Compound Interest

Percentage Accurate: 28.2% → 98.3%

Time: 26.2s

Alternatives: 15

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (/
        n
        (+ 0.01 (+ (* i -0.005) (* 0.0008333333333333334 (pow i 2.0)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * pow(i, 2.0))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * Math.pow(i, 2.0))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * math.pow(i, 2.0))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * -0.005) + Float64(0.0008333333333333334 * (i ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(N[(i * -0.005), $MachinePrecision] + N[(0.0008333333333333334 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 23.0%

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

      1. *-un-lft-identity 23.0%

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

      2. pow-to-exp 22.5%

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

      3. expm1-def 32.2%

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

      4. *-commutative 32.2%

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

      5. log1p-def 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. *-lft-identity 99.3%

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

   5. Simplified 99.3%

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

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

   1. Initial program 99.0%

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

      1. associate-*r/ 99.2%

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

      2. sub-neg 99.2%

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

      3. distribute-rgt-in 99.3%

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

      4. metadata-eval 99.3%

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

      5. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 82.3%

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

   4. Simplified 82.3%

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

      1. associate-*l/ 82.3%

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

      2. associate-/l* 82.2%

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

   6. Applied egg-rr 82.2%

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

   7. Taylor expanded in i around 0 99.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}\\ \end{array} \]

Alternative 2: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* 100.0 (/ n (/ i (expm1 i))))
     (if (<= t_0 INFINITY)
       (* t_0 100.0)
       (/
        n
        (+ 0.01 (+ (* i -0.005) (* 0.0008333333333333334 (pow i 2.0)))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * pow(i, 2.0))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * Math.pow(i, 2.0))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * math.pow(i, 2.0))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * -0.005) + Float64(0.0008333333333333334 * (i ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(n / N[(0.01 + N[(N[(i * -0.005), $MachinePrecision] + N[(0.0008333333333333334 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 23.0%

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

   2. Taylor expanded in n around inf 38.0%

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

      1. *-commutative 38.0%

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

      2. associate-/l* 37.9%

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

      3. expm1-def 80.4%

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

   4. Simplified 80.4%

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

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

   1. Initial program 99.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 82.3%

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

   4. Simplified 82.3%

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

      1. associate-*l/ 82.3%

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

      2. associate-/l* 82.2%

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

   6. Applied egg-rr 82.2%

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

   7. Taylor expanded in i around 0 99.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}\\ \end{array} \]

Alternative 3: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* 100.0 (/ n (/ i (expm1 i))))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (/
        n
        (+ 0.01 (+ (* i -0.005) (* 0.0008333333333333334 (pow i 2.0)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * pow(i, 2.0))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * Math.pow(i, 2.0))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * math.pow(i, 2.0))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * -0.005) + Float64(0.0008333333333333334 * (i ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(N[(i * -0.005), $MachinePrecision] + N[(0.0008333333333333334 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 23.0%

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

   2. Taylor expanded in n around inf 38.0%

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

      1. *-commutative 38.0%

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

      2. associate-/l* 37.9%

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

      3. expm1-def 80.4%

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

   4. Simplified 80.4%

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

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

   1. Initial program 99.0%

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

      1. associate-*r/ 99.2%

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

      2. sub-neg 99.2%

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

      3. distribute-rgt-in 99.3%

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

      4. metadata-eval 99.3%

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

      5. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 82.3%

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

   4. Simplified 82.3%

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

      1. associate-*l/ 82.3%

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

      2. associate-/l* 82.2%

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

   6. Applied egg-rr 82.2%

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

   7. Taylor expanded in i around 0 99.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{+141}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 5e+141)
   (* 100.0 (/ n (/ i (expm1 i))))
   (* 100.0 (- (pow (/ i n) (+ n -1.0)) (/ n i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 5e+141) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 100.0 * (pow((i / n), (n + -1.0)) - (n / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 5e+141) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 100.0 * (Math.pow((i / n), (n + -1.0)) - (n / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 5e+141:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 100.0 * (math.pow((i / n), (n + -1.0)) - (n / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 5e+141)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(100.0 * Float64((Float64(i / n) ^ Float64(n + -1.0)) - Float64(n / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 5e+141], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], N[(n + -1.0), $MachinePrecision]], $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.00000000000000025e141

