Compound Interest

Percentage Accurate: 28.5% → 93.3%

Time: 20.2s

Alternatives: 17

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.5% 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: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 10^{-193}:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 1e-193)
     (* n (/ (expm1 (* n (log1p (/ i n)))) (/ i 100.0)))
     (if (<= t_1 INFINITY) (* n (/ (+ (* t_0 100.0) -100.0) i)) (* n 100.0)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 1e-193) {
		tmp = n * (expm1((n * log1p((i / n)))) / (i / 100.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 1e-193) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) / (i / 100.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 1e-193:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) / (i / 100.0))
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 1e-193)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / 100.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-193], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-193

   1. Initial program 28.9%

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

      1. associate-/r/ 28.6%

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

      2. associate-*r* 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-*r/ 28.6%

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

      5. sub-neg 28.6%

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

      6. distribute-lft-in 28.6%

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

      7. fma-def 28.6%

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

      8. metadata-eval 28.6%

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

      9. metadata-eval 28.6%

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

   3. Simplified 28.6%

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

      1. fma-udef 28.6%

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

      2. metadata-eval 28.6%

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

      3. metadata-eval 28.6%

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

      4. distribute-lft-in 28.6%

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

      5. sub-neg 28.6%

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

      6. *-commutative 28.6%

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

      7. pow-to-exp 27.6%

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

      8. expm1-def 42.4%

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

      9. add-log-exp 27.6%

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

      10. pow-to-exp 28.6%

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

      11. log-pow 42.4%

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

      12. log1p-udef 97.7%

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

   5. Applied egg-rr 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. associate-/l* 97.7%

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

   7. Applied egg-rr 97.7%

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

      1. *-lft-identity 97.7%

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

   9. Simplified 97.7%

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

   if 1e-193 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.0%

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

      1. associate-/r/ 98.0%

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

      2. associate-*r* 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-*r/ 98.0%

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

      5. sub-neg 98.0%

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

      6. distribute-lft-in 98.0%

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

      7. fma-def 98.0%

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

      8. metadata-eval 98.0%

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

      9. metadata-eval 98.0%

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

   3. Simplified 98.0%

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

      1. fma-udef 98.0%

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

      2. *-commutative 98.0%

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

   5. Applied egg-rr 98.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0 76.2%

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

      1. *-commutative 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-193}:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 2: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 10^{-193}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 1e-193)
     (* 100.0 (* n (/ (expm1 (* n (log1p (/ i n)))) i)))
     (if (<= t_1 INFINITY) (* n (/ (+ (* t_0 100.0) -100.0) i)) (* n 100.0)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 1e-193) {
		tmp = 100.0 * (n * (expm1((n * log1p((i / n)))) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 1e-193) {
		tmp = 100.0 * (n * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 1e-193:
		tmp = 100.0 * (n * (math.expm1((n * math.log1p((i / n)))) / i))
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 1e-193)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-193], N[(100.0 * N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-193

   1. Initial program 28.9%

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

      1. associate-/r/ 28.6%

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

      2. pow-to-exp 27.6%

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

      3. expm1-def 42.3%

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

      4. add-log-exp 27.6%

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

      5. pow-to-exp 28.6%

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

      6. log-pow 42.3%

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

      7. log1p-udef 97.6%

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

   3. Applied egg-rr 97.6%

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

   if 1e-193 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.0%

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

      1. associate-/r/ 98.0%

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

      2. associate-*r* 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-*r/ 98.0%

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

      5. sub-neg 98.0%

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

      6. distribute-lft-in 98.0%

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

      7. fma-def 98.0%

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

      8. metadata-eval 98.0%

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

      9. metadata-eval 98.0%

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

   3. Simplified 98.0%

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

      1. fma-udef 98.0%

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

      2. *-commutative 98.0%

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

   5. Applied egg-rr 98.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0 76.2%

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

      1. *-commutative 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-193}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 3: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 10^{-193}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 1e-193)
     (* n (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) i)))
     (if (<= t_1 INFINITY) (* n (/ (+ (* t_0 100.0) -100.0) i)) (* n 100.0)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 1e-193) {
		tmp = n * (100.0 * (expm1((n * log1p((i / n)))) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 1e-193) {
		tmp = n * (100.0 * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 1e-193:
		tmp = n * (100.0 * (math.expm1((n * math.log1p((i / n)))) / i))
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 1e-193)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-193], N[(n * N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 1e-193

   1. Initial program 28.9%

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

      1. associate-/r/ 28.6%

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

      2. associate-*r* 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-*r/ 28.6%

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

      5. sub-neg 28.6%

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

      6. distribute-lft-in 28.6%

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

      7. fma-def 28.6%

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

      8. metadata-eval 28.6%

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

      9. metadata-eval 28.6%

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

   3. Simplified 28.6%

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

      1. fma-udef 28.6%

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

      2. metadata-eval 28.6%

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

      3. metadata-eval 28.6%

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

      4. distribute-lft-in 28.6%

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

      5. sub-neg 28.6%

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

      6. *-commutative 28.6%

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

      7. pow-to-exp 27.6%

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

      8. expm1-def 42.4%

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

      9. add-log-exp 27.6%

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

      10. pow-to-exp 28.6%

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

      11. log-pow 42.4%

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

      12. log1p-udef 97.7%

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

   5. Applied egg-rr 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. associate-/l* 97.7%

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

   7. Applied egg-rr 97.7%

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

      1. *-lft-identity 97.7%

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

   9. Simplified 97.7%

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

       1. associate-/r/ 97.7%

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

   11. Applied egg-rr 97.7%

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

   if 1e-193 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.0%

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

      1. associate-/r/ 98.0%

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

      2. associate-*r* 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-*r/ 98.0%

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

      5. sub-neg 98.0%

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

      6. distribute-lft-in 98.0%

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

      7. fma-def 98.0%

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

      8. metadata-eval 98.0%

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

      9. metadata-eval 98.0%

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

   3. Simplified 98.0%

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

      1. fma-udef 98.0%

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

      2. *-commutative 98.0%

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

   5. Applied egg-rr 98.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0 76.2%

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

      1. *-commutative 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-193}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 4: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{-186}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{-14}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \sqrt[3]{n \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.9)
   (* 100.0 (/ 0.0 (/ i n)))
   (if (<= i 2.9e-186)
     (* 100.0 (+ n (* (* i n) (- 0.5 (/ 0.5 n)))))
     (if (<= i 2.6e-14)
       (* 100.0 (/ (* i n) i))
       (* 100.0 (cbrt (* n (* n n))))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.9) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (i <= 2.9e-186) {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	} else if (i <= 2.6e-14) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * cbrt((n * (n * n)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.9) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (i <= 2.9e-186) {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	} else if (i <= 2.6e-14) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * Math.cbrt((n * (n * n)));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.9)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	elseif (i <= 2.9e-186)
		tmp = Float64(100.0 * Float64(n + Float64(Float64(i * n) * Float64(0.5 - Float64(0.5 / n)))));
	elseif (i <= 2.6e-14)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	else
		tmp = Float64(100.0 * cbrt(Float64(n * Float64(n * n))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.9], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.9e-186], N[(100.0 * N[(n + N[(N[(i * n), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.6e-14], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[Power[N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.8999999999999999

