 + \color{blue}{{i}^{2} \cdot -2500}}} \]

       6. unpow2 72.4%

          \[\leadsto \frac{n}{\frac{100 - i \cdot 50}{10000 + \color{blue}{\left(i \cdot i\right)} \cdot -2500}} \]

   11. Simplified 72.4%

       \[\leadsto \color{blue}{\frac{n}{\frac{100 - i \cdot 50}{10000 + \left(i \cdot i\right) \cdot -2500}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+210}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -6.8 \cdot 10^{-204}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{100 - i \cdot 50}{10000 + \left(i \cdot i\right) \cdot -2500}}\\ \end{array} \]

Alternative 9: 64.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -6 \cdot 10^{+208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.95 \cdot 10^{-200}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-122}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -6e+208)
     t_0
     (if (<= n -1.95e-200)
       (* 100.0 (/ n (+ 1.0 (* i -0.5))))
       (if (<= n 1.75e-122) (* (* n 100.0) (/ 0.0 i)) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -6e+208) {
		tmp = t_0;
	} else if (n <= -1.95e-200) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.75e-122) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-6d+208)) then
        tmp = t_0
    else if (n <= (-1.95d-200)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1.75d-122) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -6e+208) {
		tmp = t_0;
	} else if (n <= -1.95e-200) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.75e-122) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -6e+208:
		tmp = t_0
	elif n <= -1.95e-200:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1.75e-122:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -6e+208)
		tmp = t_0;
	elseif (n <= -1.95e-200)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1.75e-122)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -6e+208)
		tmp = t_0;
	elseif (n <= -1.95e-200)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1.75e-122)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6e+208], t$95$0, If[LessEqual[n, -1.95e-200], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-122], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.99999999999999989e208 or 1.7500000000000001e-122 < n

   1. Initial program 19.5%

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

      1. associate-/r/ 19.9%

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

      2. associate-*r* 19.9%

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

      3. *-commutative 19.9%

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

      4. associate-*r/ 19.9%

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

      5. sub-neg 19.9%

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

      6. distribute-lft-in 19.9%

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

      7. fma-def 19.9%

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

      8. metadata-eval 19.9%

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

      9. metadata-eval 19.9%

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

   3. Simplified 19.9%

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

   4. Taylor expanded in i around 0 69.9%

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

      1. associate-*r* 69.9%

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

      2. *-commutative 69.9%

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

      3. associate-*r/ 69.9%

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

      4. metadata-eval 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in n around inf 70.1%

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

      1. *-commutative 70.1%

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

   9. Simplified 70.1%

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

   if -5.99999999999999989e208 < n < -1.94999999999999999e-200

   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 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/l* 28.5%

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

      3. expm1-def 77.4%

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

   4. Simplified 77.4%

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

   5. Taylor expanded in i around 0 60.1%

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

      1. *-commutative 60.1%

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

   7. Simplified 60.1%

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

   if -1.94999999999999999e-200 < n < 1.7500000000000001e-122

   1. Initial program 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-/r/ 52.1%

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

      3. associate-*l* 52.1%

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

      4. sub-neg 52.1%

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

      5. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in i around 0 75.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{+208}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -1.95 \cdot 10^{-200}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-122}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 10: 63.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{+209} \lor \neg \left(n \leq 4.9 \cdot 10^{-31}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.95e+209) (not (<= n 4.9e-31)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.95e+209) || !(n <= 4.9e-31)) {
		tmp = n * (100.0 + (i * 50.0));
	} 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 ((n <= (-2.95d+209)) .or. (.not. (n <= 4.9d-31))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    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 ((n <= -2.95e+209) || !(n <= 4.9e-31)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.95e+209) or not (n <= 4.9e-31):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.95e+209) || !(n <= 4.9e-31))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	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 ((n <= -2.95e+209) || ~((n <= 4.9e-31)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.95e+209], N[Not[LessEqual[n, 4.9e-31]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.9499999999999999e209 or 4.90000000000000023e-31 < n

   1. Initial program 18.6%

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

      1. associate-/r/ 19.1%

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

      2. associate-*r* 19.1%

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

      3. *-commutative 19.1%

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

      4. associate-*r/ 19.1%

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

      5. sub-neg 19.1%

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

      6. distribute-lft-in 19.1%

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

      7. fma-def 19.1%

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

      8. metadata-eval 19.1%

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

      9. metadata-eval 19.1%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in i around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-*r/ 75.1%

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

      4. metadata-eval 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in n around inf 75.1%