   1. Initial program 20.8%

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

   2. Taylor expanded in n around inf 30.7%

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

      1. *-commutative 30.7%

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

      2. associate-/l* 30.7%

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

      3. expm1-def 83.2%

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

   4. Simplified 83.2%

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

   if 5.00000000000000025e141 < i

   1. Initial program 59.1%

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

   2. Taylor expanded in i around inf 62.7%

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

      1. div-sub 62.7%

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

      2. pow1 62.7%

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

      3. pow-div 88.6%

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

      4. clear-num 87.6%

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

   4. Applied egg-rr 87.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{+141}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right)\\ \end{array} \]

Alternative 5: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -5.2 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 10^{-77}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{+253}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{n} \cdot -0.005 + 0.01 \cdot \frac{1}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -5.2e-98)
     t_0
     (if (<= i 1e-77)
       (* n (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n))))))
       (if (<= i 5.4e+253)
         t_0
         (/ 1.0 (+ (* (/ i n) -0.005) (* 0.01 (/ 1.0 n)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -5.2e-98) {
		tmp = t_0;
	} else if (i <= 1e-77) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else if (i <= 5.4e+253) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -5.2e-98) {
		tmp = t_0;
	} else if (i <= 1e-77) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else if (i <= 5.4e+253) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -5.2e-98:
		tmp = t_0
	elif i <= 1e-77:
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))
	elif i <= 5.4e+253:
		tmp = t_0
	else:
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -5.2e-98)
		tmp = t_0;
	elseif (i <= 1e-77)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	elseif (i <= 5.4e+253)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(i / n) * -0.005) + Float64(0.01 * Float64(1.0 / n))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -5.2e-98], t$95$0, If[LessEqual[i, 1e-77], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.4e+253], t$95$0, N[(1.0 / N[(N[(N[(i / n), $MachinePrecision] * -0.005), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -5.20000000000000027e-98 or 9.9999999999999993e-78 < i < 5.40000000000000005e253

   1. Initial program 40.0%

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

   2. Taylor expanded in n around inf 54.4%

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

      1. expm1-def 70.8%

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

   4. Simplified 70.8%

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

   if -5.20000000000000027e-98 < i < 9.9999999999999993e-78

   1. Initial program 5.7%

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

      1. associate-*r/ 5.7%

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

      2. associate-/r/ 6.5%

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

      3. *-commutative 6.5%

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

      4. sub-neg 6.5%

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

      5. distribute-lft-in 6.5%

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

      6. fma-def 6.5%

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

      7. metadata-eval 6.5%

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

      8. metadata-eval 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in i around 0 91.9%

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

      1. associate-*r/ 91.9%

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

      2. metadata-eval 91.9%

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

   6. Simplified 91.9%

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

   if 5.40000000000000005e253 < i

   1. Initial program 49.7%

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

   2. Taylor expanded in n around inf 17.8%

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

      1. *-commutative 17.8%

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

      2. associate-/l* 17.8%

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

      3. expm1-def 17.8%

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

   4. Simplified 17.8%

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

      1. associate-*l/ 17.8%

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

      2. clear-num 17.8%

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

   6. Applied egg-rr 17.8%

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

      1. associate-/l/ 17.8%

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

      2. associate-/r* 17.0%

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

      3. *-commutative 17.0%

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

   8. Simplified 17.0%

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

   9. Taylor expanded in i around 0 56.4%

      \[\leadsto \frac{1}{\color{blue}{-0.005 \cdot \frac{i}{n} + 0.01 \cdot \frac{1}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.2 \cdot 10^{-98}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 10^{-77}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 5.4 \cdot 10^{+253}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{n} \cdot -0.005 + 0.01 \cdot \frac{1}{n}}\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-198} \lor \neg \left(n \leq 3.2 \cdot 10^{-130}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3.1e-198) (not (<= n 3.2e-130)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/ 0.0 (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3.1e-198) || !(n <= 3.2e-130)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -3.1e-198) || !(n <= 3.2e-130)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -3.1e-198) or not (n <= 3.2e-130):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3.1e-198) || !(n <= 3.2e-130))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -3.1e-198], N[Not[LessEqual[n, 3.2e-130]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.0999999999999998e-198 or 3.2e-130 < n