   1. Initial program 50.9%

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

   2. Taylor expanded in i around 0 30.7%

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

   if -1.8999999999999999 < i < 2.90000000000000019e-186

   1. Initial program 8.4%

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

   2. Taylor expanded in i around 0 83.3%

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

      1. associate-*r* 83.6%

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

      2. *-commutative 83.6%

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

      3. associate-*r/ 83.6%

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

      4. metadata-eval 83.6%

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

   4. Simplified 83.6%

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

   if 2.90000000000000019e-186 < i < 2.59999999999999997e-14

   1. Initial program 21.8%

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

   2. Taylor expanded in i around 0 21.8%

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

      1. +-commutative 21.8%

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

   4. Simplified 21.8%

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

      1. add-cbrt-cube 21.8%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\frac{\left(i + 1\right) - 1}{\frac{i}{n}} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}}} \]

      2. associate-/r/ 21.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      3. associate--l+ 21.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      4. metadata-eval 21.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + \color{blue}{0}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      5. associate-/r/ 21.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      6. associate--l+ 21.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      7. metadata-eval 21.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      8. associate-/r/ 21.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}} \]

      9. associate--l+ 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)} \]

      10. metadata-eval 62.5%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)} \]

   6. Applied egg-rr 62.5%

      \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)}} \]

      2. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i + 0}{i}\right)} \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      3. +-rgt-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{\color{blue}{i}}{i}\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      4. associate-*l* 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \color{blue}{\left(\frac{i + 0}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)}} \]

      5. +-rgt-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{\color{blue}{i}}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)} \]

      6. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \color{blue}{\left(n \cdot \frac{i + 0}{i}\right)}\right)\right)} \]

      7. +-rgt-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{\color{blue}{i}}{i}\right)\right)\right)} \]

   8. Simplified 62.5%

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

      1. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)}} \]

      2. *-inverses 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      3. *-un-lft-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)} \cdot \left(n \cdot \frac{i}{i}\right)} \]

      4. *-inverses 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{1}\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      5. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      6. *-un-lft-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{n}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      7. *-inverses 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \left(n \cdot \color{blue}{1}\right)} \]

      8. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      9. *-un-lft-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{n}} \]

      10. add-cbrt-cube 68.7%

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

      11. *-un-lft-identity 68.7%

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

      12. *-commutative 68.7%

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

      13. *-inverses 68.7%

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

      14. associate-*r/ 81.4%

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

   10. Applied egg-rr 81.4%

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

   if 2.59999999999999997e-14 < i

   1. Initial program 47.5%

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

   2. Taylor expanded in i around 0 19.7%

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

      1. associate-/r/ 5.8%

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

      2. *-inverses 5.8%

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

      3. *-un-lft-identity 5.8%

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

      4. add-cbrt-cube 58.4%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(n \cdot n\right) \cdot n}} \]

      5. *-un-lft-identity 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot n} \]

      6. *-commutative 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(n \cdot 1\right)}\right) \cdot n} \]

      7. *-inverses 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{\frac{i}{i}}\right)\right) \cdot n} \]

      8. *-un-lft-identity 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(1 \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \cdot n} \]

      9. *-inverses 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\frac{i}{i}} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot n} \]

      10. *-un-lft-identity 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      11. *-commutative 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \color{blue}{\left(n \cdot 1\right)}} \]

      12. *-inverses 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \color{blue}{\frac{i}{i}}\right)} \]

      13. *-commutative 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)}} \]

      14. *-inverses 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{1}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \]

      15. *-commutative 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(1 \cdot n\right)} \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \]

      16. *-un-lft-identity 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{n} \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \]

      17. *-inverses 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \]

      18. *-un-lft-identity 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)}} \]

      19. *-inverses 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \left(n \cdot \left(n \cdot \color{blue}{1}\right)\right)} \]

      20. *-commutative 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right)} \]

      21. *-un-lft-identity 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \left(n \cdot \color{blue}{n}\right)} \]

   4. Applied egg-rr 58.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.9 \cdot 10^{-186}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{-14}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \sqrt[3]{n \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 5: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.35 \cdot 10^{-56}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-186}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 1.6 \cdot 10^{-13}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \sqrt[3]{n \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.35e-56)
   (* 100.0 (/ (expm1 i) (/ i n)))
   (if (<= i 1.4e-186)
     (* 100.0 (+ n (* (* i n) (- 0.5 (/ 0.5 n)))))
     (if (<= i 1.6e-13)
       (* 100.0 (/ (* i n) i))
       (* 100.0 (cbrt (* n (* n n))))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.35e-56) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (i <= 1.4e-186) {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	} else if (i <= 1.6e-13) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * cbrt((n * (n * n)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.35e-56) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (i <= 1.4e-186) {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	} else if (i <= 1.6e-13) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * Math.cbrt((n * (n * n)));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.35e-56)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (i <= 1.4e-186)
		tmp = Float64(100.0 * Float64(n + Float64(Float64(i * n) * Float64(0.5 - Float64(0.5 / n)))));
	elseif (i <= 1.6e-13)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	else
		tmp = Float64(100.0 * cbrt(Float64(n * Float64(n * n))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.35e-56], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.4e-186], N[(100.0 * N[(n + N[(N[(i * n), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.6e-13], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[Power[N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -2.35e-56

   1. Initial program 45.8%

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

   2. Taylor expanded in n around inf 70.4%

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

      1. expm1-def 76.0%

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

   4. Simplified 76.0%

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

   if -2.35e-56 < i < 1.39999999999999992e-186

   1. Initial program 6.8%

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

   2. Taylor expanded in i around 0 86.7%

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

      1. associate-*r* 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-*r/ 87.1%

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

      4. metadata-eval 87.1%

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

   4. Simplified 87.1%

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

   if 1.39999999999999992e-186 < i < 1.6e-13

   1. Initial program 21.8%

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

   2. Taylor expanded in i around 0 21.8%

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

      1. +-commutative 21.8%

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

   4. Simplified 21.8%

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

      1. add-cbrt-cube 21.8%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\frac{\left(i + 1\right) - 1}{\frac{i}{n}} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}}} \]

