Compound Interest

Percentage Accurate: 28.3% → 94.0%

Time: 26.1s

Alternatives: 15

Speedup: 16.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= ((double) INFINITY)) {
		tmp = (n * 100.0) / (i / expm1((n * log1p((i / n)))));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

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

Python

def code(i, n):
	tmp = 0
	if ((math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= math.inf:
		tmp = (n * 100.0) / (i / math.expm1((n * math.log1p((i / n)))))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

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

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 31.4%

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

      1. *-commutative 31.4%

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

      2. associate-/r/ 31.3%

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

      3. sub-neg 31.3%

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

      4. metadata-eval 31.3%

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

      5. associate-*r* 31.2%

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

      6. *-commutative 31.2%

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

      7. clear-num 31.2%

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

      8. un-div-inv 31.3%

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

      9. metadata-eval 31.3%

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

      10. sub-neg 31.3%

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

      11. pow-to-exp 28.8%

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

      12. expm1-def 39.2%

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

      13. *-commutative 39.2%

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

      14. log1p-udef 96.2%

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

   3. Applied egg-rr 96.2%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r/ 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. clear-num 1.8%

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

      8. un-div-inv 1.8%

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

      9. metadata-eval 1.8%

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

      10. sub-neg 1.8%

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

      11. pow-to-exp 1.8%

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

      12. expm1-def 1.8%

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

      13. *-commutative 1.8%

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

      14. log1p-udef 1.8%

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

   3. Applied egg-rr 1.8%

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

   4. Taylor expanded in i around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 97.0%

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

Alternative 2: 93.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= ((double) INFINITY)) {
		tmp = expm1((n * log1p((i / n)))) * (100.0 / (i / n));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

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

Python

def code(i, n):
	tmp = 0
	if ((math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= math.inf:
		tmp = math.expm1((n * math.log1p((i / n)))) * (100.0 / (i / n))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

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

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 31.4%

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

      1. clear-num 31.4%

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

      2. un-div-inv 31.4%

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

      3. pow-to-exp 28.9%

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

      4. expm1-def 38.5%

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

      5. *-commutative 38.5%

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

      6. log1p-udef 96.2%

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

   3. Applied egg-rr 96.2%

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

      1. associate-/r/ 95.2%

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

   5. Simplified 95.2%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r/ 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. clear-num 1.8%

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

      8. un-div-inv 1.8%

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

      9. metadata-eval 1.8%

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

      10. sub-neg 1.8%

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

      11. pow-to-exp 1.8%

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

      12. expm1-def 1.8%

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

      13. *-commutative 1.8%

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

      14. log1p-udef 1.8%

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

   3. Applied egg-rr 1.8%

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

   4. Taylor expanded in i around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 96.2%

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

Alternative 3: 93.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= ((double) INFINITY)) {
		tmp = (n * 100.0) * (expm1((n * log1p((i / n)))) / i);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

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

Python

def code(i, n):
	tmp = 0
	if ((math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= math.inf:
		tmp = (n * 100.0) * (math.expm1((n * math.log1p((i / n)))) / i)
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

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

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 31.4%

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

      1. *-commutative 31.4%

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

      2. associate-/r/ 31.3%

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

      3. sub-neg 31.3%

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

      4. metadata-eval 31.3%

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

      5. associate-*r* 31.2%

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

      6. metadata-eval 31.2%

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

      7. sub-neg 31.2%

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

      8. associate-*l/ 31.3%

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

      9. associate-/l* 31.4%

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

      10. pow-to-exp 28.9%

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

      11. expm1-def 39.3%

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

      12. *-commutative 39.3%

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

      13. log1p-udef 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. associate-/r/ 96.2%

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

   5. Simplified 96.2%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r/ 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. clear-num 1.8%

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

      8. un-div-inv 1.8%

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

      9. metadata-eval 1.8%

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

      10. sub-neg 1.8%

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

      11. pow-to-exp 1.8%

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

      12. expm1-def 1.8%

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

      13. *-commutative 1.8%

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

      14. log1p-udef 1.8%

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

   3. Applied egg-rr 1.8%

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

   4. Taylor expanded in i around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 97.0%

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

Alternative 4: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -8 \cdot 10^{+18} \lor \neg \left(i \leq 0.11\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -8e+18) (not (<= i 0.11)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -8e+18) || !(i <= 0.11)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -8e+18) || !(i <= 0.11)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -8e+18) or not (i <= 0.11):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -8e+18) || !(i <= 0.11))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -8e+18], N[Not[LessEqual[i, 0.11]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -8e18 or 0.110000000000000001 < i