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

      1. *-commutative 75.1%

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

   9. Simplified 75.1%

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

   if -2.9499999999999999e209 < n < 4.90000000000000023e-31

   1. Initial program 36.3%

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

   2. Taylor expanded in n around inf 25.8%

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

      1. *-commutative 25.8%

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

      2. associate-/l* 25.8%

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

      3. expm1-def 60.9%

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

   4. Simplified 60.9%

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

   5. Taylor expanded in i around 0 56.1%

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

      1. *-commutative 56.1%

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

   7. Simplified 56.1%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{+209} \lor \neg \left(n \leq 4.9 \cdot 10^{-31}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 11: 62.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+104} \lor \neg \left(n \leq 1.92 \cdot 10^{-31}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.4e+104) (not (<= n 1.92e-31)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e+104) || !(n <= 1.92e-31)) {
		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 <= (-1.4d+104)) .or. (.not. (n <= 1.92d-31))) 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 <= -1.4e+104) || !(n <= 1.92e-31)) {
		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 <= -1.4e+104) or not (n <= 1.92e-31):
		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 <= -1.4e+104) || !(n <= 1.92e-31))
		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 <= -1.4e+104) || ~((n <= 1.92e-31)))
		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, -1.4e+104], N[Not[LessEqual[n, 1.92e-31]], $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 < -1.4e104 or 1.9200000000000001e-31 < n

   1. Initial program 20.6%

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

      1. associate-/r/ 21.1%

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

      2. associate-*r* 21.1%

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

      3. *-commutative 21.1%

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

      4. associate-*r/ 21.1%

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

      5. sub-neg 21.1%

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

      6. distribute-lft-in 21.1%

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

      7. fma-def 21.1%

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

      8. metadata-eval 21.1%

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

      9. metadata-eval 21.1%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in i around 0 69.6%

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

      1. associate-*r* 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-*r/ 69.6%

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

      4. metadata-eval 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in n around inf 69.6%

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

      1. *-commutative 69.6%

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

   9. Simplified 69.6%

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

   if -1.4e104 < n < 1.9200000000000001e-31

   1. Initial program 37.8%

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

   2. Taylor expanded in i around 0 55.2%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+104} \lor \neg \left(n \leq 1.92 \cdot 10^{-31}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 12: 57.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+30}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2e+30)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 1.5e-27) (* n 100.0) (* 50.0 (* i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2e+30) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1.5e-27) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2d+30)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 1.5d-27) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2e+30) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 1.5e-27) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2e+30:
		tmp = 100.0 * (i / (i / n))
	elif i <= 1.5e-27:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2e+30)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 1.5e-27)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2e+30)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 1.5e-27)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2e+30], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.5e-27], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2e30

   1. Initial program 66.3%

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

   2. Taylor expanded in i around 0 21.8%

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

   if -2e30 < i < 1.5000000000000001e-27

   1. Initial program 7.5%

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

   2. Taylor expanded in i around 0 85.6%

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

      1. *-commutative 85.6%

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

   4. Simplified 85.6%

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

   if 1.5000000000000001e-27 < i

   1. Initial program 51.3%

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

      1. associate-/r/ 51.5%

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

      2. associate-*r* 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-*r/ 51.5%

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

      5. sub-neg 51.5%

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

      6. distribute-lft-in 51.5%

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

      7. fma-def 51.5%

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

      8. metadata-eval 51.5%

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

      9. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in i around 0 25.7%

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

      1. associate-*r* 25.7%

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

      2. *-commutative 25.7%

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

      3. associate-*r/ 25.7%

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

      4. metadata-eval 25.7%

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

   6. Simplified 25.7%

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

   7. Taylor expanded in i around inf 25.7%

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

      1. *-commutative 25.7%

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

      2. associate-*l* 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-*r* 25.7%

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

      5. *-commutative 25.7%

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

      6. sub-neg 25.7%

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

      7. associate-*r/ 25.7%

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

      8. metadata-eval 25.7%

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

      9. distribute-neg-frac 25.7%

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

      10. metadata-eval 25.7%

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

      11. distribute-lft-out 25.7%

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

      12. associate-*l* 25.7%

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

      13. metadata-eval 25.7%

          \[\leadsto n \cdot \left(i \cdot \color{blue}{50} + \left(i \cdot 100\right) \cdot \frac{-0.5}{n}\right) \]

      14. associate-*l* 25.7%

          \[\leadsto n \cdot \left(i \cdot 50 + \color{blue}{i \cdot \left(100 \cdot \frac{-0.5}{n}\right)}\right) \]

      15. *-commutative 25.7%

          \[\leadsto n \cdot \left(i \cdot 50 + i \cdot \color{blue}{\left(\frac{-0.5}{n} \cdot 100\right)}\right) \]