      2. associate-/r/ 21.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      3. associate--l+ 21.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      4. metadata-eval 21.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + \color{blue}{0}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      5. associate-/r/ 21.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      6. associate--l+ 21.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      7. metadata-eval 21.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      8. associate-/r/ 21.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}} \]

      9. associate--l+ 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)} \]

      10. metadata-eval 62.5%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)} \]

   6. Applied egg-rr 62.5%

      \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)}} \]

      2. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i + 0}{i}\right)} \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      3. +-rgt-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{\color{blue}{i}}{i}\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      4. associate-*l* 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \color{blue}{\left(\frac{i + 0}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)}} \]

      5. +-rgt-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{\color{blue}{i}}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)} \]

      6. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \color{blue}{\left(n \cdot \frac{i + 0}{i}\right)}\right)\right)} \]

      7. +-rgt-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{\color{blue}{i}}{i}\right)\right)\right)} \]

   8. Simplified 62.5%

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

      1. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)}} \]

      2. *-inverses 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      3. *-un-lft-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)} \cdot \left(n \cdot \frac{i}{i}\right)} \]

      4. *-inverses 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{1}\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      5. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      6. *-un-lft-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{n}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      7. *-inverses 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \left(n \cdot \color{blue}{1}\right)} \]

      8. *-commutative 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      9. *-un-lft-identity 62.5%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{n}} \]

      10. add-cbrt-cube 68.7%

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

      11. *-un-lft-identity 68.7%

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

      12. *-commutative 68.7%

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

      13. *-inverses 68.7%

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

      14. associate-*r/ 81.4%

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

   10. Applied egg-rr 81.4%

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

   if 1.6e-13 < i

   1. Initial program 47.5%

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

   2. Taylor expanded in i around 0 19.7%

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

      1. associate-/r/ 5.8%

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

      2. *-inverses 5.8%

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

      3. *-un-lft-identity 5.8%

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

      4. add-cbrt-cube 58.4%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(n \cdot n\right) \cdot n}} \]

      5. *-un-lft-identity 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot n} \]

      6. *-commutative 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(n \cdot 1\right)}\right) \cdot n} \]

      7. *-inverses 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{\frac{i}{i}}\right)\right) \cdot n} \]

      8. *-un-lft-identity 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(1 \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \cdot n} \]

      9. *-inverses 58.4%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\frac{i}{i}} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot n} \]

      10. *-un-lft-identity 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      11. *-commutative 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \color{blue}{\left(n \cdot 1\right)}} \]

      12. *-inverses 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \color{blue}{\frac{i}{i}}\right)} \]

      13. *-commutative 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)}} \]

      14. *-inverses 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{1}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \]

      15. *-commutative 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(1 \cdot n\right)} \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \]

      16. *-un-lft-identity 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{n} \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \]

      17. *-inverses 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right)} \]

      18. *-un-lft-identity 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)}} \]

      19. *-inverses 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \left(n \cdot \left(n \cdot \color{blue}{1}\right)\right)} \]

      20. *-commutative 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right)} \]

      21. *-un-lft-identity 58.4%

          \[\leadsto 100 \cdot \sqrt[3]{n \cdot \left(n \cdot \color{blue}{n}\right)} \]

   4. Applied egg-rr 58.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.35 \cdot 10^{-56}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-186}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{elif}\;i \leq 1.6 \cdot 10^{-13}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \sqrt[3]{n \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.06 \cdot 10^{-247} \lor \neg \left(n \leq 3.3 \cdot 10^{-151}\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.06e-247) (not (<= n 3.3e-151)))
   (* n (* 100.0 (/ (expm1 i) i)))
   (* 100.0 (/ 0.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.06e-247) || !(n <= 3.3e-151)) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.06e-247) || !(n <= 3.3e-151)) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.06e-247) or not (n <= 3.3e-151):
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = 100.0 * (0.0 / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.06e-247) || !(n <= 3.3e-151))
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.06e-247], N[Not[LessEqual[n, 3.3e-151]], $MachinePrecision]], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.06e-247 or 3.2999999999999999e-151 < n