   1. Initial program 45.5%

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

   2. Taylor expanded in n around inf 69.7%

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

      1. expm1-def 69.7%

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

   4. Simplified 69.7%

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

   if -8e18 < i < 0.110000000000000001

   1. Initial program 11.9%

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

      1. *-commutative 11.9%

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

      2. associate-/r/ 12.5%

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

      3. sub-neg 12.5%

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

      4. metadata-eval 12.5%

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

      5. associate-*r* 12.5%

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

      6. *-commutative 12.5%

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

      7. clear-num 12.5%

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

      8. un-div-inv 12.5%

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

      9. metadata-eval 12.5%

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

      10. sub-neg 12.5%

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

      11. pow-to-exp 12.5%

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

      12. expm1-def 19.6%

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

      13. *-commutative 19.6%

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

      14. log1p-udef 70.8%

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

   3. Applied egg-rr 70.8%

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

   4. Taylor expanded in i around 0 94.3%

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

      1. sub-neg 94.3%

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

      2. associate-*r/ 94.3%

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

      3. metadata-eval 94.3%

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

      4. metadata-eval 94.3%

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

   6. Simplified 94.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8 \cdot 10^{+18} \lor \neg \left(i \leq 0.11\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 5: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.01 \lor \neg \left(n \leq 0.45\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -0.01) (not (<= n 0.45)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -0.01) || !(n <= 0.45)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -0.01) || !(n <= 0.45)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -0.01) or not (n <= 0.45):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -0.01) || !(n <= 0.45))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -0.01], N[Not[LessEqual[n, 0.45]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -0.0100000000000000002 or 0.450000000000000011 < n

   1. Initial program 20.2%

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

   2. Taylor expanded in n around inf 40.9%

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

      1. *-commutative 40.9%

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

      2. associate-/l* 40.9%

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

      3. expm1-def 96.7%

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

   4. Simplified 96.7%

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

   if -0.0100000000000000002 < n < 0.450000000000000011

   1. Initial program 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-/r/ 31.5%

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

      3. sub-neg 31.5%

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

      4. metadata-eval 31.5%

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

      5. associate-*r* 31.5%

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

      6. *-commutative 31.5%

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

      7. clear-num 31.5%

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

      8. un-div-inv 31.5%

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

      9. metadata-eval 31.5%

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

      10. sub-neg 31.5%

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

      11. pow-to-exp 31.5%

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

      12. expm1-def 51.9%

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

      13. *-commutative 51.9%

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

      14. log1p-udef 90.7%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in i around 0 82.2%

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

      1. sub-neg 82.2%

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

      2. associate-*r/ 82.2%

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

      3. metadata-eval 82.2%

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

      4. metadata-eval 82.2%

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

   6. Simplified 82.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.01 \lor \neg \left(n \leq 0.45\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 6: 66.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;n \leq -2.1 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -6.5 \cdot 10^{-213}:\\ \;\;\;\;10000 \cdot \frac{n}{100 - i \cdot 50}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-166}:\\ \;\;\;\;-100 \cdot \frac{n}{i \cdot \left(0.5 - \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n 100.0) (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666)))))))
   (if (<= n -2.1e+204)
     t_0
     (if (<= n -6.5e-213)
       (* 10000.0 (/ n (- 100.0 (* i 50.0))))
       (if (<= n 1.3e-166) (* -100.0 (/ n (* i (- 0.5 (/ 0.5 n))))) t_0)))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	double tmp;
	if (n <= -2.1e+204) {
		tmp = t_0;
	} else if (n <= -6.5e-213) {
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	} else if (n <= 1.3e-166) {
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / 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) * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0))))
    if (n <= (-2.1d+204)) then
        tmp = t_0
    else if (n <= (-6.5d-213)) then
        tmp = 10000.0d0 * (n / (100.0d0 - (i * 50.0d0)))
    else if (n <= 1.3d-166) then
        tmp = (-100.0d0) * (n / (i * (0.5d0 - (0.5d0 / 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) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	double tmp;
	if (n <= -2.1e+204) {
		tmp = t_0;
	} else if (n <= -6.5e-213) {
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	} else if (n <= 1.3e-166) {
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))
	tmp = 0
	if n <= -2.1e+204:
		tmp = t_0
	elif n <= -6.5e-213:
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)))
	elif n <= 1.3e-166:
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666)))))
	tmp = 0.0
	if (n <= -2.1e+204)
		tmp = t_0;
	elseif (n <= -6.5e-213)
		tmp = Float64(10000.0 * Float64(n / Float64(100.0 - Float64(i * 50.0))));
	elseif (n <= 1.3e-166)
		tmp = Float64(-100.0 * Float64(n / Float64(i * Float64(0.5 - Float64(0.5 / n)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	tmp = 0.0;
	if (n <= -2.1e+204)
		tmp = t_0;
	elseif (n <= -6.5e-213)
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	elseif (n <= 1.3e-166)
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.1e+204], t$95$0, If[LessEqual[n, -6.5e-213], N[(10000.0 * N[(n / N[(100.0 - N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.3e-166], N[(-100.0 * N[(n / N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.1e204 or 1.29999999999999995e-166 < n

   1. Initial program 16.0%

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

      1. *-commutative 16.0%

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

      2. associate-/r/ 16.4%

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

      3. associate-*l* 16.4%

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

      4. sub-neg 16.4%

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

      5. metadata-eval 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in n around inf 33.4%