   1. Initial program 20.1%

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

   2. Taylor expanded in n around inf 28.7%

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

      1. *-commutative 28.7%

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

      2. associate-/l* 28.7%

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

      3. expm1-def 85.0%

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

   4. Simplified 85.0%

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

   if -3.0999999999999998e-198 < n < 3.2e-130

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in i around 0 65.5%

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

      1. +-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in i around 0 75.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-198} \lor \neg \left(n \leq 3.2 \cdot 10^{-130}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 7: 65.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.8 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{-197}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 + \frac{-0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.8e+240)
   (/ (* (* i -100.0) (- n)) i)
   (if (<= n -4.4e-197)
     (/ n (+ 0.01 (* i -0.005)))
     (if (<= n 1.25e-129)
       (/ 0.0 (/ i n))
       (* 100.0 (+ n (* (+ 0.5 (/ -0.5 n)) (* i n))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.8e+240) {
		tmp = ((i * -100.0) * -n) / i;
	} else if (n <= -4.4e-197) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.25e-129) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n + ((0.5 + (-0.5 / n)) * (i * n)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-6.8d+240)) then
        tmp = ((i * (-100.0d0)) * -n) / i
    else if (n <= (-4.4d-197)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 1.25d-129) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = 100.0d0 * (n + ((0.5d0 + ((-0.5d0) / n)) * (i * n)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -6.8e+240) {
		tmp = ((i * -100.0) * -n) / i;
	} else if (n <= -4.4e-197) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.25e-129) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n + ((0.5 + (-0.5 / n)) * (i * n)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -6.8e+240:
		tmp = ((i * -100.0) * -n) / i
	elif n <= -4.4e-197:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 1.25e-129:
		tmp = 0.0 / (i / n)
	else:
		tmp = 100.0 * (n + ((0.5 + (-0.5 / n)) * (i * n)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.8e+240)
		tmp = Float64(Float64(Float64(i * -100.0) * Float64(-n)) / i);
	elseif (n <= -4.4e-197)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 1.25e-129)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 + Float64(-0.5 / n)) * Float64(i * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -6.8e+240)
		tmp = ((i * -100.0) * -n) / i;
	elseif (n <= -4.4e-197)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 1.25e-129)
		tmp = 0.0 / (i / n);
	else
		tmp = 100.0 * (n + ((0.5 + (-0.5 / n)) * (i * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.8e+240], N[(N[(N[(i * -100.0), $MachinePrecision] * (-n)), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -4.4e-197], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.25e-129], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(N[(0.5 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.80000000000000017e240

   1. Initial program 5.4%

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

      1. associate-*r/ 5.4%

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

      2. sub-neg 5.4%

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

      3. distribute-rgt-in 5.4%

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

      4. metadata-eval 5.4%

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

      5. metadata-eval 5.4%

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

      6. frac-2neg 5.4%

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

      7. *-commutative 5.4%

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

      8. fma-udef 5.4%

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

      9. neg-sub0 5.4%

         \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{-\frac{i}{n}} \]

      10. fma-udef 5.4%

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

      11. *-commutative 5.4%

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

      12. +-commutative 5.4%

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

      13. associate--r+ 5.4%

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

      14. metadata-eval 5.4%

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

      15. distribute-neg-frac 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. associate-/l* 6.1%

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

      2. *-commutative 6.1%

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

      3. associate-/l* 6.1%

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

      4. associate-/r/ 5.4%

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

      5. *-commutative 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in i around 0 11.4%