   1. Initial program 26.2%

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

      1. associate-/r/ 26.4%

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

      2. associate-*r* 26.4%

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

      3. *-commutative 26.4%

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

      4. associate-*r/ 26.4%

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

      5. sub-neg 26.4%

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

      6. distribute-lft-in 26.4%

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

      7. fma-def 26.4%

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

      8. metadata-eval 26.4%

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

      9. metadata-eval 26.4%

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

   3. Simplified 26.4%

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

      1. fma-udef 26.4%

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

      2. metadata-eval 26.4%

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

      3. metadata-eval 26.4%

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

      4. distribute-lft-in 26.4%

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

      5. sub-neg 26.4%

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

      6. *-commutative 26.4%

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

      7. pow-to-exp 21.4%

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

      8. expm1-def 32.3%

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

      9. add-log-exp 21.4%

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

      10. pow-to-exp 26.4%

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

      11. log-pow 32.3%

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

      12. log1p-udef 81.8%

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

   5. Applied egg-rr 81.8%

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

   6. Taylor expanded in n around inf 36.3%

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

      1. expm1-def 79.4%

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

   8. Simplified 79.4%

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

   if -1.06e-247 < n < 3.2999999999999999e-151

   1. Initial program 51.3%

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

   2. Taylor expanded in i around 0 71.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.06 \cdot 10^{-247} \lor \neg \left(n \leq 3.3 \cdot 10^{-151}\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 7: 66.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+56}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4e+56)
   (* n (+ 100.0 (+ (* i 50.0) (* (* i i) 16.666666666666668))))
   (if (<= n -8.5e-248)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 4.5e-155)
       (* 100.0 (/ 0.0 (/ i n)))
       (*
        100.0
        (+
         n
         (*
          n
          (+
           (* i (- 0.5 (/ 0.5 n)))
           (*
            (* i i)
            (+
             (/ 0.3333333333333333 (* n n))
             (- 0.16666666666666666 (/ 0.5 n))))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4e+56) {
		tmp = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	} else if (n <= -8.5e-248) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 4.5e-155) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = 100.0 * (n + (n * ((i * (0.5 - (0.5 / n))) + ((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4d+56)) then
        tmp = n * (100.0d0 + ((i * 50.0d0) + ((i * i) * 16.666666666666668d0)))
    else if (n <= (-8.5d-248)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 4.5d-155) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = 100.0d0 * (n + (n * ((i * (0.5d0 - (0.5d0 / n))) + ((i * i) * ((0.3333333333333333d0 / (n * n)) + (0.16666666666666666d0 - (0.5d0 / n)))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4e+56) {
		tmp = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	} else if (n <= -8.5e-248) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 4.5e-155) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = 100.0 * (n + (n * ((i * (0.5 - (0.5 / n))) + ((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4e+56:
		tmp = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)))
	elif n <= -8.5e-248:
		tmp = 100.0 * (i / (i / n))
	elif n <= 4.5e-155:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = 100.0 * (n + (n * ((i * (0.5 - (0.5 / n))) + ((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4e+56)
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i * 50.0) + Float64(Float64(i * i) * 16.666666666666668))));
	elseif (n <= -8.5e-248)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 4.5e-155)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(i * Float64(0.5 - Float64(0.5 / n))) + Float64(Float64(i * i) * Float64(Float64(0.3333333333333333 / Float64(n * n)) + Float64(0.16666666666666666 - Float64(0.5 / n))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4e+56)
		tmp = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	elseif (n <= -8.5e-248)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 4.5e-155)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = 100.0 * (n + (n * ((i * (0.5 - (0.5 / n))) + ((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4e+56], N[(n * N[(100.0 + N[(N[(i * 50.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -8.5e-248], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.5e-155], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(n * N[(N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.00000000000000037e56

   1. Initial program 29.4%

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

      1. associate-/r/ 29.9%

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

      2. associate-*r* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r/ 30.0%

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

      5. sub-neg 30.0%

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

      6. distribute-lft-in 30.0%

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

      7. fma-def 30.0%

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

      8. metadata-eval 30.0%

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

      9. metadata-eval 30.0%

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

   3. Simplified 30.0%

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

   4. Taylor expanded in i around 0 56.0%

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

      1. +-commutative 56.0%

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

      2. distribute-lft-out 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in n around inf 56.0%

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

      1. +-commutative 56.0%

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

      2. distribute-rgt-in 56.0%

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

      3. *-commutative 56.0%

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

      4. associate-*l* 56.0%

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

      5. metadata-eval 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot \color{blue}{50} + \left(0.16666666666666666 \cdot {i}^{2}\right) \cdot 100\right)\right) \]

      6. *-commutative 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{\left({i}^{2} \cdot 0.16666666666666666\right)} \cdot 100\right)\right) \]

      7. associate-*l* 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{{i}^{2} \cdot \left(0.16666666666666666 \cdot 100\right)}\right)\right) \]

      8. unpow2 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{\left(i \cdot i\right)} \cdot \left(0.16666666666666666 \cdot 100\right)\right)\right) \]

      9. metadata-eval 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot \color{blue}{16.666666666666668}\right)\right) \]

   9. Simplified 56.0%

      \[\leadsto \color{blue}{n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)} \]

   if -4.00000000000000037e56 < n < -8.5000000000000003e-248

   1. Initial program 35.5%

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

   2. Taylor expanded in i around 0 61.2%

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

   if -8.5000000000000003e-248 < n < 4.5000000000000004e-155

   1. Initial program 51.3%

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

   2. Taylor expanded in i around 0 71.3%

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

   if 4.5000000000000004e-155 < n

   1. Initial program 17.3%

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

   2. Taylor expanded in i around 0 63.7%

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

      1. distribute-lft-out 64.2%

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

      2. unpow2 64.2%

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

      3. associate--l+ 64.2%

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

      4. associate-*r/ 64.2%

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

      5. metadata-eval 64.2%

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

      6. unpow2 64.2%

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

      7. associate-*r/ 64.2%

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

      8. metadata-eval 64.2%

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

      9. associate-*r/ 64.2%

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

      10. metadata-eval 64.2%

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

   4. Simplified 64.2%

      \[\leadsto 100 \cdot \color{blue}{\left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+56}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 65.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+56}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -9.2 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-153}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e+56)
   (* n (+ 100.0 (+ (* i 50.0) (* (* i i) 16.666666666666668))))
   (if (<= n -9.2e-248)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 3e-153)
       (* 100.0 (/ 0.0 (/ i n)))
       (*
        n
        (+
         100.0
         (*
          100.0
          (+
           (*
            (* i i)
            (+
             (/ 0.3333333333333333 (* n n))
             (+ 0.16666666666666666 (/ -0.5 n))))
           (* i (- 0.5 (/ 0.5 n)))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+56) {
		tmp = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	} else if (n <= -9.2e-248) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3e-153) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (100.0 * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 + (-0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.6d+56)) then
        tmp = n * (100.0d0 + ((i * 50.0d0) + ((i * i) * 16.666666666666668d0)))
    else if (n <= (-9.2d-248)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 3d-153) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = n * (100.0d0 + (100.0d0 * (((i * i) * ((0.3333333333333333d0 / (n * n)) + (0.16666666666666666d0 + ((-0.5d0) / n)))) + (i * (0.5d0 - (0.5d0 / n))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.6e+56) {
		tmp = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	} else if (n <= -9.2e-248) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 3e-153) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (100.0 * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 + (-0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.6e+56:
		tmp = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)))
	elif n <= -9.2e-248:
		tmp = 100.0 * (i / (i / n))
	elif n <= 3e-153:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = n * (100.0 + (100.0 * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 + (-0.5 / n)))) + (i * (0.5 - (0.5 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.6e+56)
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i * 50.0) + Float64(Float64(i * i) * 16.666666666666668))));
	elseif (n <= -9.2e-248)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 3e-153)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(Float64(Float64(i * i) * Float64(Float64(0.3333333333333333 / Float64(n * n)) + Float64(0.16666666666666666 + Float64(-0.5 / n)))) + Float64(i * Float64(0.5 - Float64(0.5 / n)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.6e+56)
		tmp = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	elseif (n <= -9.2e-248)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 3e-153)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = n * (100.0 + (100.0 * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 + (-0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.6e+56], N[(n * N[(100.0 + N[(N[(i * 50.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -9.2e-248], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3e-153], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(100.0 * N[(N[(N[(i * i), $MachinePrecision] * N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.59999999999999998e56

   1. Initial program 29.4%

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

      1. associate-/r/ 29.9%

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

      2. associate-*r* 29.9%

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

      3. *-commutative 29.9%

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

      4. associate-*r/ 30.0%

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

      5. sub-neg 30.0%

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

      6. distribute-lft-in 30.0%

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

      7. fma-def 30.0%

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

      8. metadata-eval 30.0%

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

      9. metadata-eval 30.0%

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

   3. Simplified 30.0%

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

   4. Taylor expanded in i around 0 56.0%

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

      1. +-commutative 56.0%

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

      2. distribute-lft-out 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in n around inf 56.0%