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

      1. expm1-def 89.4%

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

   6. Simplified 89.4%

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

      1. div-inv 89.3%

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

   8. Applied egg-rr 89.3%

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

   9. Taylor expanded in i around 0 74.6%

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

       1. +-commutative 74.6%

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

       2. unpow2 74.6%

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

       3. associate-*r* 74.6%

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

       4. distribute-rgt-out 74.6%

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

       5. *-commutative 74.6%

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

   11. Simplified 74.6%

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

   if -2.1e204 < n < -6.5e-213

   1. Initial program 26.8%

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

   2. Taylor expanded in i around 0 57.3%

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

      1. sub-neg 57.3%

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

      2. associate-*r/ 57.3%

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

      3. metadata-eval 57.3%

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

      4. distribute-neg-frac 57.3%

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

      5. metadata-eval 57.3%

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

   4. Simplified 57.3%

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

      1. distribute-rgt-in 57.3%

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

      2. flip-+ 43.0%

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

      3. *-commutative 43.0%

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

      4. *-commutative 43.0%

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

      5. swap-sqr 42.3%

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

      6. metadata-eval 42.3%

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

      7. associate-*l* 42.3%

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

      8. associate-*l* 42.3%

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

      9. associate-*l* 42.3%

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in i around 0 53.6%

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

      1. *-commutative 53.6%

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

      2. unpow2 53.6%

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

      3. associate-*l* 54.2%

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

   9. Simplified 54.2%

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

   10. Taylor expanded in n around inf 65.5%

       \[\leadsto \color{blue}{10000 \cdot \frac{n}{100 - 50 \cdot i}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv 65.5%

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

       2. metadata-eval 65.5%

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

       3. metadata-eval 65.5%

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

       4. cancel-sign-sub-inv 65.5%

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

       5. *-commutative 65.5%

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

   12. Simplified 65.5%

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

   if -6.5e-213 < n < 1.29999999999999995e-166

   1. Initial program 61.7%

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

   2. Taylor expanded in i around 0 12.4%

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

      1. sub-neg 12.4%

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

      2. associate-*r/ 12.4%

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

      3. metadata-eval 12.4%

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

      4. distribute-neg-frac 12.4%

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

      5. metadata-eval 12.4%

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

   4. Simplified 12.4%

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

      1. distribute-rgt-in 12.4%

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

      2. flip-+ 6.3%

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

      3. *-commutative 6.3%

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

      4. *-commutative 6.3%

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

      5. swap-sqr 6.3%

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

      6. metadata-eval 6.3%

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

      7. associate-*l* 6.3%

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

      8. associate-*l* 6.3%

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

      9. associate-*l* 6.3%

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

   6. Applied egg-rr 6.3%

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

   7. Taylor expanded in i around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. associate-*l* 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in i around inf 83.4%

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

       1. associate-*r/ 83.4%

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

       2. metadata-eval 83.4%

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

   12. Simplified 83.4%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{+204}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;n \leq -6.5 \cdot 10^{-213}:\\ \;\;\;\;10000 \cdot \frac{n}{100 - i \cdot 50}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-166}:\\ \;\;\;\;-100 \cdot \frac{n}{i \cdot \left(0.5 - \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 7: 64.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -5.9 \cdot 10^{+205}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -3.2 \cdot 10^{-218}:\\ \;\;\;\;10000 \cdot \frac{n}{100 - i \cdot 50}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-166}:\\ \;\;\;\;-100 \cdot \frac{n}{i \cdot \left(0.5 - \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -5.9e+205)
     t_0
     (if (<= n -3.2e-218)
       (* 10000.0 (/ n (- 100.0 (* i 50.0))))
       (if (<= n 1.5e-166) (* -100.0 (/ n (* i (- 0.5 (/ 0.5 n))))) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -5.9e+205) {
		tmp = t_0;
	} else if (n <= -3.2e-218) {
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	} else if (n <= 1.5e-166) {
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / 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))
    if (n <= (-5.9d+205)) then
        tmp = t_0
    else if (n <= (-3.2d-218)) then
        tmp = 10000.0d0 * (n / (100.0d0 - (i * 50.0d0)))
    else if (n <= 1.5d-166) then
        tmp = (-100.0d0) * (n / (i * (0.5d0 - (0.5d0 / 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));
	double tmp;
	if (n <= -5.9e+205) {
		tmp = t_0;
	} else if (n <= -3.2e-218) {
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	} else if (n <= 1.5e-166) {
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -5.9e+205:
		tmp = t_0
	elif n <= -3.2e-218:
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)))
	elif n <= 1.5e-166:
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -5.9e+205)
		tmp = t_0;
	elseif (n <= -3.2e-218)
		tmp = Float64(10000.0 * Float64(n / Float64(100.0 - Float64(i * 50.0))));
	elseif (n <= 1.5e-166)
		tmp = Float64(-100.0 * Float64(n / Float64(i * Float64(0.5 - Float64(0.5 / n)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -5.9e+205)
		tmp = t_0;
	elseif (n <= -3.2e-218)
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	elseif (n <= 1.5e-166)
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.9e+205], t$95$0, If[LessEqual[n, -3.2e-218], N[(10000.0 * N[(n / N[(100.0 - N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5e-166], N[(-100.0 * N[(n / N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.9000000000000001e205 or 1.5000000000000001e-166 < n