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

      1. *-commutative 11.4%

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

   8. Simplified 11.4%

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

      1. *-commutative 11.4%

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

      2. frac-2neg 11.4%

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

      3. associate-*r/ 66.8%

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

      4. remove-double-neg 66.8%

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

   10. Applied egg-rr 66.8%

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

   if -6.80000000000000017e240 < n < -4.4000000000000001e-197

   1. Initial program 30.4%

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

   2. Taylor expanded in n around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 23.2%

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

      3. expm1-def 74.8%

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

   4. Simplified 74.8%

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

      1. associate-*l/ 74.9%

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

      2. associate-/l* 74.8%

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

   6. Applied egg-rr 74.8%

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

   7. Taylor expanded in i around 0 61.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 61.6%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 61.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -4.4000000000000001e-197 < n < 1.25000000000000007e-129

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in i around 0 65.5%

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

      1. +-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in i around 0 75.5%

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

   if 1.25000000000000007e-129 < n

   1. Initial program 15.0%

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

   2. Taylor expanded in i around 0 73.4%

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

      1. associate-*r* 73.4%

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

      2. cancel-sign-sub-inv 73.4%

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

      3. metadata-eval 73.4%

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

      4. associate-*r/ 73.4%

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

      5. metadata-eval 73.4%

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

   4. Simplified 73.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.8 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{-197}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 + \frac{-0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 8: 65.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -6.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.5e+241)
   (/ (* (* i -100.0) (- n)) i)
   (if (<= n -6.5e-198)
     (/ n (+ 0.01 (* i -0.005)))
     (if (<= n 3.2e-130)
       (/ 0.0 (/ i n))
       (* n (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.5e+241) {
		tmp = ((i * -100.0) * -n) / i;
	} else if (n <= -6.5e-198) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 3.2e-130) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.5d+241)) then
        tmp = ((i * (-100.0d0)) * -n) / i
    else if (n <= (-6.5d-198)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 3.2d-130) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.5e+241) {
		tmp = ((i * -100.0) * -n) / i;
	} else if (n <= -6.5e-198) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 3.2e-130) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.5e+241:
		tmp = ((i * -100.0) * -n) / i
	elif n <= -6.5e-198:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 3.2e-130:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.5e+241)
		tmp = Float64(Float64(Float64(i * -100.0) * Float64(-n)) / i);
	elseif (n <= -6.5e-198)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 3.2e-130)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.5e+241)
		tmp = ((i * -100.0) * -n) / i;
	elseif (n <= -6.5e-198)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 3.2e-130)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.5e+241], N[(N[(N[(i * -100.0), $MachinePrecision] * (-n)), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -6.5e-198], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.2e-130], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.50000000000000013e241

   1. Initial program 5.4%

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

      1. associate-*r/ 5.4%

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

      2. sub-neg 5.4%

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

      3. distribute-rgt-in 5.4%

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

      4. metadata-eval 5.4%

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

      5. metadata-eval 5.4%

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

      6. frac-2neg 5.4%

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

      7. *-commutative 5.4%

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

      8. fma-udef 5.4%

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

      9. neg-sub0 5.4%

         \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{-\frac{i}{n}} \]

      10. fma-udef 5.4%

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

      11. *-commutative 5.4%

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

      12. +-commutative 5.4%

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

      13. associate--r+ 5.4%

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

      14. metadata-eval 5.4%

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

      15. distribute-neg-frac 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. associate-/l* 6.1%

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

      2. *-commutative 6.1%

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

      3. associate-/l* 6.1%

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

      4. associate-/r/ 5.4%

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

      5. *-commutative 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in i around 0 11.4%

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

      1. *-commutative 11.4%

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

   8. Simplified 11.4%

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

      1. *-commutative 11.4%

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

      2. frac-2neg 11.4%

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

      3. associate-*r/ 66.8%

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

      4. remove-double-neg 66.8%

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

   10. Applied egg-rr 66.8%

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

   if -2.50000000000000013e241 < n < -6.5000000000000004e-198

   1. Initial program 30.4%

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

   2. Taylor expanded in n around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 23.2%

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

      3. expm1-def 74.8%

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

   4. Simplified 74.8%

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

      1. associate-*l/ 74.9%

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

      2. associate-/l* 74.8%

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

   6. Applied egg-rr 74.8%

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

   7. Taylor expanded in i around 0 61.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 61.6%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 61.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -6.5000000000000004e-198 < n < 3.2e-130

