Compound Interest

Percentage Accurate: 28.2% → 99.3%

Time: 35.8s

Alternatives: 26

Speedup: 16.0×

• 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: 28.2% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-167}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, t_0, -100\right)}{i}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{t_1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -4e-167)
     (* n (/ (fma 100.0 t_0 -100.0) i))
     (if (<= t_2 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_2 INFINITY)
         (* (/ t_1 i) (* n 100.0))
         (/ (* n 100.0) (+ 1.0 (* i -0.5))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -4e-167) {
		tmp = n * (fma(100.0, t_0, -100.0) / i);
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_1 / i) * (n * 100.0);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -4e-167)
		tmp = Float64(n * Float64(fma(100.0, t_0, -100.0) / i));
	elseif (t_2 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_1 / i) * Float64(n * 100.0));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-167], N[(n * N[(N[(100.0 * t$95$0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 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$2, Infinity], N[(N[(t$95$1 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.00000000000000001e-167

   1. Initial program 99.7%

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

      1. associate-/r/ 99.7%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. fma-def 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   if -4.00000000000000001e-167 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 20.2%

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

      1. *-un-lft-identity 20.2%

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

      2. pow-to-exp 20.2%

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

      3. expm1-def 32.3%

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

      4. *-commutative 32.3%

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

      5. log1p-udef 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   5. Simplified 99.8%

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

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

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.1%

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

   4. Simplified 81.1%

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

      1. associate-*l/ 81.1%

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

   6. Applied egg-rr 81.1%

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

   7. Taylor expanded in i around 0 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -4 \cdot 10^{-167}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 2: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := t_0 \cdot 100\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))) (t_1 (* t_0 100.0)))
   (if (<= t_0 -2e-64)
     t_1
     (if (<= t_0 0.0)
       (/ 1.0 (/ 0.01 (/ n (/ i (expm1 i)))))
       (if (<= t_0 INFINITY) t_1 (/ (* n 100.0) (+ 1.0 (* i -0.5))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -2e-64) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / (0.01 / (n / (i / expm1(i))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -2e-64) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / (0.01 / (n / (i / Math.expm1(i))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	t_1 = t_0 * 100.0
	tmp = 0
	if t_0 <= -2e-64:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = 1.0 / (0.01 / (n / (i / math.expm1(i))))
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	t_1 = Float64(t_0 * 100.0)
	tmp = 0.0
	if (t_0 <= -2e-64)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(0.01 / Float64(n / Float64(i / expm1(i)))));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-64], t$95$1, If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(0.01 / N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -1.99999999999999993e-64 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.7%

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

   if -1.99999999999999993e-64 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 22.4%

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

   2. Taylor expanded in n around inf 32.3%

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

      1. *-commutative 32.3%

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

      2. associate-/l* 32.3%

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

      3. expm1-def 79.4%

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

   4. Simplified 79.4%

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

      1. associate-*l/ 79.4%

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

   6. Applied egg-rr 79.4%

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

   7. Taylor expanded in n around 0 32.3%

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

      1. *-commutative 32.3%

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

      2. expm1-def 72.0%

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

      3. associate-/l* 79.4%

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

      4. associate-/r/ 79.3%

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

      5. associate-/l/ 79.2%

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

   9. Simplified 79.2%

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

       1. clear-num 79.4%

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

       2. inv-pow 79.4%

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

       3. *-un-lft-identity 79.4%

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

       4. times-frac 79.5%

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

       5. metadata-eval 79.5%

          \[\leadsto {\left(\frac{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}\right)}^{-1} \]

   11. Applied egg-rr 79.5%

       \[\leadsto \color{blue}{{\left(\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 79.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}}} \]

       2. associate-/l* 79.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}} \]

   13. Simplified 79.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}} \]

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.1%

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

   4. Simplified 81.1%

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

      1. associate-*l/ 81.1%

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

   6. Applied egg-rr 81.1%

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

   7. Taylor expanded in i around 0 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-64}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 3: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-64}:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{t_0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 -2e-64)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (/ 1.0 (/ 0.01 (/ n (/ i (expm1 i)))))
       (if (<= t_1 INFINITY)
         (* (/ t_0 i) (* n 100.0))
         (/ (* n 100.0) (+ 1.0 (* i -0.5))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -2e-64) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / (0.01 / (n / (i / expm1(i))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 / i) * (n * 100.0);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -2e-64) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / (0.01 / (n / (i / Math.expm1(i))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / i) * (n * 100.0);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= -2e-64:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = 1.0 / (0.01 / (n / (i / math.expm1(i))))
	elif t_1 <= math.inf:
		tmp = (t_0 / i) * (n * 100.0)
	else:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-64)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64(0.01 / Float64(n / Float64(i / expm1(i)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 / i) * Float64(n * 100.0));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-64], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(0.01 / N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -1.99999999999999993e-64

   1. Initial program 100.0%

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

   if -1.99999999999999993e-64 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 22.4%

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

   2. Taylor expanded in n around inf 32.3%

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

      1. *-commutative 32.3%

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

      2. associate-/l* 32.3%

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

      3. expm1-def 79.4%

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

   4. Simplified 79.4%

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

      1. associate-*l/ 79.4%

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

   6. Applied egg-rr 79.4%

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

   7. Taylor expanded in n around 0 32.3%

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

      1. *-commutative 32.3%

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

      2. expm1-def 72.0%

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

      3. associate-/l* 79.4%

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

      4. associate-/r/ 79.3%

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

      5. associate-/l/ 79.2%

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

   9. Simplified 79.2%

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

       1. clear-num 79.4%

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

       2. inv-pow 79.4%

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

       3. *-un-lft-identity 79.4%

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

       4. times-frac 79.5%

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

       5. metadata-eval 79.5%

          \[\leadsto {\left(\frac{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}\right)}^{-1} \]

   11. Applied egg-rr 79.5%

       \[\leadsto \color{blue}{{\left(\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 79.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}}} \]

       2. associate-/l* 79.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}} \]

   13. Simplified 79.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}} \]

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

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.1%

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

   4. Simplified 81.1%

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

      1. associate-*l/ 81.1%

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

   6. Applied egg-rr 81.1%

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

   7. Taylor expanded in i around 0 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-64}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\frac{-100 + t_0 \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{t_1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -5e-210)
     (/ (+ -100.0 (* t_0 100.0)) (/ i n))
     (if (<= t_2 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_2 INFINITY)
         (* (/ t_1 i) (* n 100.0))
         (/ (* n 100.0) (+ 1.0 (* i -0.5))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -5e-210) {
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n);
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_1 / i) * (n * 100.0);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -5e-210) {
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n);
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 / i) * (n * 100.0);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = t_0 + -1.0
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -5e-210:
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n)
	elif t_2 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_2 <= math.inf:
		tmp = (t_1 / i) * (n * 100.0)
	else:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -5e-210)
		tmp = Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / Float64(i / n));
	elseif (t_2 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_1 / i) * Float64(n * 100.0));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-210], N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 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$2, Infinity], N[(N[(t$95$1 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -5.0000000000000002e-210

   1. Initial program 99.7%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. fma-def 99.8%

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

      5. metadata-eval 99.8%

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

      6. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -5.0000000000000002e-210 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 18.8%

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

      1. *-un-lft-identity 18.8%

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

      2. pow-to-exp 18.8%

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

      3. expm1-def 31.2%

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

      4. *-commutative 31.2%

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

      5. log1p-udef 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   5. Simplified 99.8%

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

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

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 81.1%

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

   4. Simplified 81.1%

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

      1. associate-*l/ 81.1%

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

   6. Applied egg-rr 81.1%

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

   7. Taylor expanded in i around 0 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-237}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-257}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-94}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-36}:\\ \;\;\;\;100 \cdot \left(\log \left(\frac{i}{n}\right) \cdot \left(n \cdot \frac{n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.8e-237)
   (* 100.0 (/ n (/ i (expm1 i))))
   (if (<= n 2.2e-257)
     0.0
     (if (<= n 3.2e-94)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 1.95e-36)
         (* 100.0 (* (log (/ i n)) (* n (/ n i))))
         (* (* n 100.0) (/ (expm1 i) i)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.8e-237) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (n <= 2.2e-257) {
		tmp = 0.0;
	} else if (n <= 3.2e-94) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.95e-36) {
		tmp = 100.0 * (log((i / n)) * (n * (n / i)));
	} else {
		tmp = (n * 100.0) * (expm1(i) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.8e-237) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (n <= 2.2e-257) {
		tmp = 0.0;
	} else if (n <= 3.2e-94) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.95e-36) {
		tmp = 100.0 * (Math.log((i / n)) * (n * (n / i)));
	} else {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.8e-237:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif n <= 2.2e-257:
		tmp = 0.0
	elif n <= 3.2e-94:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.95e-36:
		tmp = 100.0 * (math.log((i / n)) * (n * (n / i)))
	else:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.8e-237)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (n <= 2.2e-257)
		tmp = 0.0;
	elseif (n <= 3.2e-94)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.95e-36)
		tmp = Float64(100.0 * Float64(log(Float64(i / n)) * Float64(n * Float64(n / i))));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.8e-237], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.2e-257], 0.0, If[LessEqual[n, 3.2e-94], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e-36], N[(100.0 * N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * N[(n * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -4.8e-237

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/l* 32.8%

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

      3. expm1-def 84.3%

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

   4. Simplified 84.3%

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

   if -4.8e-237 < n < 2.19999999999999988e-257

   1. Initial program 79.5%

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

      1. associate-*r/ 79.5%

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

      2. sub-neg 79.5%

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

      3. distribute-lft-in 79.5%

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

      4. fma-def 79.5%

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

      5. metadata-eval 79.5%

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

      6. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in n around inf 68.2%

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

   5. Taylor expanded in i around 0 85.7%

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

   6. Taylor expanded in i around 0 85.7%

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

   if 2.19999999999999988e-257 < n < 3.19999999999999997e-94

   1. Initial program 10.0%

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

   2. Taylor expanded in i around 0 80.2%

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

   if 3.19999999999999997e-94 < n < 1.95e-36

   1. Initial program 17.6%

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

      1. associate-*r/ 17.6%

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

      2. sub-neg 17.6%

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

      3. distribute-lft-in 17.6%

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

      4. fma-def 17.6%

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

      5. metadata-eval 17.6%

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

      6. metadata-eval 17.6%

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

   3. Simplified 17.6%

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

      1. fma-udef 17.6%

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

      2. *-commutative 17.6%

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

   5. Applied egg-rr 17.6%

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

   6. Taylor expanded in n around 0 63.9%

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

      1. mul-1-neg 63.9%

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

      2. log-rec 63.9%

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

      3. +-commutative 63.9%

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

      4. log-rec 63.9%

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

      5. unsub-neg 63.9%

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

      6. log-div 63.9%

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

   8. Simplified 63.9%

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

   9. Taylor expanded in n around 0 63.6%

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

       1. associate-/l* 63.5%

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

       2. unpow2 63.5%

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

       3. mul-1-neg 63.5%

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

       4. log-rec 63.5%

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

       5. +-commutative 63.5%

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

       6. log-rec 63.5%

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

       7. sub-neg 63.5%

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

       8. log-div 63.5%

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

       9. associate-*l/ 63.5%

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

       10. associate-/l* 63.9%

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

       11. associate-*l/ 63.9%

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

       12. associate-*r/ 64.0%

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

       13. *-commutative 64.0%

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

       14. associate-*l* 63.9%

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

   11. Simplified 63.9%

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

   if 1.95e-36 < n

   1. Initial program 20.8%

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

      1. *-commutative 20.8%

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

      2. associate-/r/ 21.2%

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

      3. associate-*l* 21.2%

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

      4. sub-neg 21.2%

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

      5. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in n around inf 32.0%

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

      1. expm1-def 93.4%

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

   6. Simplified 93.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-237}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-257}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-94}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-36}:\\ \;\;\;\;100 \cdot \left(\log \left(\frac{i}{n}\right) \cdot \left(n \cdot \frac{n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]