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

      1. +-commutative 56.0%

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

      2. distribute-rgt-in 56.0%

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

      3. *-commutative 56.0%

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

      4. associate-*l* 56.0%

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

      5. metadata-eval 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot \color{blue}{50} + \left(0.16666666666666666 \cdot {i}^{2}\right) \cdot 100\right)\right) \]

      6. *-commutative 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{\left({i}^{2} \cdot 0.16666666666666666\right)} \cdot 100\right)\right) \]

      7. associate-*l* 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{{i}^{2} \cdot \left(0.16666666666666666 \cdot 100\right)}\right)\right) \]

      8. unpow2 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{\left(i \cdot i\right)} \cdot \left(0.16666666666666666 \cdot 100\right)\right)\right) \]

      9. metadata-eval 56.0%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot \color{blue}{16.666666666666668}\right)\right) \]

   9. Simplified 56.0%

      \[\leadsto \color{blue}{n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)} \]

   if -3.59999999999999998e56 < n < -9.2000000000000001e-248

   1. Initial program 35.5%

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

   2. Taylor expanded in i around 0 61.2%

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

   if -9.2000000000000001e-248 < n < 3e-153

   1. Initial program 51.3%

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

   2. Taylor expanded in i around 0 71.3%

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

   if 3e-153 < n

   1. Initial program 17.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 17.6%

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

      7. fma-def 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 64.2%

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

      1. +-commutative 64.2%

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

      2. distribute-lft-out 64.2%

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

   6. Simplified 64.2%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+56}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -9.2 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-153}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 9: 63.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{+98}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.42 \cdot 10^{-247}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{-154}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot 100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.3e+98)
   (* 100.0 (/ (* i n) i))
   (if (<= n -1.42e-247)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 4.3e-154)
       (* 100.0 (/ 0.0 (/ i n)))
       (* n (+ 100.0 (* (- 0.5 (/ 0.5 n)) (* i 100.0))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.3e+98) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -1.42e-247) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 4.3e-154) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + ((0.5 - (0.5 / n)) * (i * 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 (n <= (-1.3d+98)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= (-1.42d-247)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 4.3d-154) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = n * (100.0d0 + ((0.5d0 - (0.5d0 / n)) * (i * 100.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.3e+98) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -1.42e-247) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 4.3e-154) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + ((0.5 - (0.5 / n)) * (i * 100.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.3e+98:
		tmp = 100.0 * ((i * n) / i)
	elif n <= -1.42e-247:
		tmp = 100.0 * (i / (i / n))
	elif n <= 4.3e-154:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = n * (100.0 + ((0.5 - (0.5 / n)) * (i * 100.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.3e+98)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -1.42e-247)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 4.3e-154)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * 100.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.3e+98)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= -1.42e-247)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 4.3e-154)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = n * (100.0 + ((0.5 - (0.5 / n)) * (i * 100.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.3e+98], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.42e-247], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.3e-154], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.3e98

   1. Initial program 19.8%

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

   2. Taylor expanded in i around 0 3.3%

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

      1. +-commutative 3.3%

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

   4. Simplified 3.3%

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

      1. add-cbrt-cube 22.9%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\frac{\left(i + 1\right) - 1}{\frac{i}{n}} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}}} \]

      2. associate-/r/ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      3. associate--l+ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      4. metadata-eval 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + \color{blue}{0}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      5. associate-/r/ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      6. associate--l+ 22.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      7. metadata-eval 22.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      8. associate-/r/ 22.7%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}} \]

      9. associate--l+ 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)} \]

      10. metadata-eval 26.2%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)} \]

   6. Applied egg-rr 26.2%

      \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)}} \]

      2. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i + 0}{i}\right)} \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      3. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{\color{blue}{i}}{i}\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      4. associate-*l* 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \color{blue}{\left(\frac{i + 0}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)}} \]

      5. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{\color{blue}{i}}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)} \]

      6. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \color{blue}{\left(n \cdot \frac{i + 0}{i}\right)}\right)\right)} \]

      7. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{\color{blue}{i}}{i}\right)\right)\right)} \]

   8. Simplified 26.2%

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

      1. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)}} \]

      2. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      3. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)} \cdot \left(n \cdot \frac{i}{i}\right)} \]

      4. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{1}\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      5. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      6. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{n}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      7. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \left(n \cdot \color{blue}{1}\right)} \]

      8. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      9. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{n}} \]

      10. add-cbrt-cube 50.6%

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

      11. *-un-lft-identity 50.6%

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

      12. *-commutative 50.6%

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

      13. *-inverses 50.6%

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

      14. associate-*r/ 61.2%

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

   10. Applied egg-rr 61.2%

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

   if -1.3e98 < n < -1.42000000000000001e-247

   1. Initial program 39.4%

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

   2. Taylor expanded in i around 0 54.6%

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

   if -1.42000000000000001e-247 < n < 4.29999999999999992e-154

   1. Initial program 51.3%

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

   2. Taylor expanded in i around 0 71.3%

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

   if 4.29999999999999992e-154 < n

   1. Initial program 17.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 17.6%

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

      7. fma-def 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 58.1%

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

      1. associate-*r* 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r/ 58.1%

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

      4. metadata-eval 58.1%

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

   6. Simplified 58.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{+98}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.42 \cdot 10^{-247}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{-154}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot 100\right)\right)\\ \end{array} \]

Alternative 10: 65.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.4 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (+ (* i 50.0) (* (* i i) 16.666666666666668))))))
   (if (<= n -3.6e+56)
     t_0
     (if (<= n -7.5e-248)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 5.4e-155) (* 100.0 (/ 0.0 (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	double tmp;
	if (n <= -3.6e+56) {
		tmp = t_0;
	} else if (n <= -7.5e-248) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.4e-155) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + ((i * 50.0d0) + ((i * i) * 16.666666666666668d0)))
    if (n <= (-3.6d+56)) then
        tmp = t_0
    else if (n <= (-7.5d-248)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 5.4d-155) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	double tmp;
	if (n <= -3.6e+56) {
		tmp = t_0;
	} else if (n <= -7.5e-248) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.4e-155) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)))
	tmp = 0
	if n <= -3.6e+56:
		tmp = t_0
	elif n <= -7.5e-248:
		tmp = 100.0 * (i / (i / n))
	elif n <= 5.4e-155:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(Float64(i * 50.0) + Float64(Float64(i * i) * 16.666666666666668))))
	tmp = 0.0
	if (n <= -3.6e+56)
		tmp = t_0;
	elseif (n <= -7.5e-248)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 5.4e-155)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + ((i * 50.0) + ((i * i) * 16.666666666666668)));
	tmp = 0.0;
	if (n <= -3.6e+56)
		tmp = t_0;
	elseif (n <= -7.5e-248)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 5.4e-155)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(N[(i * 50.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.6e+56], t$95$0, If[LessEqual[n, -7.5e-248], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.4e-155], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.59999999999999998e56 or 5.39999999999999962e-155 < n