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in i around 0 65.5%

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

      1. +-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in i around 0 75.5%

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

   if 3.2e-130 < n

   1. Initial program 15.0%

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

      1. associate-*r/ 15.1%

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

      2. associate-/r/ 15.5%

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

      3. *-commutative 15.5%

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

      4. sub-neg 15.5%

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

      5. distribute-lft-in 15.5%

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

      6. fma-def 15.5%

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

      7. metadata-eval 15.5%

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

      8. metadata-eval 15.5%

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

   3. Simplified 15.5%

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

   4. Taylor expanded in i around 0 73.4%

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

      1. associate-*r/ 73.4%

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

      2. metadata-eval 73.4%

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

   6. Simplified 73.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -6.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 9: 65.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -3.1 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{\frac{i}{n} \cdot -0.005 + 0.01 \cdot \frac{1}{n}}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.5e+240)
   (/ (* (* i -100.0) (- n)) i)
   (if (<= n -3.1e-198)
     (/ 1.0 (+ (* (/ i n) -0.005) (* 0.01 (/ 1.0 n))))
     (if (<= n 1.1e-129)
       (/ 0.0 (/ i n))
       (* n (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.5e+240) {
		tmp = ((i * -100.0) * -n) / i;
	} else if (n <= -3.1e-198) {
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	} else if (n <= 1.1e-129) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-6.5d+240)) then
        tmp = ((i * (-100.0d0)) * -n) / i
    else if (n <= (-3.1d-198)) then
        tmp = 1.0d0 / (((i / n) * (-0.005d0)) + (0.01d0 * (1.0d0 / n)))
    else if (n <= 1.1d-129) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -6.5e+240) {
		tmp = ((i * -100.0) * -n) / i;
	} else if (n <= -3.1e-198) {
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	} else if (n <= 1.1e-129) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -6.5e+240:
		tmp = ((i * -100.0) * -n) / i
	elif n <= -3.1e-198:
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)))
	elif n <= 1.1e-129:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.5e+240)
		tmp = Float64(Float64(Float64(i * -100.0) * Float64(-n)) / i);
	elseif (n <= -3.1e-198)
		tmp = Float64(1.0 / Float64(Float64(Float64(i / n) * -0.005) + Float64(0.01 * Float64(1.0 / n))));
	elseif (n <= 1.1e-129)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -6.5e+240)
		tmp = ((i * -100.0) * -n) / i;
	elseif (n <= -3.1e-198)
		tmp = 1.0 / (((i / n) * -0.005) + (0.01 * (1.0 / n)));
	elseif (n <= 1.1e-129)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.5e+240], N[(N[(N[(i * -100.0), $MachinePrecision] * (-n)), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -3.1e-198], N[(1.0 / N[(N[(N[(i / n), $MachinePrecision] * -0.005), $MachinePrecision] + N[(0.01 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.1e-129], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.50000000000000018e240

   1. Initial program 5.4%

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

      1. associate-*r/ 5.4%

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

      2. sub-neg 5.4%

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

      3. distribute-rgt-in 5.4%

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

      4. metadata-eval 5.4%

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

      5. metadata-eval 5.4%

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

      6. frac-2neg 5.4%

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

      7. *-commutative 5.4%

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

      8. fma-udef 5.4%

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

      9. neg-sub0 5.4%

         \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{-\frac{i}{n}} \]

      10. fma-udef 5.4%

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

      11. *-commutative 5.4%

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

      12. +-commutative 5.4%

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

      13. associate--r+ 5.4%

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

      14. metadata-eval 5.4%

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

      15. distribute-neg-frac 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. associate-/l* 6.1%

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

      2. *-commutative 6.1%

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

      3. associate-/l* 6.1%

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

      4. associate-/r/ 5.4%

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

      5. *-commutative 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in i around 0 11.4%