Alternative 6: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.45 \cdot 10^{+85} \lor \neg \left(i \leq 7 \cdot 10^{+106}\right):\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.45e+85) (not (<= i 7e+106)))
   (* 100.0 (/ (+ -1.0 (pow (/ i n) n)) (/ i n)))
   (/ 1.0 (/ 0.01 (/ n (/ i (expm1 i)))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.45e+85) || !(i <= 7e+106)) {
		tmp = 100.0 * ((-1.0 + pow((i / n), n)) / (i / n));
	} else {
		tmp = 1.0 / (0.01 / (n / (i / expm1(i))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.45e+85) || !(i <= 7e+106)) {
		tmp = 100.0 * ((-1.0 + Math.pow((i / n), n)) / (i / n));
	} else {
		tmp = 1.0 / (0.01 / (n / (i / Math.expm1(i))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.45e+85) or not (i <= 7e+106):
		tmp = 100.0 * ((-1.0 + math.pow((i / n), n)) / (i / n))
	else:
		tmp = 1.0 / (0.01 / (n / (i / math.expm1(i))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.45e+85) || !(i <= 7e+106))
		tmp = Float64(100.0 * Float64(Float64(-1.0 + (Float64(i / n) ^ n)) / Float64(i / n)));
	else
		tmp = Float64(1.0 / Float64(0.01 / Float64(n / Float64(i / expm1(i)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.45e+85], N[Not[LessEqual[i, 7e+106]], $MachinePrecision]], N[(100.0 * N[(N[(-1.0 + N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.01 / N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.44999999999999999e85 or 6.99999999999999962e106 < i

   1. Initial program 67.0%

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

   2. Taylor expanded in i around inf 76.3%

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

   if -1.44999999999999999e85 < i < 6.99999999999999962e106

   1. Initial program 10.8%

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

   2. Taylor expanded in n around inf 19.3%

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

      1. *-commutative 19.3%

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

      2. associate-/l* 19.3%

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

      3. expm1-def 86.4%

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

   4. Simplified 86.4%

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

      1. associate-*l/ 86.4%

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

   6. Applied egg-rr 86.4%

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

   7. Taylor expanded in n around 0 19.3%

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

      1. *-commutative 19.3%

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

      2. expm1-def 79.1%

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

      3. associate-/l* 86.4%

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

      4. associate-/r/ 86.3%

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

      5. associate-/l/ 86.2%

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

   9. Simplified 86.2%

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

       1. clear-num 86.4%

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

       2. inv-pow 86.4%

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

       3. *-un-lft-identity 86.4%

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

       4. times-frac 86.4%

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

       5. metadata-eval 86.4%

          \[\leadsto {\left(\frac{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}\right)}^{-1} \]

   11. Applied egg-rr 86.4%

       \[\leadsto \color{blue}{{\left(\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 86.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}}} \]

       2. associate-/l* 86.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}} \]

   13. Simplified 86.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.45 \cdot 10^{+85} \lor \neg \left(i \leq 7 \cdot 10^{+106}\right):\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}\\ \end{array} \]

Alternative 7: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + {\left(\frac{i}{n}\right)}^{n}\\ \mathbf{if}\;i \leq -8 \cdot 10^{+84}:\\ \;\;\;\;100 \cdot \frac{t_0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{t_0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ -1.0 (pow (/ i n) n))))
   (if (<= i -8e+84)
     (* 100.0 (/ t_0 (/ i n)))
     (if (<= i 2.8e+108)
       (/ 1.0 (/ 0.01 (/ n (/ i (expm1 i)))))
       (* (* n 100.0) (/ t_0 i))))))

C

double code(double i, double n) {
	double t_0 = -1.0 + pow((i / n), n);
	double tmp;
	if (i <= -8e+84) {
		tmp = 100.0 * (t_0 / (i / n));
	} else if (i <= 2.8e+108) {
		tmp = 1.0 / (0.01 / (n / (i / expm1(i))));
	} else {
		tmp = (n * 100.0) * (t_0 / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = -1.0 + Math.pow((i / n), n);
	double tmp;
	if (i <= -8e+84) {
		tmp = 100.0 * (t_0 / (i / n));
	} else if (i <= 2.8e+108) {
		tmp = 1.0 / (0.01 / (n / (i / Math.expm1(i))));
	} else {
		tmp = (n * 100.0) * (t_0 / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = -1.0 + math.pow((i / n), n)
	tmp = 0
	if i <= -8e+84:
		tmp = 100.0 * (t_0 / (i / n))
	elif i <= 2.8e+108:
		tmp = 1.0 / (0.01 / (n / (i / math.expm1(i))))
	else:
		tmp = (n * 100.0) * (t_0 / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(-1.0 + (Float64(i / n) ^ n))
	tmp = 0.0
	if (i <= -8e+84)
		tmp = Float64(100.0 * Float64(t_0 / Float64(i / n)));
	elseif (i <= 2.8e+108)
		tmp = Float64(1.0 / Float64(0.01 / Float64(n / Float64(i / expm1(i)))));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(t_0 / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(-1.0 + N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -8e+84], N[(100.0 * N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.8e+108], N[(1.0 / N[(0.01 / N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if i < -8.00000000000000046e84

   1. Initial program 85.7%

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

   2. Taylor expanded in i around inf 88.6%

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

   if -8.00000000000000046e84 < i < 2.7999999999999998e108

   1. Initial program 10.8%

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

   2. Taylor expanded in n around inf 19.3%

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

      1. *-commutative 19.3%

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

      2. associate-/l* 19.3%

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

      3. expm1-def 86.4%

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

   4. Simplified 86.4%

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

      1. associate-*l/ 86.4%

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

   6. Applied egg-rr 86.4%

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

   7. Taylor expanded in n around 0 19.3%

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

      1. *-commutative 19.3%

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

      2. expm1-def 79.1%

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

      3. associate-/l* 86.4%

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

      4. associate-/r/ 86.3%

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

      5. associate-/l/ 86.2%

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

   9. Simplified 86.2%

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

       1. clear-num 86.4%

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

       2. inv-pow 86.4%

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

       3. *-un-lft-identity 86.4%

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

       4. times-frac 86.4%

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

       5. metadata-eval 86.4%

          \[\leadsto {\left(\frac{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}\right)}^{-1} \]

   11. Applied egg-rr 86.4%

       \[\leadsto \color{blue}{{\left(\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 86.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}{n}}} \]

       2. associate-/l* 86.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}} \]

   13. Simplified 86.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}} \]

   if 2.7999999999999998e108 < i

   1. Initial program 51.9%

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

      1. *-commutative 51.9%

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

      2. associate-/r/ 52.4%

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

      3. associate-*l* 52.4%

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

      4. sub-neg 52.4%

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

      5. metadata-eval 52.4%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in i around inf 66.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8 \cdot 10^{+84}:\\ \;\;\;\;100 \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{\frac{0.01}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{-1 + {\left(\frac{i}{n}\right)}^{n}}{i}\\ \end{array} \]

Alternative 8: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 100\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 2.55 \cdot 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{n}{i} \cdot -200\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -1.5e-8)
     t_0
     (if (<= i 2.1e-36)
       (* n (+ 100.0 (* (* i 100.0) (- 0.5 (/ 0.5 n)))))
       (if (<= i 2.55e+242) t_0 (fabs (* (/ n i) -200.0)))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -1.5e-8) {
		tmp = t_0;
	} else if (i <= 2.1e-36) {
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))));
	} else if (i <= 2.55e+242) {
		tmp = t_0;
	} else {
		tmp = fabs(((n / i) * -200.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -1.5e-8) {
		tmp = t_0;
	} else if (i <= 2.1e-36) {
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))));
	} else if (i <= 2.55e+242) {
		tmp = t_0;
	} else {
		tmp = Math.abs(((n / i) * -200.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -1.5e-8:
		tmp = t_0
	elif i <= 2.1e-36:
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))))
	elif i <= 2.55e+242:
		tmp = t_0
	else:
		tmp = math.fabs(((n / i) * -200.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -1.5e-8)
		tmp = t_0;
	elseif (i <= 2.1e-36)
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i * 100.0) * Float64(0.5 - Float64(0.5 / n)))));
	elseif (i <= 2.55e+242)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(n / i) * -200.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.5e-8], t$95$0, If[LessEqual[i, 2.1e-36], N[(n * N[(100.0 + N[(N[(i * 100.0), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.55e+242], t$95$0, N[Abs[N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.49999999999999987e-8 or 2.09999999999999991e-36 < i < 2.54999999999999998e242

   1. Initial program 55.4%

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

   2. Taylor expanded in n around inf 62.3%

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

      1. expm1-def 65.4%

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

   4. Simplified 65.4%

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

   if -1.49999999999999987e-8 < i < 2.09999999999999991e-36

   1. Initial program 6.8%

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

      1. associate-/r/ 7.3%

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

      2. associate-*r* 7.3%

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

      3. *-commutative 7.3%

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

      4. associate-*r/ 7.3%

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

      5. sub-neg 7.3%

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

      6. distribute-lft-in 7.3%

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

      7. fma-def 7.3%

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

      8. metadata-eval 7.3%

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

      9. metadata-eval 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in i around 0 89.7%

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

      1. associate-*r* 89.7%

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

      2. *-commutative 89.7%

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

      3. associate-*r/ 89.7%

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

      4. metadata-eval 89.7%

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

   6. Simplified 89.7%

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

   if 2.54999999999999998e242 < i

   1. Initial program 45.0%

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

   2. Taylor expanded in n around inf 19.3%

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

      1. *-commutative 19.3%

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

      2. associate-/l* 19.3%

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

      3. expm1-def 19.3%

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

   4. Simplified 19.3%

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

   5. Taylor expanded in i around 0 59.8%

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

      1. *-commutative 59.8%

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

   7. Simplified 59.8%

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

   8. Taylor expanded in i around inf 59.8%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 59.7%

         \[\leadsto \color{blue}{\sqrt{-200 \cdot \frac{n}{i}} \cdot \sqrt{-200 \cdot \frac{n}{i}}} \]

      2. sqrt-unprod 59.0%

         \[\leadsto \color{blue}{\sqrt{\left(-200 \cdot \frac{n}{i}\right) \cdot \left(-200 \cdot \frac{n}{i}\right)}} \]

      3. pow2 59.0%

         \[\leadsto \sqrt{\color{blue}{{\left(-200 \cdot \frac{n}{i}\right)}^{2}}} \]

   10. Applied egg-rr 59.0%

       \[\leadsto \color{blue}{\sqrt{{\left(-200 \cdot \frac{n}{i}\right)}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 59.0%

          \[\leadsto \sqrt{\color{blue}{\left(-200 \cdot \frac{n}{i}\right) \cdot \left(-200 \cdot \frac{n}{i}\right)}} \]

       2. rem-sqrt-square 60.1%

          \[\leadsto \color{blue}{\left|-200 \cdot \frac{n}{i}\right|} \]

       3. *-commutative 60.1%

          \[\leadsto \left|\color{blue}{\frac{n}{i} \cdot -200}\right| \]

   12. Simplified 60.1%

       \[\leadsto \color{blue}{\left|\frac{n}{i} \cdot -200\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 100\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 2.55 \cdot 10^{+242}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{n}{i} \cdot -200\right|\\ \end{array} \]