   1. Initial program 16.0%

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

   2. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/l* 33.5%

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

      3. expm1-def 89.5%

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

   4. Simplified 89.5%

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

   5. Taylor expanded in i around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. distribute-rgt-out 71.7%

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

   7. Simplified 71.7%

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

   if -5.9000000000000001e205 < n < -3.2000000000000001e-218

   1. Initial program 26.8%

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

   2. Taylor expanded in i around 0 57.3%

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

      1. sub-neg 57.3%

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

      2. associate-*r/ 57.3%

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

      3. metadata-eval 57.3%

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

      4. distribute-neg-frac 57.3%

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

      5. metadata-eval 57.3%

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

   4. Simplified 57.3%

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

      1. distribute-rgt-in 57.3%

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

      2. flip-+ 43.0%

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

      3. *-commutative 43.0%

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

      4. *-commutative 43.0%

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

      5. swap-sqr 42.3%

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

      6. metadata-eval 42.3%

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

      7. associate-*l* 42.3%

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

      8. associate-*l* 42.3%

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

      9. associate-*l* 42.3%

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in i around 0 53.6%

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

      1. *-commutative 53.6%

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

      2. unpow2 53.6%

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

      3. associate-*l* 54.2%

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

   9. Simplified 54.2%

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

   10. Taylor expanded in n around inf 65.5%

       \[\leadsto \color{blue}{10000 \cdot \frac{n}{100 - 50 \cdot i}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv 65.5%

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

       2. metadata-eval 65.5%

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

       3. metadata-eval 65.5%

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

       4. cancel-sign-sub-inv 65.5%

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

       5. *-commutative 65.5%

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

   12. Simplified 65.5%

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

   if -3.2000000000000001e-218 < n < 1.5000000000000001e-166

   1. Initial program 61.7%

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

   2. Taylor expanded in i around 0 12.4%

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

      1. sub-neg 12.4%

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

      2. associate-*r/ 12.4%

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

      3. metadata-eval 12.4%

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

      4. distribute-neg-frac 12.4%

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

      5. metadata-eval 12.4%

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

   4. Simplified 12.4%

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

      1. distribute-rgt-in 12.4%

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

      2. flip-+ 6.3%

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

      3. *-commutative 6.3%

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

      4. *-commutative 6.3%

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

      5. swap-sqr 6.3%

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

      6. metadata-eval 6.3%

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

      7. associate-*l* 6.3%

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

      8. associate-*l* 6.3%

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

      9. associate-*l* 6.3%

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

   6. Applied egg-rr 6.3%

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

   7. Taylor expanded in i around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. associate-*l* 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in i around inf 83.4%

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

       1. associate-*r/ 83.4%

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

       2. metadata-eval 83.4%

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

   12. Simplified 83.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{+205}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -3.2 \cdot 10^{-218}:\\ \;\;\;\;10000 \cdot \frac{n}{100 - i \cdot 50}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-166}:\\ \;\;\;\;-100 \cdot \frac{n}{i \cdot \left(0.5 - \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 8: 64.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.15 \cdot 10^{+204}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{-219}:\\ \;\;\;\;10000 \cdot \frac{n}{100 - i \cdot 50}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-166}:\\ \;\;\;\;-100 \cdot \frac{n}{i \cdot \left(0.5 - \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 - 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.15e+204)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n -3.7e-219)
     (* 10000.0 (/ n (- 100.0 (* i 50.0))))
     (if (<= n 1.65e-166)
       (* -100.0 (/ n (* i (- 0.5 (/ 0.5 n)))))
       (* 100.0 (+ n (* i (- (* n 0.5) 0.5))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.15e+204) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= -3.7e-219) {
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	} else if (n <= 1.65e-166) {
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))));
	} else {
		tmp = 100.0 * (n + (i * ((n * 0.5) - 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.15d+204)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= (-3.7d-219)) then
        tmp = 10000.0d0 * (n / (100.0d0 - (i * 50.0d0)))
    else if (n <= 1.65d-166) then
        tmp = (-100.0d0) * (n / (i * (0.5d0 - (0.5d0 / n))))
    else
        tmp = 100.0d0 * (n + (i * ((n * 0.5d0) - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.15e+204) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= -3.7e-219) {
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	} else if (n <= 1.65e-166) {
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))));
	} else {
		tmp = 100.0 * (n + (i * ((n * 0.5) - 0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.15e+204:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= -3.7e-219:
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)))
	elif n <= 1.65e-166:
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))))
	else:
		tmp = 100.0 * (n + (i * ((n * 0.5) - 0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.15e+204)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= -3.7e-219)
		tmp = Float64(10000.0 * Float64(n / Float64(100.0 - Float64(i * 50.0))));
	elseif (n <= 1.65e-166)
		tmp = Float64(-100.0 * Float64(n / Float64(i * Float64(0.5 - Float64(0.5 / n)))));
	else
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(Float64(n * 0.5) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.15e+204)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= -3.7e-219)
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	elseif (n <= 1.65e-166)
		tmp = -100.0 * (n / (i * (0.5 - (0.5 / n))));
	else
		tmp = 100.0 * (n + (i * ((n * 0.5) - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.15e+204], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -3.7e-219], N[(10000.0 * N[(n / N[(100.0 - N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e-166], N[(-100.0 * N[(n / N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.15e204

   1. Initial program 9.3%

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

   2. Taylor expanded in n around inf 56.4%

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

      1. *-commutative 56.4%

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

      2. associate-/l* 56.4%

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

      3. expm1-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in i around 0 65.0%