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

      1. *-commutative 11.4%

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

   8. Simplified 11.4%

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

      1. *-commutative 11.4%

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

      2. frac-2neg 11.4%

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

      3. associate-*r/ 66.8%

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

      4. remove-double-neg 66.8%

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

   10. Applied egg-rr 66.8%

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

   if -6.50000000000000018e240 < n < -3.0999999999999998e-198

   1. Initial program 30.4%

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

   2. Taylor expanded in n around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 23.2%

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

      3. expm1-def 74.8%

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

   4. Simplified 74.8%

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

      1. associate-*l/ 74.9%

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

      2. clear-num 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. associate-/l/ 63.6%

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

      2. associate-/r* 59.9%

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

      3. *-commutative 59.9%

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

   8. Simplified 59.9%

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

   9. Taylor expanded in i around 0 61.9%

      \[\leadsto \frac{1}{\color{blue}{-0.005 \cdot \frac{i}{n} + 0.01 \cdot \frac{1}{n}}} \]

   if -3.0999999999999998e-198 < n < 1.10000000000000001e-129

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in i around 0 65.5%

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

      1. +-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in i around 0 75.5%

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

   if 1.10000000000000001e-129 < n

   1. Initial program 15.0%

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

      1. associate-*r/ 15.1%

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

      2. associate-/r/ 15.5%

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

      3. *-commutative 15.5%

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

      4. sub-neg 15.5%

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

      5. distribute-lft-in 15.5%

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

      6. fma-def 15.5%

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

      7. metadata-eval 15.5%

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

      8. metadata-eval 15.5%

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

   3. Simplified 15.5%

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

   4. Taylor expanded in i around 0 73.4%

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

      1. associate-*r/ 73.4%

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

      2. metadata-eval 73.4%

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

   6. Simplified 73.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -3.1 \cdot 10^{-198}:\\ \;\;\;\;\frac{1}{\frac{i}{n} \cdot -0.005 + 0.01 \cdot \frac{1}{n}}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 10: 65.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{+241}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.42 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{+30}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* (* i -100.0) (- n)) i)))
   (if (<= n -3.8e+241)
     t_0
     (if (<= n -8.5e-197)
       (/ n (+ 0.01 (* i -0.005)))
       (if (<= n 1.42e-154)
         (/ 0.0 (/ i n))
         (if (<= n 4.1e+30) (* 100.0 (/ i (/ i n))) t_0))))))

C

double code(double i, double n) {
	double t_0 = ((i * -100.0) * -n) / i;
	double tmp;
	if (n <= -3.8e+241) {
		tmp = t_0;
	} else if (n <= -8.5e-197) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.42e-154) {
		tmp = 0.0 / (i / n);
	} else if (n <= 4.1e+30) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((i * (-100.0d0)) * -n) / i
    if (n <= (-3.8d+241)) then
        tmp = t_0
    else if (n <= (-8.5d-197)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 1.42d-154) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 4.1d+30) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = ((i * -100.0) * -n) / i;
	double tmp;
	if (n <= -3.8e+241) {
		tmp = t_0;
	} else if (n <= -8.5e-197) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.42e-154) {
		tmp = 0.0 / (i / n);
	} else if (n <= 4.1e+30) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((i * -100.0) * -n) / i
	tmp = 0
	if n <= -3.8e+241:
		tmp = t_0
	elif n <= -8.5e-197:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 1.42e-154:
		tmp = 0.0 / (i / n)
	elif n <= 4.1e+30:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(i * -100.0) * Float64(-n)) / i)
	tmp = 0.0
	if (n <= -3.8e+241)
		tmp = t_0;
	elseif (n <= -8.5e-197)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 1.42e-154)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 4.1e+30)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = ((i * -100.0) * -n) / i;
	tmp = 0.0;
	if (n <= -3.8e+241)
		tmp = t_0;
	elseif (n <= -8.5e-197)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 1.42e-154)
		tmp = 0.0 / (i / n);
	elseif (n <= 4.1e+30)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(i * -100.0), $MachinePrecision] * (-n)), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -3.8e+241], t$95$0, If[LessEqual[n, -8.5e-197], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.42e-154], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.1e+30], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.79999999999999972e241 or 4.10000000000000005e30 < n