Alternative 9: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.3 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.02 \cdot 10^{-33}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 100\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \left(n \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{n}{i} \cdot -200\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.3e-8)
   (* 100.0 (/ (expm1 i) (/ i n)))
   (if (<= i 1.02e-33)
     (* n (+ 100.0 (* (* i 100.0) (- 0.5 (/ 0.5 n)))))
     (if (<= i 2.9e+242)
       (* (expm1 i) (* n (/ 100.0 i)))
       (fabs (* (/ n i) -200.0))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.3e-8) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (i <= 1.02e-33) {
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))));
	} else if (i <= 2.9e+242) {
		tmp = expm1(i) * (n * (100.0 / i));
	} else {
		tmp = fabs(((n / i) * -200.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -4.3e-8) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (i <= 1.02e-33) {
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))));
	} else if (i <= 2.9e+242) {
		tmp = Math.expm1(i) * (n * (100.0 / i));
	} else {
		tmp = Math.abs(((n / i) * -200.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -4.3e-8:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif i <= 1.02e-33:
		tmp = n * (100.0 + ((i * 100.0) * (0.5 - (0.5 / n))))
	elif i <= 2.9e+242:
		tmp = math.expm1(i) * (n * (100.0 / i))
	else:
		tmp = math.fabs(((n / i) * -200.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.3e-8)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (i <= 1.02e-33)
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i * 100.0) * Float64(0.5 - Float64(0.5 / n)))));
	elseif (i <= 2.9e+242)
		tmp = Float64(expm1(i) * Float64(n * Float64(100.0 / i)));
	else
		tmp = abs(Float64(Float64(n / i) * -200.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.3e-8], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.02e-33], N[(n * N[(100.0 + N[(N[(i * 100.0), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.9e+242], N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -4.3000000000000001e-8

   1. Initial program 69.0%

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

   2. Taylor expanded in n around inf 72.8%

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

      1. expm1-def 73.3%

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

   4. Simplified 73.3%

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

   if -4.3000000000000001e-8 < i < 1.02e-33

   1. Initial program 6.8%

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

      1. associate-/r/ 7.3%

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

      2. associate-*r* 7.3%

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

      3. *-commutative 7.3%

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

      4. associate-*r/ 7.3%

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

      5. sub-neg 7.3%

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

      6. distribute-lft-in 7.3%

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

      7. fma-def 7.3%

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

      8. metadata-eval 7.3%

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

      9. metadata-eval 7.3%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in i around 0 89.8%

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

      1. associate-*r* 89.8%

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

      2. *-commutative 89.8%

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

      3. associate-*r/ 89.8%

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

      4. metadata-eval 89.8%

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

   6. Simplified 89.8%

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

   if 1.02e-33 < i < 2.89999999999999997e242

   1. Initial program 42.0%

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

   2. Taylor expanded in n around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-/l* 52.5%

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

      3. expm1-def 56.4%

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

   4. Simplified 56.4%

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

      1. associate-*l/ 56.4%

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

   6. Applied egg-rr 56.4%

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

   7. Taylor expanded in n around 0 52.5%

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

      1. *-commutative 52.5%

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

      2. expm1-def 56.4%

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

      3. associate-/l* 56.4%

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

      4. associate-/r/ 56.3%

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

      5. associate-/l/ 56.3%

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

   9. Simplified 56.3%

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

   10. Taylor expanded in n around 0 52.5%

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

       1. associate-*r/ 52.5%

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

       2. expm1-def 56.3%

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

       3. *-commutative 56.3%

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

       4. associate-*r* 56.4%

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

       5. associate-*r/ 56.2%

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

       6. *-commutative 56.2%

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

       7. associate-*r* 56.3%

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

       8. associate-*l/ 56.3%

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

       9. *-commutative 56.3%

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

       10. associate-*l/ 56.3%

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

   12. Simplified 56.3%

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

   if 2.89999999999999997e242 < i

   1. Initial program 45.0%

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

   2. Taylor expanded in n around inf 19.3%

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

      1. *-commutative 19.3%

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

      2. associate-/l* 19.3%

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

      3. expm1-def 19.3%

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

   4. Simplified 19.3%

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

   5. Taylor expanded in i around 0 59.8%

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

      1. *-commutative 59.8%

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

   7. Simplified 59.8%

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

   8. Taylor expanded in i around inf 59.8%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 59.7%

         \[\leadsto \color{blue}{\sqrt{-200 \cdot \frac{n}{i}} \cdot \sqrt{-200 \cdot \frac{n}{i}}} \]

      2. sqrt-unprod 59.0%

         \[\leadsto \color{blue}{\sqrt{\left(-200 \cdot \frac{n}{i}\right) \cdot \left(-200 \cdot \frac{n}{i}\right)}} \]

      3. pow2 59.0%

         \[\leadsto \sqrt{\color{blue}{{\left(-200 \cdot \frac{n}{i}\right)}^{2}}} \]

   10. Applied egg-rr 59.0%

       \[\leadsto \color{blue}{\sqrt{{\left(-200 \cdot \frac{n}{i}\right)}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 59.0%

          \[\leadsto \sqrt{\color{blue}{\left(-200 \cdot \frac{n}{i}\right) \cdot \left(-200 \cdot \frac{n}{i}\right)}} \]

       2. rem-sqrt-square 60.1%

          \[\leadsto \color{blue}{\left|-200 \cdot \frac{n}{i}\right|} \]

       3. *-commutative 60.1%

          \[\leadsto \left|\color{blue}{\frac{n}{i} \cdot -200}\right| \]

   12. Simplified 60.1%

       \[\leadsto \color{blue}{\left|\frac{n}{i} \cdot -200\right|} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.3 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.02 \cdot 10^{-33}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 100\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \left(n \cdot \frac{100}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{n}{i} \cdot -200\right|\\ \end{array} \]

Alternative 10: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{-229} \lor \neg \left(n \leq 3.4 \cdot 10^{-213}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.4e-229) (not (<= n 3.4e-213)))
   (* 100.0 (/ n (/ i (expm1 i))))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.4e-229) || !(n <= 3.4e-213)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.4e-229) || !(n <= 3.4e-213)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.4e-229) or not (n <= 3.4e-213):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.4e-229) || !(n <= 3.4e-213))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.4e-229], N[Not[LessEqual[n, 3.4e-213]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.4e-229 or 3.4000000000000002e-213 < n

   1. Initial program 24.1%

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

   2. Taylor expanded in n around inf 27.6%

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

      1. *-commutative 27.6%

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

      2. associate-/l* 27.6%

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

      3. expm1-def 82.0%

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

   4. Simplified 82.0%

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

   if -2.4e-229 < n < 3.4000000000000002e-213

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

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

   5. Taylor expanded in i around 0 89.1%

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

   6. Taylor expanded in i around 0 89.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{-229} \lor \neg \left(n \leq 3.4 \cdot 10^{-213}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-226}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.5e-226)
   (* 100.0 (/ n (/ i (expm1 i))))
   (if (<= n 3.4e-213) 0.0 (* (* n 100.0) (/ (expm1 i) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.5e-226) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (n <= 3.4e-213) {
		tmp = 0.0;
	} else {
		tmp = (n * 100.0) * (expm1(i) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.5e-226) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (n <= 3.4e-213) {
		tmp = 0.0;
	} else {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.5e-226:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif n <= 3.4e-213:
		tmp = 0.0
	else:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.5e-226)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (n <= 3.4e-213)
		tmp = 0.0;
	else
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.5e-226], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.4e-213], 0.0, N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.5000000000000007e-226

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. associate-/l* 32.8%

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

      3. expm1-def 84.3%

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

   4. Simplified 84.3%

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

   if -9.5000000000000007e-226 < n < 3.4000000000000002e-213

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

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

   5. Taylor expanded in i around 0 89.1%

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

   6. Taylor expanded in i around 0 89.1%

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

   if 3.4000000000000002e-213 < n

   1. Initial program 18.5%

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

      1. *-commutative 18.5%

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

      2. associate-/r/ 18.8%

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

      3. associate-*l* 18.8%

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

      4. sub-neg 18.8%

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

      5. metadata-eval 18.8%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in n around inf 22.3%

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

      1. expm1-def 79.7%

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

   6. Simplified 79.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-226}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]

Alternative 12: 67.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n \cdot 100}{1 + i \cdot -0.5}\\ t_1 := 0.5 - \frac{0.5}{n}\\ t_2 := i \cdot \left(n \cdot t_1\right)\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{+198}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-236}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-260}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 6.7 \cdot 10^{+143}:\\ \;\;\;\;100 \cdot \frac{n \cdot n - t_2 \cdot t_2}{n - t_2}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 100\right) \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* n 100.0) (+ 1.0 (* i -0.5))))
        (t_1 (- 0.5 (/ 0.5 n)))
        (t_2 (* i (* n t_1))))
   (if (<= n -2.5e+198)
     (* n (+ 100.0 (* 100.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
     (if (<= n -1.65e-236)
       t_0
       (if (<= n 5e-260)
         0.0
         (if (<= n 9.2e-25)
           t_0
           (if (<= n 6.7e+143)
             (* 100.0 (/ (- (* n n) (* t_2 t_2)) (- n t_2)))
             (* n (+ 100.0 (* (* i 100.0) t_1))))))))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) / (1.0 + (i * -0.5));
	double t_1 = 0.5 - (0.5 / n);
	double t_2 = i * (n * t_1);
	double tmp;
	if (n <= -2.5e+198) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -1.65e-236) {
		tmp = t_0;
	} else if (n <= 5e-260) {
		tmp = 0.0;
	} else if (n <= 9.2e-25) {
		tmp = t_0;
	} else if (n <= 6.7e+143) {
		tmp = 100.0 * (((n * n) - (t_2 * t_2)) / (n - t_2));
	} else {
		tmp = n * (100.0 + ((i * 100.0) * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    t_1 = 0.5d0 - (0.5d0 / n)
    t_2 = i * (n * t_1)
    if (n <= (-2.5d+198)) then
        tmp = n * (100.0d0 + (100.0d0 * (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= (-1.65d-236)) then
        tmp = t_0
    else if (n <= 5d-260) then
        tmp = 0.0d0
    else if (n <= 9.2d-25) then
        tmp = t_0
    else if (n <= 6.7d+143) then
        tmp = 100.0d0 * (((n * n) - (t_2 * t_2)) / (n - t_2))
    else
        tmp = n * (100.0d0 + ((i * 100.0d0) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) / (1.0 + (i * -0.5));
	double t_1 = 0.5 - (0.5 / n);
	double t_2 = i * (n * t_1);
	double tmp;
	if (n <= -2.5e+198) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -1.65e-236) {
		tmp = t_0;
	} else if (n <= 5e-260) {
		tmp = 0.0;
	} else if (n <= 9.2e-25) {
		tmp = t_0;
	} else if (n <= 6.7e+143) {
		tmp = 100.0 * (((n * n) - (t_2 * t_2)) / (n - t_2));
	} else {
		tmp = n * (100.0 + ((i * 100.0) * t_1));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) / (1.0 + (i * -0.5))
	t_1 = 0.5 - (0.5 / n)
	t_2 = i * (n * t_1)
	tmp = 0
	if n <= -2.5e+198:
		tmp = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= -1.65e-236:
		tmp = t_0
	elif n <= 5e-260:
		tmp = 0.0
	elif n <= 9.2e-25:
		tmp = t_0
	elif n <= 6.7e+143:
		tmp = 100.0 * (((n * n) - (t_2 * t_2)) / (n - t_2))
	else:
		tmp = n * (100.0 + ((i * 100.0) * t_1))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)))
	t_1 = Float64(0.5 - Float64(0.5 / n))
	t_2 = Float64(i * Float64(n * t_1))
	tmp = 0.0
	if (n <= -2.5e+198)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= -1.65e-236)
		tmp = t_0;
	elseif (n <= 5e-260)
		tmp = 0.0;
	elseif (n <= 9.2e-25)
		tmp = t_0;
	elseif (n <= 6.7e+143)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * n) - Float64(t_2 * t_2)) / Float64(n - t_2)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i * 100.0) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n * 100.0) / (1.0 + (i * -0.5));
	t_1 = 0.5 - (0.5 / n);
	t_2 = i * (n * t_1);
	tmp = 0.0;
	if (n <= -2.5e+198)
		tmp = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= -1.65e-236)
		tmp = t_0;
	elseif (n <= 5e-260)
		tmp = 0.0;
	elseif (n <= 9.2e-25)
		tmp = t_0;
	elseif (n <= 6.7e+143)
		tmp = 100.0 * (((n * n) - (t_2 * t_2)) / (n - t_2));
	else
		tmp = n * (100.0 + ((i * 100.0) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.5e+198], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.65e-236], t$95$0, If[LessEqual[n, 5e-260], 0.0, If[LessEqual[n, 9.2e-25], t$95$0, If[LessEqual[n, 6.7e+143], N[(100.0 * N[(N[(N[(n * n), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(n - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(N[(i * 100.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -2.50000000000000024e198