   1. Initial program 21.9%

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

      1. associate-/r/ 22.2%

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

      2. associate-*r* 22.2%

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

      3. *-commutative 22.2%

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

      4. associate-*r/ 22.2%

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

      5. sub-neg 22.2%

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

      6. distribute-lft-in 22.3%

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

      7. fma-def 22.2%

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

      8. metadata-eval 22.2%

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

      9. metadata-eval 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in i around 0 61.1%

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

      1. +-commutative 61.1%

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

      2. distribute-lft-out 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in n around inf 60.3%

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

      1. +-commutative 60.3%

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

      2. distribute-rgt-in 60.3%

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

      3. *-commutative 60.3%

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

      4. associate-*l* 60.3%

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

      5. metadata-eval 60.3%

         \[\leadsto n \cdot \left(100 + \left(i \cdot \color{blue}{50} + \left(0.16666666666666666 \cdot {i}^{2}\right) \cdot 100\right)\right) \]

      6. *-commutative 60.3%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{\left({i}^{2} \cdot 0.16666666666666666\right)} \cdot 100\right)\right) \]

      7. associate-*l* 60.3%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{{i}^{2} \cdot \left(0.16666666666666666 \cdot 100\right)}\right)\right) \]

      8. unpow2 60.3%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \color{blue}{\left(i \cdot i\right)} \cdot \left(0.16666666666666666 \cdot 100\right)\right)\right) \]

      9. metadata-eval 60.3%

         \[\leadsto n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot \color{blue}{16.666666666666668}\right)\right) \]

   9. Simplified 60.3%

      \[\leadsto \color{blue}{n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)} \]

   if -3.59999999999999998e56 < n < -7.4999999999999994e-248

   1. Initial program 35.5%

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

   2. Taylor expanded in i around 0 61.2%

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

   if -7.4999999999999994e-248 < n < 5.39999999999999962e-155

   1. Initial program 51.3%

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

   2. Taylor expanded in i around 0 71.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+56}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.4 \cdot 10^{-155}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + \left(i \cdot 50 + \left(i \cdot i\right) \cdot 16.666666666666668\right)\right)\\ \end{array} \]

Alternative 11: 63.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+98}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-147}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 - 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.6e+98)
   (* 100.0 (/ (* i n) i))
   (if (<= n -8.5e-248)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 1.75e-147)
       (* 100.0 (/ 0.0 (/ i n)))
       (+ (* n 100.0) (* i (- (* n 50.0) 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.6e+98) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -8.5e-248) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.75e-147) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = (n * 100.0) + (i * ((n * 50.0) - 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.6d+98)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= (-8.5d-248)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 1.75d-147) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = (n * 100.0d0) + (i * ((n * 50.0d0) - 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.6e+98) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -8.5e-248) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.75e-147) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = (n * 100.0) + (i * ((n * 50.0) - 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.6e+98:
		tmp = 100.0 * ((i * n) / i)
	elif n <= -8.5e-248:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.75e-147:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = (n * 100.0) + (i * ((n * 50.0) - 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.6e+98)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -8.5e-248)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.75e-147)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(n * 50.0) - 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.6e+98)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= -8.5e-248)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 1.75e-147)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = (n * 100.0) + (i * ((n * 50.0) - 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.6e+98], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -8.5e-248], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-147], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(n * 50.0), $MachinePrecision] - 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.6000000000000001e98

   1. Initial program 19.8%

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

   2. Taylor expanded in i around 0 3.3%

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

      1. +-commutative 3.3%

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

   4. Simplified 3.3%

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

      1. add-cbrt-cube 22.9%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\frac{\left(i + 1\right) - 1}{\frac{i}{n}} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}}} \]

      2. associate-/r/ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      3. associate--l+ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      4. metadata-eval 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + \color{blue}{0}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      5. associate-/r/ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      6. associate--l+ 22.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      7. metadata-eval 22.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      8. associate-/r/ 22.7%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}} \]

      9. associate--l+ 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)} \]

      10. metadata-eval 26.2%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)} \]

   6. Applied egg-rr 26.2%

      \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)}} \]

      2. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i + 0}{i}\right)} \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      3. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{\color{blue}{i}}{i}\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      4. associate-*l* 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \color{blue}{\left(\frac{i + 0}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)}} \]

      5. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{\color{blue}{i}}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)} \]

      6. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \color{blue}{\left(n \cdot \frac{i + 0}{i}\right)}\right)\right)} \]

      7. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{\color{blue}{i}}{i}\right)\right)\right)} \]

   8. Simplified 26.2%

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

      1. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)}} \]

      2. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      3. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)} \cdot \left(n \cdot \frac{i}{i}\right)} \]

      4. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{1}\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      5. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      6. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{n}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      7. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \left(n \cdot \color{blue}{1}\right)} \]

      8. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      9. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{n}} \]

      10. add-cbrt-cube 50.6%

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

      11. *-un-lft-identity 50.6%

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

      12. *-commutative 50.6%

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

      13. *-inverses 50.6%

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

      14. associate-*r/ 61.2%

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

   10. Applied egg-rr 61.2%

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

   if -1.6000000000000001e98 < n < -8.5000000000000003e-248

   1. Initial program 39.4%

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

   2. Taylor expanded in i around 0 54.6%

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

   if -8.5000000000000003e-248 < n < 1.75000000000000002e-147

   1. Initial program 51.3%

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

   2. Taylor expanded in i around 0 71.3%

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

   if 1.75000000000000002e-147 < n

   1. Initial program 17.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 17.6%

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

      7. fma-def 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 58.1%

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

      1. associate-*r* 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r/ 58.1%

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

      4. metadata-eval 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in n around 0 58.1%

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

      1. fma-def 58.1%

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

      2. *-commutative 58.1%

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

      3. *-commutative 58.1%

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

   9. Simplified 58.1%

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

   10. Taylor expanded in i around 0 58.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+98}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-248}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-147}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 - 50\right)\\ \end{array} \]