   1. Initial program 13.9%

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

      1. associate-*r/ 13.9%

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

      2. sub-neg 13.9%

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

      3. distribute-rgt-in 13.9%

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

      4. metadata-eval 13.9%

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

      5. metadata-eval 13.9%

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

      6. frac-2neg 13.9%

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

      7. *-commutative 13.9%

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

      8. fma-udef 13.9%

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

      9. neg-sub0 13.9%

         \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{-\frac{i}{n}} \]

      10. fma-udef 13.9%

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

      11. *-commutative 13.9%

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

      12. +-commutative 13.9%

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

      13. associate--r+ 13.9%

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

      14. metadata-eval 13.9%

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

      15. distribute-neg-frac 13.9%

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

   3. Applied egg-rr 13.9%

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

      1. associate-/l* 14.5%

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

      2. *-commutative 14.5%

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

      3. associate-/l* 14.5%

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

      4. associate-/r/ 13.9%

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

      5. *-commutative 13.9%

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

   5. Simplified 13.9%

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

   6. Taylor expanded in i around 0 24.8%

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

      1. *-commutative 24.8%

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

   8. Simplified 24.8%

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

      1. *-commutative 24.8%

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

      2. frac-2neg 24.8%

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

      3. associate-*r/ 70.0%

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

      4. remove-double-neg 70.0%

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

   10. Applied egg-rr 70.0%

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

   if -3.79999999999999972e241 < n < -8.5e-197

   1. Initial program 30.4%

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

   2. Taylor expanded in n around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 23.2%

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

      3. expm1-def 74.8%

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

   4. Simplified 74.8%

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

      1. associate-*l/ 74.9%

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

      2. associate-/l* 74.8%

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

   6. Applied egg-rr 74.8%

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

   7. Taylor expanded in i around 0 61.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 61.6%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 61.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -8.5e-197 < n < 1.42e-154

   1. Initial program 57.6%

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

      1. associate-*r/ 57.6%

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

      2. sub-neg 57.6%

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

      3. distribute-rgt-in 57.6%

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

      4. metadata-eval 57.6%

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

      5. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in i around 0 67.1%

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

      1. +-commutative 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in i around 0 78.2%

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

   if 1.42e-154 < n < 4.10000000000000005e30

   1. Initial program 11.2%

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

   2. Taylor expanded in i around 0 79.3%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.42 \cdot 10^{-154}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{+30}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \end{array} \]

Alternative 11: 65.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.4 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -3.1 \cdot 10^{-198}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8.4e+240)
   (/ (* (* i -100.0) (- n)) i)
   (if (<= n -3.1e-198)
     (/ n (+ 0.01 (* i -0.005)))
     (if (<= n 3.2e-130) (/ 0.0 (/ i n)) (* 100.0 (+ n (* 0.5 (* i n))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8.4e+240) {
		tmp = ((i * -100.0) * -n) / i;
	} else if (n <= -3.1e-198) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 3.2e-130) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n + (0.5 * (i * n)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-8.4d+240)) then
        tmp = ((i * (-100.0d0)) * -n) / i
    else if (n <= (-3.1d-198)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 3.2d-130) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = 100.0d0 * (n + (0.5d0 * (i * n)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8.4e+240) {
		tmp = ((i * -100.0) * -n) / i;
	} else if (n <= -3.1e-198) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 3.2e-130) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = 100.0 * (n + (0.5 * (i * n)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8.4e+240:
		tmp = ((i * -100.0) * -n) / i
	elif n <= -3.1e-198:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 3.2e-130:
		tmp = 0.0 / (i / n)
	else:
		tmp = 100.0 * (n + (0.5 * (i * n)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8.4e+240)
		tmp = Float64(Float64(Float64(i * -100.0) * Float64(-n)) / i);
	elseif (n <= -3.1e-198)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 3.2e-130)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n + Float64(0.5 * Float64(i * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -8.4e+240)
		tmp = ((i * -100.0) * -n) / i;
	elseif (n <= -3.1e-198)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 3.2e-130)
		tmp = 0.0 / (i / n);
	else
		tmp = 100.0 * (n + (0.5 * (i * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8.4e+240], N[(N[(N[(i * -100.0), $MachinePrecision] * (-n)), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -3.1e-198], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.2e-130], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(0.5 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.3999999999999996e240