   1. Initial program 11.4%

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

      1. associate-/r/ 12.2%

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

      2. associate-*r* 12.2%

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

      3. *-commutative 12.2%

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

      4. associate-*r/ 12.2%

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

      5. sub-neg 12.2%

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

      6. distribute-lft-in 12.2%

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

      7. fma-def 12.2%

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

      8. metadata-eval 12.2%

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

      9. metadata-eval 12.2%

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

   3. Simplified 12.2%

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

   4. Taylor expanded in i around 0 89.6%

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

      1. distribute-lft-out 89.6%

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

      2. associate-*r/ 89.6%

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

      3. metadata-eval 89.6%

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

      4. unpow2 89.6%

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

      5. associate--l+ 89.6%

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

      6. associate-*r/ 89.6%

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

      7. metadata-eval 89.6%

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

      8. unpow2 89.6%

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

      9. associate-*r/ 89.6%

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

      10. metadata-eval 89.6%

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

   6. Simplified 89.6%

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

   7. Taylor expanded in n around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. unpow2 89.6%

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

   9. Simplified 89.6%

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

   10. Taylor expanded in n around inf 89.6%

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

       1. +-commutative 89.6%

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

       2. *-commutative 89.6%

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

       3. *-commutative 89.6%

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

       4. unpow2 89.6%

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

       5. associate-*r* 89.6%

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

       6. distribute-lft-out 89.6%

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

   12. Simplified 89.6%

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

   if -2.50000000000000024e198 < n < -1.6500000000000001e-236 or 5.0000000000000003e-260 < n < 9.1999999999999997e-25

   1. Initial program 27.7%

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

   2. Taylor expanded in n around inf 21.8%

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

      1. *-commutative 21.8%

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

      2. associate-/l* 21.8%

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

      3. expm1-def 69.6%

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

   4. Simplified 69.6%

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

      1. associate-*l/ 69.6%

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

   6. Applied egg-rr 69.6%

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

   7. Taylor expanded in i around 0 64.4%

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

      1. *-commutative 64.4%

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

   9. Simplified 64.4%

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

   if -1.6500000000000001e-236 < n < 5.0000000000000003e-260

   1. Initial program 79.5%

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

      1. associate-*r/ 79.5%

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

      2. sub-neg 79.5%

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

      3. distribute-lft-in 79.5%

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

      4. fma-def 79.5%

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

      5. metadata-eval 79.5%

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

      6. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in n around inf 68.2%

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

   5. Taylor expanded in i around 0 85.7%

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

   6. Taylor expanded in i around 0 85.7%

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

   if 9.1999999999999997e-25 < n < 6.7000000000000002e143

   1. Initial program 25.9%

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

   2. Taylor expanded in i around 0 70.7%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*r/ 70.7%

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

      4. metadata-eval 70.7%

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

   4. Simplified 70.7%

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

      1. flip-+ 90.0%

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

      2. associate-*l* 90.0%

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

      3. associate-*l* 90.0%

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

      4. associate-*l* 90.0%

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

   6. Applied egg-rr 90.0%

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

   if 6.7000000000000002e143 < n

   1. Initial program 17.7%

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

      1. associate-/r/ 18.3%

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

      2. associate-*r* 18.3%

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

      3. *-commutative 18.3%

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

      4. associate-*r/ 18.3%

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

      5. sub-neg 18.3%

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

      6. distribute-lft-in 18.3%

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

      7. fma-def 18.3%

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

      8. metadata-eval 18.3%

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

      9. metadata-eval 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in i around 0 82.8%

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

      1. associate-*r* 82.8%

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

      2. *-commutative 82.8%

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

      3. associate-*r/ 82.8%

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

      4. metadata-eval 82.8%

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

   6. Simplified 82.8%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(i \cdot 100\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{+198}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-260}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 6.7 \cdot 10^{+143}:\\ \;\;\;\;100 \cdot \frac{n \cdot n - \left(i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right) \cdot \left(i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{n - i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 100\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]

Alternative 13: 65.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+198}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-225}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot i\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.5e+198)
   (* n (+ 100.0 (* 100.0 (* i (+ 0.5 (* i 0.16666666666666666))))))
   (if (<= n -4.5e-225)
     (/ (* n 100.0) (+ 1.0 (* i -0.5)))
     (if (<= n 3e-213)
       0.0
       (* n (/ (* 100.0 (+ i (* (- 0.5 (/ 0.5 n)) (* i i)))) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.5e+198) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -4.5e-225) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 3e-213) {
		tmp = 0.0;
	} else {
		tmp = n * ((100.0 * (i + ((0.5 - (0.5 / n)) * (i * i)))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.5d+198)) then
        tmp = n * (100.0d0 + (100.0d0 * (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    else if (n <= (-4.5d-225)) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    else if (n <= 3d-213) then
        tmp = 0.0d0
    else
        tmp = n * ((100.0d0 * (i + ((0.5d0 - (0.5d0 / n)) * (i * i)))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.5e+198) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	} else if (n <= -4.5e-225) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 3e-213) {
		tmp = 0.0;
	} else {
		tmp = n * ((100.0 * (i + ((0.5 - (0.5 / n)) * (i * i)))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.5e+198:
		tmp = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))))
	elif n <= -4.5e-225:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	elif n <= 3e-213:
		tmp = 0.0
	else:
		tmp = n * ((100.0 * (i + ((0.5 - (0.5 / n)) * (i * i)))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.5e+198)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))));
	elseif (n <= -4.5e-225)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	elseif (n <= 3e-213)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(Float64(100.0 * Float64(i + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * i)))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.5e+198)
		tmp = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	elseif (n <= -4.5e-225)
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	elseif (n <= 3e-213)
		tmp = 0.0;
	else
		tmp = n * ((100.0 * (i + ((0.5 - (0.5 / n)) * (i * i)))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.5e+198], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -4.5e-225], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3e-213], 0.0, N[(n * N[(N[(100.0 * N[(i + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.5e198

   1. Initial program 11.4%

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

      1. associate-/r/ 12.2%

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

      2. associate-*r* 12.2%

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

      3. *-commutative 12.2%

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

      4. associate-*r/ 12.2%

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

      5. sub-neg 12.2%

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

      6. distribute-lft-in 12.2%

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

      7. fma-def 12.2%

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

      8. metadata-eval 12.2%

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

      9. metadata-eval 12.2%

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

   3. Simplified 12.2%

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

   4. Taylor expanded in i around 0 89.6%

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

      1. distribute-lft-out 89.6%

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

      2. associate-*r/ 89.6%

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

      3. metadata-eval 89.6%

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

      4. unpow2 89.6%

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

      5. associate--l+ 89.6%

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

      6. associate-*r/ 89.6%

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

      7. metadata-eval 89.6%

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

      8. unpow2 89.6%

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

      9. associate-*r/ 89.6%

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

      10. metadata-eval 89.6%

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

   6. Simplified 89.6%

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

   7. Taylor expanded in n around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. unpow2 89.6%

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

   9. Simplified 89.6%

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

   10. Taylor expanded in n around inf 89.6%

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

       1. +-commutative 89.6%

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

       2. *-commutative 89.6%

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

       3. *-commutative 89.6%

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

       4. unpow2 89.6%

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

       5. associate-*r* 89.6%

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

       6. distribute-lft-out 89.6%

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

   12. Simplified 89.6%

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

   if -9.5e198 < n < -4.5e-225

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/l* 31.0%

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

      3. expm1-def 80.6%

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

   4. Simplified 80.6%

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

      1. associate-*l/ 80.6%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in i around 0 63.7%

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

      1. *-commutative 63.7%

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

   9. Simplified 63.7%

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

   if -4.5e-225 < n < 2.99999999999999986e-213

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

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

   5. Taylor expanded in i around 0 89.1%

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

   6. Taylor expanded in i around 0 89.1%

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

   if 2.99999999999999986e-213 < n

   1. Initial program 18.5%

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

      1. associate-/r/ 18.8%

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

      2. associate-*r* 18.8%

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

      3. *-commutative 18.8%

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

      4. associate-*r/ 18.8%

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

      5. sub-neg 18.8%

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

      6. distribute-lft-in 18.8%

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

      7. fma-def 18.8%

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

      8. metadata-eval 18.8%

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

      9. metadata-eval 18.8%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in i around 0 71.0%

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

      1. distribute-lft-out 71.0%

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

      2. unpow2 71.0%

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

      3. associate-*r/ 71.0%

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

      4. metadata-eval 71.0%

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

   6. Simplified 71.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+198}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-225}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot i\right)\right)}{i}\\ \end{array} \]

Alternative 14: 65.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;n \leq -5.5 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (* n (+ 100.0 (* 100.0 (* i (+ 0.5 (* i 0.16666666666666666))))))))
   (if (<= n -5.5e+198)
     t_0
     (if (<= n -5.2e-232)
       (/ (* n 100.0) (+ 1.0 (* i -0.5)))
       (if (<= n 3.4e-213) 0.0 t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	double tmp;
	if (n <= -5.5e+198) {
		tmp = t_0;
	} else if (n <= -5.2e-232) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 3.4e-213) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (100.0d0 * (i * (0.5d0 + (i * 0.16666666666666666d0)))))
    if (n <= (-5.5d+198)) then
        tmp = t_0
    else if (n <= (-5.2d-232)) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    else if (n <= 3.4d-213) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	double tmp;
	if (n <= -5.5e+198) {
		tmp = t_0;
	} else if (n <= -5.2e-232) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 3.4e-213) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))))
	tmp = 0
	if n <= -5.5e+198:
		tmp = t_0
	elif n <= -5.2e-232:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	elif n <= 3.4e-213:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))))
	tmp = 0.0
	if (n <= -5.5e+198)
		tmp = t_0;
	elseif (n <= -5.2e-232)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	elseif (n <= 3.4e-213)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (100.0 * (i * (0.5 + (i * 0.16666666666666666)))));
	tmp = 0.0;
	if (n <= -5.5e+198)
		tmp = t_0;
	elseif (n <= -5.2e-232)
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	elseif (n <= 3.4e-213)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.5e+198], t$95$0, If[LessEqual[n, -5.2e-232], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.4e-213], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.5000000000000004e198 or 3.4000000000000002e-213 < n

   1. Initial program 17.1%

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

      1. associate-/r/ 17.5%

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

      2. associate-*r* 17.5%

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

      3. *-commutative 17.5%

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

      4. associate-*r/ 17.5%

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

      5. sub-neg 17.5%

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

      6. distribute-lft-in 17.5%

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

      7. fma-def 17.5%

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

      8. metadata-eval 17.5%

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

      9. metadata-eval 17.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in i around 0 69.8%

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

      1. distribute-lft-out 69.8%

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

      2. associate-*r/ 69.8%

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

      3. metadata-eval 69.8%

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

      4. unpow2 69.8%

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

      5. associate--l+ 69.8%

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

      6. associate-*r/ 69.8%

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

      7. metadata-eval 69.8%

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

      8. unpow2 69.8%

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

      9. associate-*r/ 69.8%

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

      10. metadata-eval 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in n around inf 73.9%

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

      1. *-commutative 73.9%

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

      2. unpow2 73.9%

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

   9. Simplified 73.9%

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

   10. Taylor expanded in n around inf 74.0%

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

       1. +-commutative 74.0%

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

       2. *-commutative 74.0%

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

       3. *-commutative 74.0%

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

       4. unpow2 74.0%

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

       5. associate-*r* 74.0%

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

       6. distribute-lft-out 74.0%

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

   12. Simplified 74.0%

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

   if -5.5000000000000004e198 < n < -5.19999999999999992e-232

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/l* 31.0%

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

      3. expm1-def 80.6%

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

   4. Simplified 80.6%

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

      1. associate-*l/ 80.6%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in i around 0 63.7%