Alternative 12: 63.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{+98}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-247}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-157}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.3e+98)
   (* 100.0 (/ (* i n) i))
   (if (<= n -1.5e-247)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 5.8e-157)
       (* 100.0 (/ 0.0 (/ i n)))
       (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.3e+98) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -1.5e-247) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.8e-157) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.3d+98)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= (-1.5d-247)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 5.8d-157) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.3e+98) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -1.5e-247) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 5.8e-157) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.3e+98:
		tmp = 100.0 * ((i * n) / i)
	elif n <= -1.5e-247:
		tmp = 100.0 * (i / (i / n))
	elif n <= 5.8e-157:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.3e+98)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -1.5e-247)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 5.8e-157)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.3e+98)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= -1.5e-247)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 5.8e-157)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.3e+98], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.5e-247], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-157], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.3e98

   1. Initial program 19.8%

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

   2. Taylor expanded in i around 0 3.3%

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

      1. +-commutative 3.3%

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

   4. Simplified 3.3%

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

      1. add-cbrt-cube 22.9%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\frac{\left(i + 1\right) - 1}{\frac{i}{n}} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}}} \]

      2. associate-/r/ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      3. associate--l+ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      4. metadata-eval 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + \color{blue}{0}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      5. associate-/r/ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      6. associate--l+ 22.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      7. metadata-eval 22.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      8. associate-/r/ 22.7%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}} \]

      9. associate--l+ 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)} \]

      10. metadata-eval 26.2%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)} \]

   6. Applied egg-rr 26.2%

      \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)}} \]

      2. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i + 0}{i}\right)} \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      3. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{\color{blue}{i}}{i}\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      4. associate-*l* 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \color{blue}{\left(\frac{i + 0}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)}} \]

      5. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{\color{blue}{i}}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)} \]

      6. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \color{blue}{\left(n \cdot \frac{i + 0}{i}\right)}\right)\right)} \]

      7. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{\color{blue}{i}}{i}\right)\right)\right)} \]

   8. Simplified 26.2%

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

      1. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)}} \]

      2. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      3. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)} \cdot \left(n \cdot \frac{i}{i}\right)} \]

      4. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{1}\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      5. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      6. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{n}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      7. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \left(n \cdot \color{blue}{1}\right)} \]

      8. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      9. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{n}} \]

      10. add-cbrt-cube 50.6%

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

      11. *-un-lft-identity 50.6%

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

      12. *-commutative 50.6%

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

      13. *-inverses 50.6%

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

      14. associate-*r/ 61.2%

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

   10. Applied egg-rr 61.2%

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

   if -1.3e98 < n < -1.4999999999999999e-247

   1. Initial program 39.4%

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

   2. Taylor expanded in i around 0 54.6%

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

   if -1.4999999999999999e-247 < n < 5.79999999999999977e-157

   1. Initial program 51.3%

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

   2. Taylor expanded in i around 0 71.3%

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

   if 5.79999999999999977e-157 < n

   1. Initial program 17.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 17.6%

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

      7. fma-def 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 58.1%

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

      1. associate-*r* 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r/ 58.1%

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

      4. metadata-eval 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in n around inf 57.6%

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

      1. *-commutative 57.6%

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

   9. Simplified 57.6%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{+98}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-247}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-157}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 13: 56.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -10000 \lor \neg \left(i \leq 1.45 \cdot 10^{-35}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -10000.0) (not (<= i 1.45e-35)))
   (* 100.0 (/ i (/ i n)))
   (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -10000.0) || !(i <= 1.45e-35)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-10000.0d0)) .or. (.not. (i <= 1.45d-35))) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -10000.0) || !(i <= 1.45e-35)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -10000.0) or not (i <= 1.45e-35):
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -10000.0) || !(i <= 1.45e-35))
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -10000.0) || ~((i <= 1.45e-35)))
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -10000.0], N[Not[LessEqual[i, 1.45e-35]], $MachinePrecision]], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1e4 or 1.4500000000000001e-35 < i

   1. Initial program 46.7%

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

   2. Taylor expanded in i around 0 23.5%

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

   if -1e4 < i < 1.4500000000000001e-35

   1. Initial program 13.4%

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

   2. Taylor expanded in i around 0 78.9%

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

      1. *-commutative 78.9%

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

   4. Simplified 78.9%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -10000 \lor \neg \left(i \leq 1.45 \cdot 10^{-35}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 14: 61.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{+96} \lor \neg \left(n \leq 5 \cdot 10^{-40}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.95e+96) (not (<= n 5e-40)))
   (* 100.0 (/ (* i n) i))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.95e+96) || !(n <= 5e-40)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.95d+96)) .or. (.not. (n <= 5d-40))) then
        tmp = 100.0d0 * ((i * n) / i)
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.95e+96) || !(n <= 5e-40)) {
		tmp = 100.0 * ((i * n) / i);
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.95e+96) or not (n <= 5e-40):
		tmp = 100.0 * ((i * n) / i)
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.95e+96) || !(n <= 5e-40))
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2.95e+96) || ~((n <= 5e-40)))
		tmp = 100.0 * ((i * n) / i);
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.95e+96], N[Not[LessEqual[n, 5e-40]], $MachinePrecision]], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.95000000000000014e96 or 4.99999999999999965e-40 < n

   1. Initial program 19.1%

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

   2. Taylor expanded in i around 0 3.7%

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

      1. +-commutative 3.7%

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

   4. Simplified 3.7%

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

      1. add-cbrt-cube 19.2%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\frac{\left(i + 1\right) - 1}{\frac{i}{n}} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}}} \]

      2. associate-/r/ 19.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      3. associate--l+ 19.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      4. metadata-eval 19.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + \color{blue}{0}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      5. associate-/r/ 19.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      6. associate--l+ 19.1%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      7. metadata-eval 19.1%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      8. associate-/r/ 19.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}} \]

      9. associate--l+ 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)} \]

      10. metadata-eval 37.8%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)} \]

   6. Applied egg-rr 37.8%

      \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)}} \]

      2. *-commutative 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i + 0}{i}\right)} \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      3. +-rgt-identity 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{\color{blue}{i}}{i}\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      4. associate-*l* 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \color{blue}{\left(\frac{i + 0}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)}} \]

      5. +-rgt-identity 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{\color{blue}{i}}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)} \]

      6. *-commutative 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \color{blue}{\left(n \cdot \frac{i + 0}{i}\right)}\right)\right)} \]

      7. +-rgt-identity 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{\color{blue}{i}}{i}\right)\right)\right)} \]

   8. Simplified 37.8%

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

      1. *-commutative 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)}} \]

      2. *-inverses 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      3. *-un-lft-identity 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)} \cdot \left(n \cdot \frac{i}{i}\right)} \]