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

      1. associate-*r* 65.0%

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

      2. distribute-rgt-out 65.0%

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

   7. Simplified 65.0%

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

   if -2.15e204 < n < -3.7e-219

   1. Initial program 26.8%

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

   2. Taylor expanded in i around 0 57.3%

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

      1. sub-neg 57.3%

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

      2. associate-*r/ 57.3%

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

      3. metadata-eval 57.3%

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

      4. distribute-neg-frac 57.3%

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

      5. metadata-eval 57.3%

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

   4. Simplified 57.3%

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

      1. distribute-rgt-in 57.3%

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

      2. flip-+ 43.0%

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

      3. *-commutative 43.0%

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

      4. *-commutative 43.0%

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

      5. swap-sqr 42.3%

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

      6. metadata-eval 42.3%

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

      7. associate-*l* 42.3%

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

      8. associate-*l* 42.3%

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

      9. associate-*l* 42.3%

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in i around 0 53.6%

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

      1. *-commutative 53.6%

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

      2. unpow2 53.6%

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

      3. associate-*l* 54.2%

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

   9. Simplified 54.2%

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

   10. Taylor expanded in n around inf 65.5%

       \[\leadsto \color{blue}{10000 \cdot \frac{n}{100 - 50 \cdot i}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv 65.5%

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

       2. metadata-eval 65.5%

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

       3. metadata-eval 65.5%

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

       4. cancel-sign-sub-inv 65.5%

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

       5. *-commutative 65.5%

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

   12. Simplified 65.5%

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

   if -3.7e-219 < n < 1.65000000000000009e-166

   1. Initial program 61.7%

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

   2. Taylor expanded in i around 0 12.4%

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

      1. sub-neg 12.4%

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

      2. associate-*r/ 12.4%

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

      3. metadata-eval 12.4%

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

      4. distribute-neg-frac 12.4%

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

      5. metadata-eval 12.4%

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

   4. Simplified 12.4%

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

      1. distribute-rgt-in 12.4%

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

      2. flip-+ 6.3%

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

      3. *-commutative 6.3%

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

      4. *-commutative 6.3%

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

      5. swap-sqr 6.3%

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

      6. metadata-eval 6.3%

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

      7. associate-*l* 6.3%

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

      8. associate-*l* 6.3%

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

      9. associate-*l* 6.3%

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

   6. Applied egg-rr 6.3%

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

   7. Taylor expanded in i around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. associate-*l* 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in i around inf 83.4%

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

       1. associate-*r/ 83.4%

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

       2. metadata-eval 83.4%

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

   12. Simplified 83.4%

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

   if 1.65000000000000009e-166 < n

   1. Initial program 18.1%

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

   2. Taylor expanded in i around 0 73.9%

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

      1. sub-neg 73.9%

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

      2. associate-*r/ 73.9%

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

      3. metadata-eval 73.9%

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

      4. distribute-neg-frac 73.9%

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

      5. metadata-eval 73.9%

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

   4. Simplified 73.9%

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

   5. Taylor expanded in n around 0 73.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.15 \cdot 10^{+204}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{-219}:\\ \;\;\;\;10000 \cdot \frac{n}{100 - i \cdot 50}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-166}:\\ \;\;\;\;-100 \cdot \frac{n}{i \cdot \left(0.5 - \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 - 0.5\right)\right)\\ \end{array} \]

Alternative 9: 73.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{+204} \lor \neg \left(n \leq 0.45\right):\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.75e+204) (not (<= n 0.45)))
   (* (* n 100.0) (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666)))))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.75e+204) || !(n <= 0.45)) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.75d+204)) .or. (.not. (n <= 0.45d0))) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0))))
    else
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((0.5d0 / n) + (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.75e+204) || !(n <= 0.45)) {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.75e+204) or not (n <= 0.45):
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.75e+204) || !(n <= 0.45))
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666)))));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.75e+204) || ~((n <= 0.45)))
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	else
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.75e+204], N[Not[LessEqual[n, 0.45]], $MachinePrecision]], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.74999999999999995e204 or 0.450000000000000011 < n

   1. Initial program 18.7%

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

      1. *-commutative 18.7%

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

      2. associate-/r/ 19.3%

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

      3. associate-*l* 19.3%

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

      4. sub-neg 19.3%

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

      5. metadata-eval 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in n around inf 43.7%