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

      1. *-commutative 63.7%

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

   9. Simplified 63.7%

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

   if -5.19999999999999992e-232 < n < 3.4000000000000002e-213

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

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

   5. Taylor expanded in i around 0 89.1%

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

   6. Taylor expanded in i around 0 89.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{+198}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]

Alternative 15: 63.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.4 \cdot 10^{-227}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-215}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (+ n (* 0.5 (* i n))))))
   (if (<= n -3.6e+198)
     t_0
     (if (<= n -5.4e-227)
       (/ n (+ 0.01 (* i -0.005)))
       (if (<= n 9.5e-215) 0.0 t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n + (0.5 * (i * n)));
	double tmp;
	if (n <= -3.6e+198) {
		tmp = t_0;
	} else if (n <= -5.4e-227) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 9.5e-215) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n + (0.5d0 * (i * n)))
    if (n <= (-3.6d+198)) then
        tmp = t_0
    else if (n <= (-5.4d-227)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 9.5d-215) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n + (0.5 * (i * n)));
	double tmp;
	if (n <= -3.6e+198) {
		tmp = t_0;
	} else if (n <= -5.4e-227) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 9.5e-215) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n + (0.5 * (i * n)))
	tmp = 0
	if n <= -3.6e+198:
		tmp = t_0
	elif n <= -5.4e-227:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 9.5e-215:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n + Float64(0.5 * Float64(i * n))))
	tmp = 0.0
	if (n <= -3.6e+198)
		tmp = t_0;
	elseif (n <= -5.4e-227)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 9.5e-215)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n + (0.5 * (i * n)));
	tmp = 0.0;
	if (n <= -3.6e+198)
		tmp = t_0;
	elseif (n <= -5.4e-227)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 9.5e-215)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n + N[(0.5 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.6e+198], t$95$0, If[LessEqual[n, -5.4e-227], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e-215], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6000000000000002e198 or 9.5000000000000007e-215 < n

   1. Initial program 17.1%

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

   2. Taylor expanded in n around inf 25.4%

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

      1. *-commutative 25.4%

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

      2. associate-/l* 25.4%

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

      3. expm1-def 82.9%

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

   4. Simplified 82.9%

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

   5. Taylor expanded in i around 0 73.0%

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

      1. *-commutative 73.0%

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

   7. Simplified 73.0%

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

   if -3.6000000000000002e198 < n < -5.3999999999999999e-227

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/l* 31.0%

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

      3. expm1-def 80.6%

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

   4. Simplified 80.6%

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

      1. associate-*l/ 80.6%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in n around 0 31.0%

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

      1. *-commutative 31.0%

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

      2. expm1-def 73.1%

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

      3. associate-/l* 80.6%

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

      4. associate-/r/ 80.4%

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

      5. associate-/l/ 80.3%

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

   9. Simplified 80.3%

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

   10. Taylor expanded in i around 0 63.6%

       \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   11. Step-by-step derivation

       1. *-commutative 63.6%

          \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   12. Simplified 63.6%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -5.3999999999999999e-227 < n < 9.5000000000000007e-215

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

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

   5. Taylor expanded in i around 0 89.1%

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

   6. Taylor expanded in i around 0 89.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+198}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq -5.4 \cdot 10^{-227}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-215}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 16: 63.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \mathbf{if}\;n \leq -5 \cdot 10^{+199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-227}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (+ n (* 0.5 (* i n))))))
   (if (<= n -5e+199)
     t_0
     (if (<= n -1.35e-227)
       (* 100.0 (/ n (+ 1.0 (* i -0.5))))
       (if (<= n 1e-213) 0.0 t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n + (0.5 * (i * n)));
	double tmp;
	if (n <= -5e+199) {
		tmp = t_0;
	} else if (n <= -1.35e-227) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1e-213) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n + (0.5d0 * (i * n)))
    if (n <= (-5d+199)) then
        tmp = t_0
    else if (n <= (-1.35d-227)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1d-213) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n + (0.5 * (i * n)));
	double tmp;
	if (n <= -5e+199) {
		tmp = t_0;
	} else if (n <= -1.35e-227) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1e-213) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n + (0.5 * (i * n)))
	tmp = 0
	if n <= -5e+199:
		tmp = t_0
	elif n <= -1.35e-227:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1e-213:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n + Float64(0.5 * Float64(i * n))))
	tmp = 0.0
	if (n <= -5e+199)
		tmp = t_0;
	elseif (n <= -1.35e-227)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1e-213)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n + (0.5 * (i * n)));
	tmp = 0.0;
	if (n <= -5e+199)
		tmp = t_0;
	elseif (n <= -1.35e-227)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1e-213)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n + N[(0.5 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5e+199], t$95$0, If[LessEqual[n, -1.35e-227], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1e-213], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.9999999999999998e199 or 9.9999999999999995e-214 < n

   1. Initial program 17.1%

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

   2. Taylor expanded in n around inf 25.4%

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

      1. *-commutative 25.4%

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

      2. associate-/l* 25.4%

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

      3. expm1-def 82.9%

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

   4. Simplified 82.9%

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

   5. Taylor expanded in i around 0 73.0%

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

      1. *-commutative 73.0%

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

   7. Simplified 73.0%

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

   if -4.9999999999999998e199 < n < -1.35e-227

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/l* 31.0%

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

      3. expm1-def 80.6%

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

   4. Simplified 80.6%

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

   5. Taylor expanded in i around 0 63.7%

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

      1. *-commutative 63.7%

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

   7. Simplified 63.7%

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

   if -1.35e-227 < n < 9.9999999999999995e-214

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

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

   5. Taylor expanded in i around 0 89.1%

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

   6. Taylor expanded in i around 0 89.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+199}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-227}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 17: 63.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+205}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \mathbf{elif}\;n \leq -6.5 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-214}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.9e+205)
   (+ (* n 100.0) (* (* i n) 50.0))
   (if (<= n -6.5e-234)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 7.5e-214) 0.0 (* 100.0 (+ n (* 0.5 (* i n))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.9e+205) {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	} else if (n <= -6.5e-234) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7.5e-214) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + (0.5 * (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 <= (-1.9d+205)) then
        tmp = (n * 100.0d0) + ((i * n) * 50.0d0)
    else if (n <= (-6.5d-234)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 7.5d-214) then
        tmp = 0.0d0
    else
        tmp = 100.0d0 * (n + (0.5d0 * (i * n)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.9e+205) {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	} else if (n <= -6.5e-234) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7.5e-214) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + (0.5 * (i * n)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.9e+205:
		tmp = (n * 100.0) + ((i * n) * 50.0)
	elif n <= -6.5e-234:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 7.5e-214:
		tmp = 0.0
	else:
		tmp = 100.0 * (n + (0.5 * (i * n)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.9e+205)
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(i * n) * 50.0));
	elseif (n <= -6.5e-234)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 7.5e-214)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n + Float64(0.5 * Float64(i * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.9e+205)
		tmp = (n * 100.0) + ((i * n) * 50.0);
	elseif (n <= -6.5e-234)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 7.5e-214)
		tmp = 0.0;
	else
		tmp = 100.0 * (n + (0.5 * (i * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.9e+205], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -6.5e-234], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.5e-214], 0.0, N[(100.0 * N[(n + N[(0.5 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.9e205

   1. Initial program 11.4%

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

   2. Taylor expanded in n around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 38.6%

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

      3. expm1-def 96.3%

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

   4. Simplified 96.3%

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

   5. Taylor expanded in i around 0 86.1%

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

   if -1.9e205 < n < -6.4999999999999994e-234

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/l* 31.0%

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

      3. expm1-def 80.6%

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

   4. Simplified 80.6%

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

   5. Taylor expanded in i around 0 63.7%

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

      1. *-commutative 63.7%

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

   7. Simplified 63.7%

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

   if -6.4999999999999994e-234 < n < 7.49999999999999966e-214

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

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

   5. Taylor expanded in i around 0 89.1%

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

   6. Taylor expanded in i around 0 89.1%

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

   if 7.49999999999999966e-214 < n

   1. Initial program 18.5%

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

   2. Taylor expanded in n around inf 22.3%

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

      1. *-commutative 22.3%

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

      2. associate-/l* 22.3%

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

      3. expm1-def 79.7%

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

   4. Simplified 79.7%

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

   5. Taylor expanded in i around 0 69.9%

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

      1. *-commutative 69.9%

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

   7. Simplified 69.9%

      \[\leadsto \color{blue}{\left(n + \left(n \cdot i\right) \cdot 0.5\right)} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+205}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \mathbf{elif}\;n \leq -6.5 \cdot 10^{-234}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-214}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 18: 63.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.85 \cdot 10^{+200}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{-232}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.85e+200)
   (+ (* n 100.0) (* (* i n) 50.0))
   (if (<= n -1.3e-232)
     (/ (* n 100.0) (+ 1.0 (* i -0.5)))
     (if (<= n 2.2e-213) 0.0 (* 100.0 (+ n (* 0.5 (* i n))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.85e+200) {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	} else if (n <= -1.3e-232) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 2.2e-213) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + (0.5 * (i * n)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.85d+200)) then
        tmp = (n * 100.0d0) + ((i * n) * 50.0d0)
    else if (n <= (-1.3d-232)) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    else if (n <= 2.2d-213) then
        tmp = 0.0d0
    else
        tmp = 100.0d0 * (n + (0.5d0 * (i * n)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.85e+200) {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	} else if (n <= -1.3e-232) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 2.2e-213) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + (0.5 * (i * n)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.85e+200:
		tmp = (n * 100.0) + ((i * n) * 50.0)
	elif n <= -1.3e-232:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	elif n <= 2.2e-213:
		tmp = 0.0
	else:
		tmp = 100.0 * (n + (0.5 * (i * n)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.85e+200)
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(i * n) * 50.0));
	elseif (n <= -1.3e-232)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	elseif (n <= 2.2e-213)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n + Float64(0.5 * Float64(i * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.85e+200)
		tmp = (n * 100.0) + ((i * n) * 50.0);
	elseif (n <= -1.3e-232)
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	elseif (n <= 2.2e-213)
		tmp = 0.0;
	else
		tmp = 100.0 * (n + (0.5 * (i * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.85e+200], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.3e-232], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.2e-213], 0.0, N[(100.0 * N[(n + N[(0.5 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.85000000000000003e200

   1. Initial program 11.4%

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

   2. Taylor expanded in n around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 38.6%

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

      3. expm1-def 96.3%

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

   4. Simplified 96.3%

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

   5. Taylor expanded in i around 0 86.1%

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

   if -2.85000000000000003e200 < n < -1.29999999999999998e-232

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/l* 31.0%

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

      3. expm1-def 80.6%

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

   4. Simplified 80.6%

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

      1. associate-*l/ 80.6%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in i around 0 63.7%