      4. *-inverses 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{1}\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      5. *-commutative 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      6. *-un-lft-identity 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{n}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      7. *-inverses 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \left(n \cdot \color{blue}{1}\right)} \]

      8. *-commutative 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      9. *-un-lft-identity 37.8%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{n}} \]

      10. add-cbrt-cube 49.2%

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

      11. *-un-lft-identity 49.2%

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

      12. *-commutative 49.2%

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

      13. *-inverses 49.2%

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

      14. associate-*r/ 58.4%

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

   10. Applied egg-rr 58.4%

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

   if -2.95000000000000014e96 < n < 4.99999999999999965e-40

   1. Initial program 39.5%

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

   2. Taylor expanded in i around 0 52.8%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{+96} \lor \neg \left(n \leq 5 \cdot 10^{-40}\right):\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 15: 62.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+98}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.6e+98)
   (* 100.0 (/ (* i n) i))
   (if (<= n 1.5) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.6e+98) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.6d+98)) then
        tmp = 100.0d0 * ((i * n) / i)
    else if (n <= 1.5d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.6e+98) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.6e+98:
		tmp = 100.0 * ((i * n) / i)
	elif n <= 1.5:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.6e+98)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 1.5)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.6e+98)
		tmp = 100.0 * ((i * n) / i);
	elseif (n <= 1.5)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.6e+98], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.6000000000000001e98

   1. Initial program 19.8%

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

   2. Taylor expanded in i around 0 3.3%

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

      1. +-commutative 3.3%

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

   4. Simplified 3.3%

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

      1. add-cbrt-cube 22.9%

         \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\frac{\left(i + 1\right) - 1}{\frac{i}{n}} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}}} \]

      2. associate-/r/ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)} \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      3. associate--l+ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      4. metadata-eval 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + \color{blue}{0}}{i} \cdot n\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      5. associate-/r/ 22.9%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      6. associate--l+ 22.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      7. metadata-eval 22.6%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)\right) \cdot \frac{\left(i + 1\right) - 1}{\frac{i}{n}}} \]

      8. associate-/r/ 22.7%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \color{blue}{\left(\frac{\left(i + 1\right) - 1}{i} \cdot n\right)}} \]

      9. associate--l+ 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{\color{blue}{i + \left(1 - 1\right)}}{i} \cdot n\right)} \]

      10. metadata-eval 26.2%

          \[\leadsto 100 \cdot \sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + \color{blue}{0}}{i} \cdot n\right)} \]

   6. Applied egg-rr 26.2%

      \[\leadsto 100 \cdot \color{blue}{\sqrt[3]{\left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)}} \]

      2. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \frac{i + 0}{i}\right)} \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      3. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{\color{blue}{i}}{i}\right) \cdot \left(\left(\frac{i + 0}{i} \cdot n\right) \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)} \]

      4. associate-*l* 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \color{blue}{\left(\frac{i + 0}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)}} \]

      5. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{\color{blue}{i}}{i} \cdot \left(n \cdot \left(\frac{i + 0}{i} \cdot n\right)\right)\right)} \]

      6. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \color{blue}{\left(n \cdot \frac{i + 0}{i}\right)}\right)\right)} \]

      7. +-rgt-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \frac{i}{i}\right) \cdot \left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{\color{blue}{i}}{i}\right)\right)\right)} \]

   8. Simplified 26.2%

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

      1. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(\frac{i}{i} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)}} \]

      2. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(\color{blue}{1} \cdot \left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      3. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\color{blue}{\left(n \cdot \left(n \cdot \frac{i}{i}\right)\right)} \cdot \left(n \cdot \frac{i}{i}\right)} \]

      4. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \left(n \cdot \color{blue}{1}\right)\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      5. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{\left(1 \cdot n\right)}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      6. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot \color{blue}{n}\right) \cdot \left(n \cdot \frac{i}{i}\right)} \]

      7. *-inverses 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \left(n \cdot \color{blue}{1}\right)} \]

      8. *-commutative 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{\left(1 \cdot n\right)}} \]

      9. *-un-lft-identity 26.2%

         \[\leadsto 100 \cdot \sqrt[3]{\left(n \cdot n\right) \cdot \color{blue}{n}} \]

      10. add-cbrt-cube 50.6%

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

      11. *-un-lft-identity 50.6%

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

      12. *-commutative 50.6%

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

      13. *-inverses 50.6%

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

      14. associate-*r/ 61.2%

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

   10. Applied egg-rr 61.2%

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

   if -1.6000000000000001e98 < n < 1.5

   1. Initial program 36.9%

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

   2. Taylor expanded in i around 0 53.1%

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

   if 1.5 < n

   1. Initial program 19.8%

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

      1. associate-/r/ 20.2%

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

      2. associate-*r* 20.2%

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

      3. *-commutative 20.2%

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

      4. associate-*r/ 20.2%

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

      5. sub-neg 20.2%

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

      6. distribute-lft-in 20.2%

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

      7. fma-def 20.2%

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

      8. metadata-eval 20.2%

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

      9. metadata-eval 20.2%

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

   3. Simplified 20.2%

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

   4. Taylor expanded in i around 0 58.1%

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

      1. associate-*r* 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r/ 58.1%

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

      4. metadata-eval 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in n around inf 58.1%

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

      1. *-commutative 58.1%

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

   9. Simplified 58.1%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+98}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 16: 2.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

   1. associate-/r/ 30.1%

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

   2. associate-*r* 30.1%

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

   3. *-commutative 30.1%

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

   4. associate-*r/ 30.1%

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

   5. sub-neg 30.1%

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

   6. distribute-lft-in 30.1%

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

   7. fma-def 30.1%

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

   8. metadata-eval 30.1%

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

   9. metadata-eval 30.1%

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

3. Simplified 30.1%

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

4. Taylor expanded in i around 0 47.0%

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

   1. associate-*r* 47.0%

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

   2. *-commutative 47.0%

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

   3. associate-*r/ 47.0%

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

   4. metadata-eval 47.0%

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

6. Simplified 47.0%

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

7. Taylor expanded in n around 0 3.3%

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

   1. *-commutative 3.3%

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

9. Simplified 3.3%

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

10. Final simplification 3.3%

    \[\leadsto i \cdot -50 \]

Alternative 17: 48.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

2. Taylor expanded in i around 0 43.5%

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

   1. *-commutative 43.5%

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

4. Simplified 43.5%

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

5. Final simplification 43.5%

   \[\leadsto n \cdot 100 \]

Developer target: 34.6% 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 2023199 
(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))))