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

      1. expm1-def 98.1%

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

   6. Simplified 98.1%

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

      1. div-inv 98.0%

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

   8. Applied egg-rr 98.0%

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

   9. Taylor expanded in i around 0 78.1%

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

       1. +-commutative 78.1%

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

       2. unpow2 78.1%

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

       3. associate-*r* 78.1%

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

       4. distribute-rgt-out 78.1%

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

       5. *-commutative 78.1%

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

   11. Simplified 78.1%

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

   if -1.74999999999999995e204 < n < 0.450000000000000011

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-/r/ 29.7%

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

      3. sub-neg 29.7%

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

      4. metadata-eval 29.7%

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

      5. associate-*r* 29.7%

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

      6. *-commutative 29.7%

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

      7. clear-num 29.7%

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

      8. un-div-inv 29.7%

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

      9. metadata-eval 29.7%

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

      10. sub-neg 29.7%

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

      11. pow-to-exp 27.6%

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

      12. expm1-def 42.4%

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

      13. *-commutative 42.4%

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

      14. log1p-udef 85.6%

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

   3. Applied egg-rr 85.6%

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

   4. Taylor expanded in i around 0 78.1%

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

      1. sub-neg 78.1%

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

      2. associate-*r/ 78.1%

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

      3. metadata-eval 78.1%

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

      4. metadata-eval 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{+204} \lor \neg \left(n \leq 0.45\right):\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 10: 73.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{+204}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + i \cdot \left(i \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;n \leq 0.45:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.1e+204)
   (* 100.0 (+ n (* n (+ (* i 0.5) (* i (* i 0.25))))))
   (if (<= n 0.45)
     (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))
     (* (* n 100.0) (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.1e+204) {
		tmp = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.25)))));
	} else if (n <= 0.45) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.1d+204)) then
        tmp = 100.0d0 * (n + (n * ((i * 0.5d0) + (i * (i * 0.25d0)))))
    else if (n <= 0.45d0) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((0.5d0 / n) + (-0.5d0))))
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.1e+204) {
		tmp = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.25)))));
	} else if (n <= 0.45) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.1e+204:
		tmp = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.25)))))
	elif n <= 0.45:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	else:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.1e+204)
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(i * 0.5) + Float64(i * Float64(i * 0.25))))));
	elseif (n <= 0.45)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.1e+204)
		tmp = 100.0 * (n + (n * ((i * 0.5) + (i * (i * 0.25)))));
	elseif (n <= 0.45)
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	else
		tmp = (n * 100.0) * (1.0 + (i * (0.5 + (i * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.1e+204], N[(100.0 * N[(n + N[(n * N[(N[(i * 0.5), $MachinePrecision] + N[(i * N[(i * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.45], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.09999999999999975e204

   1. Initial program 9.3%

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

   2. Taylor expanded in n around inf 56.4%

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

      1. *-commutative 56.4%

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

      2. associate-/l* 56.4%

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

      3. expm1-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in i around 0 52.6%

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

      1. *-commutative 52.6%

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

   7. Simplified 52.6%

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

   8. Taylor expanded in i around 0 64.9%

      \[\leadsto \color{blue}{\left(n + \left(0.25 \cdot \left({i}^{2} \cdot n\right) + 0.5 \cdot \left(i \cdot n\right)\right)\right)} \cdot 100 \]
   9. Step-by-step derivation

      1. +-commutative 64.9%

         \[\leadsto \left(n + \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + 0.25 \cdot \left({i}^{2} \cdot n\right)\right)}\right) \cdot 100 \]

      2. associate-*r* 64.9%

         \[\leadsto \left(n + \left(\color{blue}{\left(0.5 \cdot i\right) \cdot n} + 0.25 \cdot \left({i}^{2} \cdot n\right)\right)\right) \cdot 100 \]

      3. associate-*r* 64.9%

         \[\leadsto \left(n + \left(\left(0.5 \cdot i\right) \cdot n + \color{blue}{\left(0.25 \cdot {i}^{2}\right) \cdot n}\right)\right) \cdot 100 \]

      4. distribute-rgt-out 66.3%

         \[\leadsto \left(n + \color{blue}{n \cdot \left(0.5 \cdot i + 0.25 \cdot {i}^{2}\right)}\right) \cdot 100 \]

      5. unpow2 66.3%

         \[\leadsto \left(n + n \cdot \left(0.5 \cdot i + 0.25 \cdot \color{blue}{\left(i \cdot i\right)}\right)\right) \cdot 100 \]

      6. associate-*r* 66.3%

         \[\leadsto \left(n + n \cdot \left(0.5 \cdot i + \color{blue}{\left(0.25 \cdot i\right) \cdot i}\right)\right) \cdot 100 \]

   10. Simplified 66.3%

       \[\leadsto \color{blue}{\left(n + n \cdot \left(0.5 \cdot i + \left(0.25 \cdot i\right) \cdot i\right)\right)} \cdot 100 \]

   if -4.09999999999999975e204 < n < 0.450000000000000011

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-/r/ 29.7%

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

      3. sub-neg 29.7%

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

      4. metadata-eval 29.7%

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

      5. associate-*r* 29.7%

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

      6. *-commutative 29.7%

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

      7. clear-num 29.7%

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

      8. un-div-inv 29.7%

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

      9. metadata-eval 29.7%

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

      10. sub-neg 29.7%

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

      11. pow-to-exp 27.6%

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

      12. expm1-def 42.4%

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

      13. *-commutative 42.4%

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

      14. log1p-udef 85.6%

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

   3. Applied egg-rr 85.6%

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

   4. Taylor expanded in i around 0 78.1%

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

      1. sub-neg 78.1%

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

      2. associate-*r/ 78.1%

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

      3. metadata-eval 78.1%

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

      4. metadata-eval 78.1%

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

   6. Simplified 78.1%

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

   if 0.450000000000000011 < n

   1. Initial program 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/r/ 23.8%

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

      3. associate-*l* 23.8%

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

      4. sub-neg 23.8%

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

      5. metadata-eval 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in n around inf 37.8%