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

      1. *-commutative 63.7%

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

   9. Simplified 63.7%

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

   if -1.29999999999999998e-232 < n < 2.2000000000000001e-213

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

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

   5. Taylor expanded in i around 0 89.1%

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

   6. Taylor expanded in i around 0 89.1%

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

   if 2.2000000000000001e-213 < n

   1. Initial program 18.5%

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

   2. Taylor expanded in n around inf 22.3%

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

      1. *-commutative 22.3%

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

      2. associate-/l* 22.3%

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

      3. expm1-def 79.7%

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

   4. Simplified 79.7%

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

   5. Taylor expanded in i around 0 69.9%

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

      1. *-commutative 69.9%

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

   7. Simplified 69.9%

      \[\leadsto \color{blue}{\left(n + \left(n \cdot i\right) \cdot 0.5\right)} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.85 \cdot 10^{+200}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{-232}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 19: 64.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+202}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-225}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.7e+202)
   (* n (+ 100.0 (* 100.0 (* i (* i 0.16666666666666666)))))
   (if (<= n -4e-225)
     (/ (* n 100.0) (+ 1.0 (* i -0.5)))
     (if (<= n 1.05e-213) 0.0 (* 100.0 (+ n (* 0.5 (* i n))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.7e+202) {
		tmp = n * (100.0 + (100.0 * (i * (i * 0.16666666666666666))));
	} else if (n <= -4e-225) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 1.05e-213) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + (0.5 * (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 <= (-3.7d+202)) then
        tmp = n * (100.0d0 + (100.0d0 * (i * (i * 0.16666666666666666d0))))
    else if (n <= (-4d-225)) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * (-0.5d0)))
    else if (n <= 1.05d-213) then
        tmp = 0.0d0
    else
        tmp = 100.0d0 * (n + (0.5d0 * (i * n)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.7e+202) {
		tmp = n * (100.0 + (100.0 * (i * (i * 0.16666666666666666))));
	} else if (n <= -4e-225) {
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	} else if (n <= 1.05e-213) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + (0.5 * (i * n)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.7e+202:
		tmp = n * (100.0 + (100.0 * (i * (i * 0.16666666666666666))))
	elif n <= -4e-225:
		tmp = (n * 100.0) / (1.0 + (i * -0.5))
	elif n <= 1.05e-213:
		tmp = 0.0
	else:
		tmp = 100.0 * (n + (0.5 * (i * n)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.7e+202)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(i * 0.16666666666666666)))));
	elseif (n <= -4e-225)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * -0.5)));
	elseif (n <= 1.05e-213)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n + Float64(0.5 * Float64(i * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.7e+202)
		tmp = n * (100.0 + (100.0 * (i * (i * 0.16666666666666666))));
	elseif (n <= -4e-225)
		tmp = (n * 100.0) / (1.0 + (i * -0.5));
	elseif (n <= 1.05e-213)
		tmp = 0.0;
	else
		tmp = 100.0 * (n + (0.5 * (i * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.7e+202], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -4e-225], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-213], 0.0, N[(100.0 * N[(n + N[(0.5 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.6999999999999999e202

   1. Initial program 11.4%

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

      1. associate-/r/ 12.2%

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

      2. associate-*r* 12.2%

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

      3. *-commutative 12.2%

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

      4. associate-*r/ 12.2%

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

      5. sub-neg 12.2%

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

      6. distribute-lft-in 12.2%

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

      7. fma-def 12.2%

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

      8. metadata-eval 12.2%

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

      9. metadata-eval 12.2%

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

   3. Simplified 12.2%

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

   4. Taylor expanded in i around 0 89.6%

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

      1. distribute-lft-out 89.6%

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

      2. associate-*r/ 89.6%

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

      3. metadata-eval 89.6%

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

      4. unpow2 89.6%

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

      5. associate--l+ 89.6%

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

      6. associate-*r/ 89.6%

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

      7. metadata-eval 89.6%

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

      8. unpow2 89.6%

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

      9. associate-*r/ 89.6%

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

      10. metadata-eval 89.6%

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

   6. Simplified 89.6%

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

   7. Taylor expanded in n around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. unpow2 89.6%

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

   9. Simplified 89.6%

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

   10. Taylor expanded in i around inf 88.5%

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

       1. *-commutative 88.5%

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

       2. unpow2 88.5%

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

       3. associate-*r* 88.5%

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

   12. Simplified 88.5%

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

   if -3.6999999999999999e202 < n < -3.9999999999999998e-225

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/l* 31.0%

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

      3. expm1-def 80.6%

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

   4. Simplified 80.6%

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

      1. associate-*l/ 80.6%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in i around 0 63.7%

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

      1. *-commutative 63.7%

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

   9. Simplified 63.7%

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

   if -3.9999999999999998e-225 < n < 1.0499999999999999e-213

   1. Initial program 60.5%

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

      1. associate-*r/ 60.5%

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

      2. sub-neg 60.5%

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

      3. distribute-lft-in 60.5%

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

      4. fma-def 60.5%

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

      5. metadata-eval 60.5%

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

      6. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in n around inf 52.0%

      \[\leadsto \frac{\color{blue}{100 \cdot e^{i} - 100}}{\frac{i}{n}} \]

   5. Taylor expanded in i around 0 89.1%

      \[\leadsto \frac{\color{blue}{100} - 100}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 89.1%

      \[\leadsto \color{blue}{0} \]

   if 1.0499999999999999e-213 < n

   1. Initial program 18.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 22.3%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 22.3%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 22.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 79.7%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   5. Taylor expanded in i around 0 69.9%

      \[\leadsto \color{blue}{\left(n + 0.5 \cdot \left(n \cdot i\right)\right)} \cdot 100 \]
   6. Step-by-step derivation

      1. *-commutative 69.9%

         \[\leadsto \left(n + \color{blue}{\left(n \cdot i\right) \cdot 0.5}\right) \cdot 100 \]

   7. Simplified 69.9%

      \[\leadsto \color{blue}{\left(n + \left(n \cdot i\right) \cdot 0.5\right)} \cdot 100 \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+202}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-225}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-213}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 20: 61.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2600000000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+45}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 2.3 \cdot 10^{+247}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2600000000.0)
   0.0
   (if (<= i 2.8e+45)
     (* n 100.0)
     (if (<= i 2.3e+247)
       (* 16.666666666666668 (* n (* i i)))
       (* (/ n i) -200.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2600000000.0) {
		tmp = 0.0;
	} else if (i <= 2.8e+45) {
		tmp = n * 100.0;
	} else if (i <= 2.3e+247) {
		tmp = 16.666666666666668 * (n * (i * i));
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2600000000.0d0)) then
        tmp = 0.0d0
    else if (i <= 2.8d+45) then
        tmp = n * 100.0d0
    else if (i <= 2.3d+247) then
        tmp = 16.666666666666668d0 * (n * (i * i))
    else
        tmp = (n / i) * (-200.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2600000000.0) {
		tmp = 0.0;
	} else if (i <= 2.8e+45) {
		tmp = n * 100.0;
	} else if (i <= 2.3e+247) {
		tmp = 16.666666666666668 * (n * (i * i));
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2600000000.0:
		tmp = 0.0
	elif i <= 2.8e+45:
		tmp = n * 100.0
	elif i <= 2.3e+247:
		tmp = 16.666666666666668 * (n * (i * i))
	else:
		tmp = (n / i) * -200.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2600000000.0)
		tmp = 0.0;
	elseif (i <= 2.8e+45)
		tmp = Float64(n * 100.0);
	elseif (i <= 2.3e+247)
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	else
		tmp = Float64(Float64(n / i) * -200.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2600000000.0)
		tmp = 0.0;
	elseif (i <= 2.8e+45)
		tmp = n * 100.0;
	elseif (i <= 2.3e+247)
		tmp = 16.666666666666668 * (n * (i * i));
	else
		tmp = (n / i) * -200.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2600000000.0], 0.0, If[LessEqual[i, 2.8e+45], N[(n * 100.0), $MachinePrecision], If[LessEqual[i, 2.3e+247], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -2.6e9

   1. Initial program 73.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 73.3%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 73.3%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 73.3%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. fma-def 73.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{\frac{i}{n}} \]

      5. metadata-eval 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{\frac{i}{n}} \]

      6. metadata-eval 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]

   4. Taylor expanded in n around inf 71.6%

      \[\leadsto \frac{\color{blue}{100 \cdot e^{i} - 100}}{\frac{i}{n}} \]

   5. Taylor expanded in i around 0 30.4%

      \[\leadsto \frac{\color{blue}{100} - 100}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 30.4%

      \[\leadsto \color{blue}{0} \]

   if -2.6e9 < i < 2.7999999999999999e45

   1. Initial program 7.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 83.5%

      \[\leadsto \color{blue}{100 \cdot n} \]
   3. Step-by-step derivation

      1. *-commutative 83.5%

         \[\leadsto \color{blue}{n \cdot 100} \]

   4. Simplified 83.5%

      \[\leadsto \color{blue}{n \cdot 100} \]

   if 2.7999999999999999e45 < i < 2.29999999999999991e247

   1. Initial program 48.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 48.8%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 48.8%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 48.8%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 48.8%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 48.8%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 48.8%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. fma-def 48.8%

         \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{i} \]

      8. metadata-eval 48.8%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{i} \]

      9. metadata-eval 48.8%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 48.8%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]

   4. Taylor expanded in i around 0 46.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out 46.1%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      2. associate-*r/ 46.1%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      3. metadata-eval 46.1%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      4. unpow2 46.1%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. associate--l+ 46.1%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      6. associate-*r/ 46.1%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. metadata-eval 46.1%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      8. unpow2 46.1%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      9. associate-*r/ 46.1%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right)\right) \]

      10. metadata-eval 46.1%

          \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right)\right) \]

   6. Simplified 46.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

   7. Taylor expanded in i around inf 46.2%

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 46.2%

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)} \]

      2. unpow2 46.2%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) \]

      3. associate--l+ 46.2%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      4. associate-*r/ 46.2%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. metadata-eval 46.2%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      6. unpow2 46.2%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      7. associate-*r/ 46.2%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \]

      8. metadata-eval 46.2%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right) \]

      9. associate-*l* 46.2%

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

      10. associate-*r* 46.2%

          \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)} \]

      11. unpow2 46.2%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{\color{blue}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      12. metadata-eval 46.2%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{\color{blue}{0.3333333333333333 \cdot 1}}{{n}^{2}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      13. associate-*r/ 46.2%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\color{blue}{0.3333333333333333 \cdot \frac{1}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      14. metadata-eval 46.2%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \frac{\color{blue}{0.5 \cdot 1}}{n}\right)\right)\right) \]

      15. associate-*r/ 46.2%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \color{blue}{0.5 \cdot \frac{1}{n}}\right)\right)\right) \]

   9. Simplified 46.2%

      \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{\frac{0.3333333333333333}{n}}{n} + \frac{-0.5}{n}\right)\right)\right)} \]

   10. Taylor expanded in n around inf 46.4%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot {i}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 46.4%

          \[\leadsto 16.666666666666668 \cdot \left(n \cdot \color{blue}{\left(i \cdot i\right)}\right) \]

   12. Simplified 46.4%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)} \]

   if 2.29999999999999991e247 < i

   1. Initial program 43.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 12.3%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 12.3%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 12.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 12.3%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   5. Taylor expanded in i around 0 61.9%

      \[\leadsto \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \cdot 100 \]
   6. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \frac{n \cdot 100}{1 + \color{blue}{i \cdot -0.5}} \]

   7. Simplified 61.9%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]

   8. Taylor expanded in i around inf 61.9%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2600000000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+45}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 2.3 \cdot 10^{+247}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \]

Alternative 21: 61.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -380000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 1.15 \cdot 10^{+250}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -380000.0)
   0.0
   (if (<= i 1.9e-9)
     (* 100.0 (+ n (* i -0.5)))
     (if (<= i 1.15e+250)
       (* 16.666666666666668 (* n (* i i)))
       (* (/ n i) -200.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -380000.0) {
		tmp = 0.0;
	} else if (i <= 1.9e-9) {
		tmp = 100.0 * (n + (i * -0.5));
	} else if (i <= 1.15e+250) {
		tmp = 16.666666666666668 * (n * (i * i));
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-380000.0d0)) then
        tmp = 0.0d0
    else if (i <= 1.9d-9) then
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    else if (i <= 1.15d+250) then
        tmp = 16.666666666666668d0 * (n * (i * i))
    else
        tmp = (n / i) * (-200.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -380000.0) {
		tmp = 0.0;
	} else if (i <= 1.9e-9) {
		tmp = 100.0 * (n + (i * -0.5));
	} else if (i <= 1.15e+250) {
		tmp = 16.666666666666668 * (n * (i * i));
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -380000.0:
		tmp = 0.0
	elif i <= 1.9e-9:
		tmp = 100.0 * (n + (i * -0.5))
	elif i <= 1.15e+250:
		tmp = 16.666666666666668 * (n * (i * i))
	else:
		tmp = (n / i) * -200.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -380000.0)
		tmp = 0.0;
	elseif (i <= 1.9e-9)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	elseif (i <= 1.15e+250)
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	else
		tmp = Float64(Float64(n / i) * -200.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -380000.0)
		tmp = 0.0;
	elseif (i <= 1.9e-9)
		tmp = 100.0 * (n + (i * -0.5));
	elseif (i <= 1.15e+250)
		tmp = 16.666666666666668 * (n * (i * i));
	else
		tmp = (n / i) * -200.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -380000.0], 0.0, If[LessEqual[i, 1.9e-9], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.15e+250], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -3.8e5