      16. distribute-lft-in 25.7%

          \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

      17. *-commutative 25.7%

          \[\leadsto \color{blue}{\left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right) \cdot n} \]

      18. associate-*r* 25.8%

          \[\leadsto \color{blue}{i \cdot \left(\left(50 + \frac{-0.5}{n} \cdot 100\right) \cdot n\right)} \]

      19. *-commutative 25.8%

          \[\leadsto i \cdot \color{blue}{\left(n \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   9. Simplified 25.8%

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

   10. Taylor expanded in n around inf 26.5%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+30}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 13: 53.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.5e-27) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.5e-27) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.5d-27) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.5e-27) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.5e-27:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.5e-27)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.5e-27)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.5e-27], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.5000000000000001e-27

   1. Initial program 20.1%

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

   2. Taylor expanded in i around 0 68.4%

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

      1. *-commutative 68.4%

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

   4. Simplified 68.4%

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

   if 1.5000000000000001e-27 < i

   1. Initial program 51.3%

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

      1. associate-/r/ 51.5%

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

      2. associate-*r* 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-*r/ 51.5%

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

      5. sub-neg 51.5%

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

      6. distribute-lft-in 51.5%

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

      7. fma-def 51.5%

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

      8. metadata-eval 51.5%

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

      9. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in i around 0 25.7%

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

      1. associate-*r* 25.7%

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

      2. *-commutative 25.7%

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

      3. associate-*r/ 25.7%

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

      4. metadata-eval 25.7%

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

   6. Simplified 25.7%

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

   7. Taylor expanded in i around inf 25.7%

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

      1. *-commutative 25.7%

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

      2. associate-*l* 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-*r* 25.7%

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

      5. *-commutative 25.7%

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

      6. sub-neg 25.7%

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

      7. associate-*r/ 25.7%

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

      8. metadata-eval 25.7%

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

      9. distribute-neg-frac 25.7%

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

      10. metadata-eval 25.7%

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

      11. distribute-lft-out 25.7%

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

      12. associate-*l* 25.7%

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

      13. metadata-eval 25.7%

          \[\leadsto n \cdot \left(i \cdot \color{blue}{50} + \left(i \cdot 100\right) \cdot \frac{-0.5}{n}\right) \]

      14. associate-*l* 25.7%

          \[\leadsto n \cdot \left(i \cdot 50 + \color{blue}{i \cdot \left(100 \cdot \frac{-0.5}{n}\right)}\right) \]

      15. *-commutative 25.7%

          \[\leadsto n \cdot \left(i \cdot 50 + i \cdot \color{blue}{\left(\frac{-0.5}{n} \cdot 100\right)}\right) \]

      16. distribute-lft-in 25.7%

          \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

      17. *-commutative 25.7%

          \[\leadsto \color{blue}{\left(i \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right) \cdot n} \]

      18. associate-*r* 25.8%

          \[\leadsto \color{blue}{i \cdot \left(\left(50 + \frac{-0.5}{n} \cdot 100\right) \cdot n\right)} \]

      19. *-commutative 25.8%

          \[\leadsto i \cdot \color{blue}{\left(n \cdot \left(50 + \frac{-0.5}{n} \cdot 100\right)\right)} \]

   9. Simplified 25.8%

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

   10. Taylor expanded in n around inf 26.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 14: 2.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

   1. associate-/r/ 29.4%

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

   2. associate-*r* 29.4%

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

   3. *-commutative 29.4%

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

   4. associate-*r/ 29.4%

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

   5. sub-neg 29.4%

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

   6. distribute-lft-in 29.4%

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

   7. fma-def 29.4%

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

   8. metadata-eval 29.4%

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

   9. metadata-eval 29.4%

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

3. Simplified 29.4%

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

4. Taylor expanded in i around 0 55.7%

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

   1. associate-*r* 55.7%

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

   2. *-commutative 55.7%

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

   3. associate-*r/ 55.7%

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

   4. metadata-eval 55.7%

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

6. Simplified 55.7%

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

7. Taylor expanded in n around 0 2.8%

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

   1. *-commutative 2.8%

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

9. Simplified 2.8%

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

10. Final simplification 2.8%

    \[\leadsto i \cdot -50 \]

Alternative 15: 49.2% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

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

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 29.1%

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

2. Taylor expanded in i around 0 50.0%

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

   1. *-commutative 50.0%

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

4. Simplified 50.0%

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

5. Final simplification 50.0%

   \[\leadsto n \cdot 100 \]

Developer target: 33.4% 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 2023274 
(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))))