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

      1. expm1-def 97.3%

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

   6. Simplified 97.3%

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

      1. div-inv 97.2%

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

   8. Applied egg-rr 97.2%

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

   9. Taylor expanded in i around 0 83.7%

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

       1. +-commutative 83.7%

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

       2. unpow2 83.7%

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

       3. associate-*r* 83.7%

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

       4. distribute-rgt-out 83.7%

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

       5. *-commutative 83.7%

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

   11. Simplified 83.7%

       \[\leadsto \color{blue}{\left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{+204}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + i \cdot \left(i \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;n \leq 0.45:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 11: 64.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{+205}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-198}:\\ \;\;\;\;10000 \cdot \frac{n}{100 - i \cdot 50}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -5.2e+205)
     t_0
     (if (<= n -3.8e-198)
       (* 10000.0 (/ n (- 100.0 (* i 50.0))))
       (if (<= n 1.75e-166) 0.0 t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -5.2e+205) {
		tmp = t_0;
	} else if (n <= -3.8e-198) {
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	} else if (n <= 1.75e-166) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-5.2d+205)) then
        tmp = t_0
    else if (n <= (-3.8d-198)) then
        tmp = 10000.0d0 * (n / (100.0d0 - (i * 50.0d0)))
    else if (n <= 1.75d-166) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -5.2e+205) {
		tmp = t_0;
	} else if (n <= -3.8e-198) {
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	} else if (n <= 1.75e-166) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -5.2e+205:
		tmp = t_0
	elif n <= -3.8e-198:
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)))
	elif n <= 1.75e-166:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -5.2e+205)
		tmp = t_0;
	elseif (n <= -3.8e-198)
		tmp = Float64(10000.0 * Float64(n / Float64(100.0 - Float64(i * 50.0))));
	elseif (n <= 1.75e-166)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -5.2e+205)
		tmp = t_0;
	elseif (n <= -3.8e-198)
		tmp = 10000.0 * (n / (100.0 - (i * 50.0)));
	elseif (n <= 1.75e-166)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.2e+205], t$95$0, If[LessEqual[n, -3.8e-198], N[(10000.0 * N[(n / N[(100.0 - N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-166], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e205 or 1.75e-166 < n

   1. Initial program 16.0%

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

   2. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/l* 33.5%

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

      3. expm1-def 89.5%

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

   4. Simplified 89.5%

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

   5. Taylor expanded in i around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. distribute-rgt-out 71.7%

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

   7. Simplified 71.7%

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

   if -5.1999999999999998e205 < n < -3.8000000000000002e-198

   1. Initial program 25.0%

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

   2. Taylor expanded in i around 0 58.3%

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

      1. sub-neg 58.3%

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

      2. associate-*r/ 58.3%

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

      3. metadata-eval 58.3%

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

      4. distribute-neg-frac 58.3%

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

      5. metadata-eval 58.3%

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

   4. Simplified 58.3%

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

      1. distribute-rgt-in 58.3%

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

      2. flip-+ 44.8%

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

      3. *-commutative 44.8%

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

      4. *-commutative 44.8%

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

      5. swap-sqr 44.0%

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

      6. metadata-eval 44.0%

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

      7. associate-*l* 44.0%

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

      8. associate-*l* 44.0%

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

      9. associate-*l* 44.0%

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

   6. Applied egg-rr 44.0%

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

   7. Taylor expanded in i around 0 52.9%

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

      1. *-commutative 52.9%

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

      2. unpow2 52.9%

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

      3. associate-*l* 53.5%

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

   9. Simplified 53.5%

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

   10. Taylor expanded in n around inf 65.4%

       \[\leadsto \color{blue}{10000 \cdot \frac{n}{100 - 50 \cdot i}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv 65.4%

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

       2. metadata-eval 65.4%

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

       3. metadata-eval 65.4%

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

       4. cancel-sign-sub-inv 65.4%

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

       5. *-commutative 65.4%

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

   12. Simplified 65.4%

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

   if -3.8000000000000002e-198 < n < 1.75e-166

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-/r/ 62.6%

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

      3. associate-*l* 62.6%

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

      4. sub-neg 62.6%

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

      5. metadata-eval 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in i around 0 81.7%

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

   5. Taylor expanded in i around 0 81.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{+205}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-198}:\\ \;\;\;\;10000 \cdot \frac{n}{100 - i \cdot 50}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 12: 63.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.45 \lor \neg \left(i \leq 4 \cdot 10^{-9}\right):\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -0.45) (not (<= i 4e-9)))
   (* 200.0 (/ (* n n) i))
   (* 100.0 (+ n (* i -0.5)))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -0.45) || !(i <= 4e-9)) {
		tmp = 200.0 * ((n * n) / i);
	} else {
		tmp = 100.0 * (n + (i * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-0.45d0)) .or. (.not. (i <= 4d-9))) then
        tmp = 200.0d0 * ((n * n) / i)
    else
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -0.45) || !(i <= 4e-9)) {
		tmp = 200.0 * ((n * n) / i);
	} else {
		tmp = 100.0 * (n + (i * -0.5));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -0.45) or not (i <= 4e-9):
		tmp = 200.0 * ((n * n) / i)
	else:
		tmp = 100.0 * (n + (i * -0.5))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -0.45) || !(i <= 4e-9))
		tmp = Float64(200.0 * Float64(Float64(n * n) / i));
	else
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -0.45) || ~((i <= 4e-9)))
		tmp = 200.0 * ((n * n) / i);
	else
		tmp = 100.0 * (n + (i * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -0.45], N[Not[LessEqual[i, 4e-9]], $MachinePrecision]], N[(200.0 * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -0.450000000000000011 or 4.00000000000000025e-9 < 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 20.3%