   1. Initial program 73.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 73.3%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 73.3%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 73.3%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. fma-def 73.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{\frac{i}{n}} \]

      5. metadata-eval 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{\frac{i}{n}} \]

      6. metadata-eval 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]

   4. Taylor expanded in n around inf 71.6%

      \[\leadsto \frac{\color{blue}{100 \cdot e^{i} - 100}}{\frac{i}{n}} \]

   5. Taylor expanded in i around 0 30.4%

      \[\leadsto \frac{\color{blue}{100} - 100}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 30.4%

      \[\leadsto \color{blue}{0} \]

   if -3.8e5 < i < 1.90000000000000006e-9

   1. Initial program 6.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 88.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n + n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 88.3%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(n \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right) \]

      2. *-commutative 88.3%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(i \cdot n\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \]

      3. associate-*r/ 88.3%

         \[\leadsto 100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \]

      4. metadata-eval 88.3%

         \[\leadsto 100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \]

   4. Simplified 88.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)} \]

   5. Taylor expanded in n around 0 87.4%

      \[\leadsto 100 \cdot \left(n + \color{blue}{-0.5 \cdot i}\right) \]
   6. Step-by-step derivation

      1. *-commutative 87.4%

         \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot -0.5}\right) \]

   7. Simplified 87.4%

      \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot -0.5}\right) \]

   if 1.90000000000000006e-9 < i < 1.1500000000000001e250

   1. Initial program 44.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 45.0%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 45.1%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 45.1%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 45.0%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 45.0%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 45.0%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. fma-def 45.0%

         \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{i} \]

      8. metadata-eval 45.0%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{i} \]

      9. metadata-eval 45.0%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]

   4. Taylor expanded in i around 0 40.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out 40.7%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      2. associate-*r/ 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      3. metadata-eval 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      4. unpow2 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. associate--l+ 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      6. associate-*r/ 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. metadata-eval 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      8. unpow2 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      9. associate-*r/ 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right)\right) \]

      10. metadata-eval 40.7%

          \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right)\right) \]

   6. Simplified 40.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

   7. Taylor expanded in i around inf 38.9%

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 38.9%

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)} \]

      2. unpow2 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) \]

      3. associate--l+ 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      4. associate-*r/ 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. metadata-eval 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      6. unpow2 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      7. associate-*r/ 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \]

      8. metadata-eval 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right) \]

      9. associate-*l* 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

      10. associate-*r* 38.9%

          \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)} \]

      11. unpow2 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{\color{blue}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      12. metadata-eval 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{\color{blue}{0.3333333333333333 \cdot 1}}{{n}^{2}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      13. associate-*r/ 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\color{blue}{0.3333333333333333 \cdot \frac{1}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      14. metadata-eval 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \frac{\color{blue}{0.5 \cdot 1}}{n}\right)\right)\right) \]

      15. associate-*r/ 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \color{blue}{0.5 \cdot \frac{1}{n}}\right)\right)\right) \]

   9. Simplified 38.9%

      \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{\frac{0.3333333333333333}{n}}{n} + \frac{-0.5}{n}\right)\right)\right)} \]

   10. Taylor expanded in n around inf 40.1%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot {i}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 40.1%

          \[\leadsto 16.666666666666668 \cdot \left(n \cdot \color{blue}{\left(i \cdot i\right)}\right) \]

   12. Simplified 40.1%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)} \]

   if 1.1500000000000001e250 < i

   1. Initial program 43.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 12.3%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 12.3%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 12.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 12.3%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   5. Taylor expanded in i around 0 61.9%

      \[\leadsto \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \cdot 100 \]
   6. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \frac{n \cdot 100}{1 + \color{blue}{i \cdot -0.5}} \]

   7. Simplified 61.9%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]

   8. Taylor expanded in i around inf 61.9%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -380000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 1.15 \cdot 10^{+250}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \]

Alternative 22: 61.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -6000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{+250}:\\ \;\;\;\;n \cdot \left(\left(i \cdot i\right) \cdot 16.666666666666668\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -6000000000.0)
   0.0
   (if (<= i 1.9e-9)
     (* 100.0 (+ n (* i -0.5)))
     (if (<= i 5.2e+250)
       (* n (* (* i i) 16.666666666666668))
       (* (/ n i) -200.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -6000000000.0) {
		tmp = 0.0;
	} else if (i <= 1.9e-9) {
		tmp = 100.0 * (n + (i * -0.5));
	} else if (i <= 5.2e+250) {
		tmp = n * ((i * i) * 16.666666666666668);
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-6000000000.0d0)) then
        tmp = 0.0d0
    else if (i <= 1.9d-9) then
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    else if (i <= 5.2d+250) then
        tmp = n * ((i * i) * 16.666666666666668d0)
    else
        tmp = (n / i) * (-200.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -6000000000.0) {
		tmp = 0.0;
	} else if (i <= 1.9e-9) {
		tmp = 100.0 * (n + (i * -0.5));
	} else if (i <= 5.2e+250) {
		tmp = n * ((i * i) * 16.666666666666668);
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -6000000000.0:
		tmp = 0.0
	elif i <= 1.9e-9:
		tmp = 100.0 * (n + (i * -0.5))
	elif i <= 5.2e+250:
		tmp = n * ((i * i) * 16.666666666666668)
	else:
		tmp = (n / i) * -200.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -6000000000.0)
		tmp = 0.0;
	elseif (i <= 1.9e-9)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	elseif (i <= 5.2e+250)
		tmp = Float64(n * Float64(Float64(i * i) * 16.666666666666668));
	else
		tmp = Float64(Float64(n / i) * -200.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -6000000000.0)
		tmp = 0.0;
	elseif (i <= 1.9e-9)
		tmp = 100.0 * (n + (i * -0.5));
	elseif (i <= 5.2e+250)
		tmp = n * ((i * i) * 16.666666666666668);
	else
		tmp = (n / i) * -200.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -6000000000.0], 0.0, If[LessEqual[i, 1.9e-9], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.2e+250], N[(n * N[(N[(i * i), $MachinePrecision] * 16.666666666666668), $MachinePrecision]), $MachinePrecision], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -6e9

   1. Initial program 73.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 73.3%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 73.3%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 73.3%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. fma-def 73.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{\frac{i}{n}} \]

      5. metadata-eval 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{\frac{i}{n}} \]

      6. metadata-eval 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]

   4. Taylor expanded in n around inf 71.6%

      \[\leadsto \frac{\color{blue}{100 \cdot e^{i} - 100}}{\frac{i}{n}} \]

   5. Taylor expanded in i around 0 30.4%

      \[\leadsto \frac{\color{blue}{100} - 100}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 30.4%

      \[\leadsto \color{blue}{0} \]

   if -6e9 < i < 1.90000000000000006e-9

   1. Initial program 6.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 88.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n + n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 88.3%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(n \cdot i\right) \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right) \]

      2. *-commutative 88.3%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(i \cdot n\right)} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) \]

      3. associate-*r/ 88.3%

         \[\leadsto 100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \]

      4. metadata-eval 88.3%

         \[\leadsto 100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right)\right) \]

   4. Simplified 88.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)} \]

   5. Taylor expanded in n around 0 87.4%

      \[\leadsto 100 \cdot \left(n + \color{blue}{-0.5 \cdot i}\right) \]
   6. Step-by-step derivation

      1. *-commutative 87.4%

         \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot -0.5}\right) \]

   7. Simplified 87.4%

      \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot -0.5}\right) \]

   if 1.90000000000000006e-9 < i < 5.20000000000000023e250

   1. Initial program 44.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 45.0%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 45.1%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 45.1%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 45.0%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 45.0%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 45.0%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. fma-def 45.0%

         \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{i} \]

      8. metadata-eval 45.0%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{i} \]

      9. metadata-eval 45.0%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]

   4. Taylor expanded in i around 0 40.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out 40.7%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      2. associate-*r/ 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      3. metadata-eval 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      4. unpow2 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. associate--l+ 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      6. associate-*r/ 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. metadata-eval 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      8. unpow2 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      9. associate-*r/ 40.7%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right)\right) \]

      10. metadata-eval 40.7%

          \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right)\right) \]

   6. Simplified 40.7%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

   7. Taylor expanded in i around inf 38.9%

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 38.9%

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)} \]

      2. unpow2 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) \]

      3. associate--l+ 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      4. associate-*r/ 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. metadata-eval 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      6. unpow2 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      7. associate-*r/ 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \]

      8. metadata-eval 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right) \]

      9. associate-*l* 38.9%

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

      10. associate-*r* 38.9%

          \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)} \]

      11. unpow2 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{\color{blue}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      12. metadata-eval 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{\color{blue}{0.3333333333333333 \cdot 1}}{{n}^{2}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      13. associate-*r/ 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\color{blue}{0.3333333333333333 \cdot \frac{1}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      14. metadata-eval 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \frac{\color{blue}{0.5 \cdot 1}}{n}\right)\right)\right) \]

      15. associate-*r/ 38.9%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \color{blue}{0.5 \cdot \frac{1}{n}}\right)\right)\right) \]

   9. Simplified 38.9%

      \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{\frac{0.3333333333333333}{n}}{n} + \frac{-0.5}{n}\right)\right)\right)} \]

   10. Taylor expanded in n around inf 40.1%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot {i}^{2}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 40.1%

          \[\leadsto \color{blue}{\left(n \cdot {i}^{2}\right) \cdot 16.666666666666668} \]

       2. associate-*l* 40.1%

          \[\leadsto \color{blue}{n \cdot \left({i}^{2} \cdot 16.666666666666668\right)} \]

       3. unpow2 40.1%

          \[\leadsto n \cdot \left(\color{blue}{\left(i \cdot i\right)} \cdot 16.666666666666668\right) \]

   12. Simplified 40.1%

       \[\leadsto \color{blue}{n \cdot \left(\left(i \cdot i\right) \cdot 16.666666666666668\right)} \]

   if 5.20000000000000023e250 < i

   1. Initial program 43.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 12.3%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 12.3%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 12.3%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 12.3%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]

   5. Taylor expanded in i around 0 61.9%

      \[\leadsto \frac{n}{\color{blue}{1 + -0.5 \cdot i}} \cdot 100 \]
   6. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \frac{n \cdot 100}{1 + \color{blue}{i \cdot -0.5}} \]

   7. Simplified 61.9%

      \[\leadsto \frac{n}{\color{blue}{1 + i \cdot -0.5}} \cdot 100 \]

   8. Taylor expanded in i around inf 61.9%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 5.2 \cdot 10^{+250}:\\ \;\;\;\;n \cdot \left(\left(i \cdot i\right) \cdot 16.666666666666668\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \]