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

      1. sub-neg 20.3%

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

      2. associate-*r/ 20.3%

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

      3. metadata-eval 20.3%

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

      4. distribute-neg-frac 20.3%

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

      5. metadata-eval 20.3%

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

   4. Simplified 20.3%

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

      1. distribute-rgt-in 20.3%

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

      2. flip-+ 7.8%

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

      3. *-commutative 7.8%

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

      4. *-commutative 7.8%

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

      5. swap-sqr 7.8%

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

      6. metadata-eval 7.8%

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

      7. associate-*l* 7.8%

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

      8. associate-*l* 7.8%

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

      9. associate-*l* 7.8%

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

   6. Applied egg-rr 7.8%

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

   7. Taylor expanded in i around 0 29.2%

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

      1. *-commutative 29.2%

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

      2. unpow2 29.2%

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

      3. associate-*l* 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in n around 0 36.5%

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

       1. unpow2 36.5%

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

   12. Simplified 36.5%

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

   if -0.450000000000000011 < i < 4.00000000000000025e-9

   1. Initial program 10.3%

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

   2. Taylor expanded in i around 0 86.6%

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

      1. sub-neg 86.6%

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

      2. associate-*r/ 86.6%

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

      3. metadata-eval 86.6%

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

      4. distribute-neg-frac 86.6%

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

      5. metadata-eval 86.6%

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

   4. Simplified 86.6%

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

   5. Taylor expanded in n around 0 86.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.45 \lor \neg \left(i \leq 4 \cdot 10^{-9}\right):\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \end{array} \]

Alternative 13: 62.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.6 \cdot 10^{-197} \lor \neg \left(n \leq 2 \cdot 10^{-166}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -7.6e-197) (not (<= n 2e-166))) (* n (+ 100.0 (* i 50.0))) 0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -7.6e-197) || !(n <= 2e-166)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-7.6d-197)) .or. (.not. (n <= 2d-166))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -7.6e-197) || !(n <= 2e-166)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -7.6e-197) or not (n <= 2e-166):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -7.6e-197) || !(n <= 2e-166))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -7.6e-197) || ~((n <= 2e-166)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -7.6e-197], N[Not[LessEqual[n, 2e-166]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -7.5999999999999998e-197 or 2.00000000000000008e-166 < n

   1. Initial program 18.7%

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

   2. Taylor expanded in n around inf 30.9%

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

      1. *-commutative 30.9%

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

      2. associate-/l* 30.9%

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

      3. expm1-def 86.6%

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

   4. Simplified 86.6%

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

   5. Taylor expanded in i around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. distribute-rgt-out 67.6%

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

   7. Simplified 67.6%

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

   if -7.5999999999999998e-197 < n < 2.00000000000000008e-166

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-/r/ 62.6%

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

      3. associate-*l* 62.6%

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

      4. sub-neg 62.6%

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

      5. metadata-eval 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in i around 0 81.7%

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

   5. Taylor expanded in i around 0 81.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.6 \cdot 10^{-197} \lor \neg \left(n \leq 2 \cdot 10^{-166}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 55.9% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-197}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-164}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9e-197) (* n 100.0) (if (<= n 1.35e-164) 0.0 (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9e-197) {
		tmp = n * 100.0;
	} else if (n <= 1.35e-164) {
		tmp = 0.0;
	} 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 (n <= (-9d-197)) then
        tmp = n * 100.0d0
    else if (n <= 1.35d-164) then
        tmp = 0.0d0
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9e-197) {
		tmp = n * 100.0;
	} else if (n <= 1.35e-164) {
		tmp = 0.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9e-197:
		tmp = n * 100.0
	elif n <= 1.35e-164:
		tmp = 0.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9e-197)
		tmp = Float64(n * 100.0);
	elseif (n <= 1.35e-164)
		tmp = 0.0;
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9e-197)
		tmp = n * 100.0;
	elseif (n <= 1.35e-164)
		tmp = 0.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9e-197], N[(n * 100.0), $MachinePrecision], If[LessEqual[n, 1.35e-164], 0.0, N[(n * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.0000000000000002e-197 or 1.3500000000000001e-164 < n

   1. Initial program 18.7%

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

   2. Taylor expanded in i around 0 59.9%

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

      1. *-commutative 59.9%

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

   4. Simplified 59.9%

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

   if -9.0000000000000002e-197 < n < 1.3500000000000001e-164

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-/r/ 62.6%

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

      3. associate-*l* 62.6%

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

      4. sub-neg 62.6%

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

      5. metadata-eval 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in i around 0 81.7%

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

   5. Taylor expanded in i around 0 81.7%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-197}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-164}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 15: 17.6% accurate, 114.0× speedup

Math

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

FPCore

(FPCore (i n) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

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

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 24.8%

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

   1. *-commutative 24.8%

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

   2. associate-/r/ 25.1%

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

   3. associate-*l* 25.0%

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

   4. sub-neg 25.0%

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

   5. metadata-eval 25.0%

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

3. Simplified 25.0%

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

4. Taylor expanded in i around 0 17.3%

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

5. Taylor expanded in i around 0 17.3%

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

6. Final simplification 17.3%

   \[\leadsto 0 \]

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 2023275 
(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))))