Alternative 23: 61.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.9 \lor \neg \left(i \leq 4.3 \cdot 10^{+250}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot 0.16666666666666666\right) \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i 1.9) (not (<= i 4.3e+250)))
   (/ n (+ 0.01 (* i -0.005)))
   (* (* i 0.16666666666666666) (* i (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= 1.9) || !(i <= 4.3e+250)) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = (i * 0.16666666666666666) * (i * (n * 100.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= 1.9d0) .or. (.not. (i <= 4.3d+250))) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else
        tmp = (i * 0.16666666666666666d0) * (i * (n * 100.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= 1.9) || !(i <= 4.3e+250)) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = (i * 0.16666666666666666) * (i * (n * 100.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= 1.9) or not (i <= 4.3e+250):
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = (i * 0.16666666666666666) * (i * (n * 100.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= 1.9) || !(i <= 4.3e+250))
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(Float64(i * 0.16666666666666666) * Float64(i * Float64(n * 100.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= 1.9) || ~((i <= 4.3e+250)))
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = (i * 0.16666666666666666) * (i * (n * 100.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, 1.9], N[Not[LessEqual[i, 4.3e+250]], $MachinePrecision]], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * 0.16666666666666666), $MachinePrecision] * N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.8999999999999999 or 4.3e250 < i

   1. Initial program 23.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 23.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 23.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 23.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 81.5%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   5. Step-by-step derivation

      1. associate-*l/ 81.5%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   6. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   7. Taylor expanded in n around 0 23.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   8. Step-by-step derivation

      1. *-commutative 23.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. expm1-def 75.2%

         \[\leadsto \frac{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100 \]

      3. associate-/l* 81.5%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

      4. associate-/r/ 81.4%

         \[\leadsto \color{blue}{\frac{n}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{100}}} \]

      5. associate-/l/ 81.3%

         \[\leadsto \frac{n}{\color{blue}{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}} \]

   9. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}} \]

   10. Taylor expanded in i around 0 73.1%

       \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   11. Step-by-step derivation

       1. *-commutative 73.1%

          \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   12. Simplified 73.1%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if 1.8999999999999999 < i < 4.3e250

   1. Initial program 44.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 44.8%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 44.8%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 44.8%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 44.8%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 44.8%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 44.8%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. fma-def 44.8%

         \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{i} \]

      8. metadata-eval 44.8%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{i} \]

      9. metadata-eval 44.8%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 44.8%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]

   4. Taylor expanded in i around 0 40.3%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out 40.3%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      2. associate-*r/ 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      3. metadata-eval 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      4. unpow2 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. associate--l+ 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      6. associate-*r/ 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. metadata-eval 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      8. unpow2 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      9. associate-*r/ 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right)\right) \]

      10. metadata-eval 40.3%

          \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right)\right) \]

   6. Simplified 40.3%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

   7. Taylor expanded in i around inf 40.4%

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 40.4%

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)} \]

      2. unpow2 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) \]

      3. associate--l+ 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      4. associate-*r/ 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. metadata-eval 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      6. unpow2 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      7. associate-*r/ 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \]

      8. metadata-eval 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right) \]

      9. associate-*l* 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

      10. associate-*r* 40.4%

          \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)} \]

      11. unpow2 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{\color{blue}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      12. metadata-eval 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{\color{blue}{0.3333333333333333 \cdot 1}}{{n}^{2}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      13. associate-*r/ 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\color{blue}{0.3333333333333333 \cdot \frac{1}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      14. metadata-eval 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \frac{\color{blue}{0.5 \cdot 1}}{n}\right)\right)\right) \]

      15. associate-*r/ 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \color{blue}{0.5 \cdot \frac{1}{n}}\right)\right)\right) \]

   9. Simplified 40.4%

      \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{\frac{0.3333333333333333}{n}}{n} + \frac{-0.5}{n}\right)\right)\right)} \]

   10. Taylor expanded in n around inf 40.7%

       \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot i\right)} \]
   11. Step-by-step derivation

       1. *-commutative 40.7%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \color{blue}{\left(i \cdot 0.16666666666666666\right)} \]

   12. Simplified 40.7%

       \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \color{blue}{\left(i \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.9 \lor \neg \left(i \leq 4.3 \cdot 10^{+250}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot 0.16666666666666666\right) \cdot \left(i \cdot \left(n \cdot 100\right)\right)\\ \end{array} \]

Alternative 24: 61.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.9 \lor \neg \left(i \leq 9 \cdot 10^{+250}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i 1.9) (not (<= i 9e+250)))
   (/ n (+ 0.01 (* i -0.005)))
   (* 16.666666666666668 (* n (* i i)))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= 1.9) || !(i <= 9e+250)) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= 1.9d0) .or. (.not. (i <= 9d+250))) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else
        tmp = 16.666666666666668d0 * (n * (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= 1.9) || !(i <= 9e+250)) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= 1.9) or not (i <= 9e+250):
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = 16.666666666666668 * (n * (i * i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= 1.9) || !(i <= 9e+250))
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= 1.9) || ~((i <= 9e+250)))
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = 16.666666666666668 * (n * (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, 1.9], N[Not[LessEqual[i, 9e+250]], $MachinePrecision]], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.8999999999999999 or 8.99999999999999993e250 < i

   1. Initial program 23.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf 23.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative 23.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* 23.9%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{e^{i} - 1}}} \cdot 100 \]

      3. expm1-def 81.5%

         \[\leadsto \frac{n}{\frac{i}{\color{blue}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

   4. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
   5. Step-by-step derivation

      1. associate-*l/ 81.5%

         \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   6. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]

   7. Taylor expanded in n around 0 23.9%

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   8. Step-by-step derivation

      1. *-commutative 23.9%

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. expm1-def 75.2%

         \[\leadsto \frac{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100 \]

      3. associate-/l* 81.5%

         \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \cdot 100 \]

      4. associate-/r/ 81.4%

         \[\leadsto \color{blue}{\frac{n}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{100}}} \]

      5. associate-/l/ 81.3%

         \[\leadsto \frac{n}{\color{blue}{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}} \]

   9. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{100 \cdot \mathsf{expm1}\left(i\right)}}} \]

   10. Taylor expanded in i around 0 73.1%

       \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   11. Step-by-step derivation

       1. *-commutative 73.1%

          \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   12. Simplified 73.1%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if 1.8999999999999999 < i < 8.99999999999999993e250

   1. Initial program 44.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-/r/ 44.8%

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      2. associate-*r* 44.8%

         \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n} \]

      3. *-commutative 44.8%

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)} \]

      4. associate-*r/ 44.8%

         \[\leadsto n \cdot \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i}} \]

      5. sub-neg 44.8%

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{i} \]

      6. distribute-lft-in 44.8%

         \[\leadsto n \cdot \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{i} \]

      7. fma-def 44.8%

         \[\leadsto n \cdot \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{i} \]

      8. metadata-eval 44.8%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{i} \]

      9. metadata-eval 44.8%

         \[\leadsto n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{i} \]

   3. Simplified 44.8%

      \[\leadsto \color{blue}{n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}} \]

   4. Taylor expanded in i around 0 40.3%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out 40.3%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      2. associate-*r/ 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      3. metadata-eval 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{\color{blue}{0.5}}{n}\right) + {i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      4. unpow2 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. associate--l+ 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right)\right) \]

      6. associate-*r/ 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      7. metadata-eval 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      8. unpow2 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right) \]

      9. associate-*r/ 40.3%

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right)\right) \]

      10. metadata-eval 40.3%

          \[\leadsto n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right)\right) \]

   6. Simplified 40.3%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

   7. Taylor expanded in i around inf 40.4%

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 40.4%

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)} \]

      2. unpow2 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) \]

      3. associate--l+ 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      4. associate-*r/ 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. metadata-eval 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      6. unpow2 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      7. associate-*r/ 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \]

      8. metadata-eval 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right)\right) \]

      9. associate-*l* 40.4%

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)} \]

      10. associate-*r* 40.4%

          \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)} \]

      11. unpow2 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{0.3333333333333333}{\color{blue}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      12. metadata-eval 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\frac{\color{blue}{0.3333333333333333 \cdot 1}}{{n}^{2}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      13. associate-*r/ 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(\color{blue}{0.3333333333333333 \cdot \frac{1}{{n}^{2}}} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right) \]

      14. metadata-eval 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \frac{\color{blue}{0.5 \cdot 1}}{n}\right)\right)\right) \]

      15. associate-*r/ 40.4%

          \[\leadsto \left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - \color{blue}{0.5 \cdot \frac{1}{n}}\right)\right)\right) \]

   9. Simplified 40.4%

      \[\leadsto \color{blue}{\left(\left(100 \cdot n\right) \cdot i\right) \cdot \left(i \cdot \left(0.16666666666666666 + \left(\frac{\frac{0.3333333333333333}{n}}{n} + \frac{-0.5}{n}\right)\right)\right)} \]

   10. Taylor expanded in n around inf 40.7%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot {i}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 40.7%

          \[\leadsto 16.666666666666668 \cdot \left(n \cdot \color{blue}{\left(i \cdot i\right)}\right) \]

   12. Simplified 40.7%

       \[\leadsto \color{blue}{16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.9 \lor \neg \left(i \leq 9 \cdot 10^{+250}\right):\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 25: 59.4% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -105000000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -105000000.0) 0.0 (if (<= i 1.9e-9) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -105000000.0) {
		tmp = 0.0;
	} else if (i <= 1.9e-9) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-105000000.0d0)) then
        tmp = 0.0d0
    else if (i <= 1.9d-9) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -105000000.0) {
		tmp = 0.0;
	} else if (i <= 1.9e-9) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -105000000.0:
		tmp = 0.0
	elif i <= 1.9e-9:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -105000000.0)
		tmp = 0.0;
	elseif (i <= 1.9e-9)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -105000000.0)
		tmp = 0.0;
	elseif (i <= 1.9e-9)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -105000000.0], 0.0, If[LessEqual[i, 1.9e-9], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.05e8 or 1.90000000000000006e-9 < i

   1. Initial program 57.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Step-by-step derivation

      1. associate-*r/ 57.9%

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      2. sub-neg 57.9%

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

      3. distribute-lft-in 57.9%

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

      4. fma-def 57.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{\frac{i}{n}} \]

      5. metadata-eval 57.9%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{\frac{i}{n}} \]

      6. metadata-eval 57.9%

         \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]

   4. Taylor expanded in n around inf 58.6%

      \[\leadsto \frac{\color{blue}{100 \cdot e^{i} - 100}}{\frac{i}{n}} \]

   5. Taylor expanded in i around 0 28.9%

      \[\leadsto \frac{\color{blue}{100} - 100}{\frac{i}{n}} \]

   6. Taylor expanded in i around 0 28.9%

      \[\leadsto \color{blue}{0} \]

   if -1.05e8 < i < 1.90000000000000006e-9

   1. Initial program 6.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0 87.3%

      \[\leadsto \color{blue}{100 \cdot n} \]
   3. Step-by-step derivation

      1. *-commutative 87.3%

         \[\leadsto \color{blue}{n \cdot 100} \]

   4. Simplified 87.3%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -105000000:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 26: 18.3% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 27.1%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Step-by-step derivation

   1. associate-*r/ 27.1%

      \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

   2. sub-neg 27.1%

      \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(-1\right)\right)}}{\frac{i}{n}} \]

   3. distribute-lft-in 27.1%

      \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(-1\right)}}{\frac{i}{n}} \]

   4. fma-def 27.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \left(-1\right)\right)}}{\frac{i}{n}} \]

   5. metadata-eval 27.1%

      \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, 100 \cdot \color{blue}{-1}\right)}{\frac{i}{n}} \]

   6. metadata-eval 27.1%

      \[\leadsto \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, \color{blue}{-100}\right)}{\frac{i}{n}} \]

3. Simplified 27.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{\frac{i}{n}}} \]

4. Taylor expanded in n around inf 29.2%

   \[\leadsto \frac{\color{blue}{100 \cdot e^{i} - 100}}{\frac{i}{n}} \]

5. Taylor expanded in i around 0 15.5%

   \[\leadsto \frac{\color{blue}{100} - 100}{\frac{i}{n}} \]

6. Taylor expanded in i around 0 15.8%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 15.8%

   \[\leadsto 0 \]

Developer target: 34.3% 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 2023279 
(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))))