   1. Initial program 5.4%

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

      1. associate-*r/ 5.4%

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

      2. sub-neg 5.4%

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

      3. distribute-rgt-in 5.4%

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

      4. metadata-eval 5.4%

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

      5. metadata-eval 5.4%

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

      6. frac-2neg 5.4%

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

      7. *-commutative 5.4%

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

      8. fma-udef 5.4%

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

      9. neg-sub0 5.4%

         \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}{-\frac{i}{n}} \]

      10. fma-udef 5.4%

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

      11. *-commutative 5.4%

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

      12. +-commutative 5.4%

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

      13. associate--r+ 5.4%

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

      14. metadata-eval 5.4%

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

      15. distribute-neg-frac 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. associate-/l* 6.1%

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

      2. *-commutative 6.1%

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

      3. associate-/l* 6.1%

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

      4. associate-/r/ 5.4%

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

      5. *-commutative 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in i around 0 11.4%

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

      1. *-commutative 11.4%

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

   8. Simplified 11.4%

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

      1. *-commutative 11.4%

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

      2. frac-2neg 11.4%

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

      3. associate-*r/ 66.8%

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

      4. remove-double-neg 66.8%

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

   10. Applied egg-rr 66.8%

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

   if -8.3999999999999996e240 < n < -3.0999999999999998e-198

   1. Initial program 30.4%

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

   2. Taylor expanded in n around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 23.2%

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

      3. expm1-def 74.8%

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

   4. Simplified 74.8%

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

      1. associate-*l/ 74.9%

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

      2. associate-/l* 74.8%

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

   6. Applied egg-rr 74.8%

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

   7. Taylor expanded in i around 0 61.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 61.6%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 61.6%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -3.0999999999999998e-198 < n < 3.2e-130

   1. Initial program 51.9%

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

      1. associate-*r/ 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-rgt-in 51.9%

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

      4. metadata-eval 51.9%

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

      5. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in i around 0 65.5%

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

      1. +-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in i around 0 75.5%

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

   if 3.2e-130 < n

   1. Initial program 15.0%

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

   2. Taylor expanded in n around inf 27.9%

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

      1. *-commutative 27.9%

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

      2. associate-/l* 27.9%

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

      3. expm1-def 90.9%

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

   4. Simplified 90.9%

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

   5. Taylor expanded in i around 0 73.4%

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

      1. *-commutative 73.4%

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

      2. *-commutative 73.4%

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

   7. Simplified 73.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.4 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(i \cdot -100\right) \cdot \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq -3.1 \cdot 10^{-198}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + 0.5 \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 12: 56.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+53} \lor \neg \left(i \leq 0.05\right):\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -5e+53) (not (<= i 0.05))) (* 100.0 (* i (/ n i))) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -5e+53) || !(i <= 0.05)) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-5d+53)) .or. (.not. (i <= 0.05d0))) then
        tmp = 100.0d0 * (i * (n / i))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -5e+53) || !(i <= 0.05)) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -5e+53) or not (i <= 0.05):
		tmp = 100.0 * (i * (n / i))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -5e+53) || !(i <= 0.05))
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -5e+53) || ~((i <= 0.05)))
		tmp = 100.0 * (i * (n / i));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -5e+53], N[Not[LessEqual[i, 0.05]], $MachinePrecision]], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -5.0000000000000004e53 or 0.050000000000000003 < i

   1. Initial program 54.4%

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

   2. Taylor expanded in i around 0 25.9%

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

      1. div-inv 25.9%

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

      2. *-commutative 25.9%

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

      3. clear-num 23.9%

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

   4. Applied egg-rr 23.9%

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

   if -5.0000000000000004e53 < i < 0.050000000000000003

   1. Initial program 7.3%

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

   2. Taylor expanded in i around 0 82.7%

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

      1. *-commutative 82.7%

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

   4. Simplified 82.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+53} \lor \neg \left(i \leq 0.05\right):\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 13: 56.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{+53} \lor \neg \left(i \leq 6.8 \cdot 10^{-10}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1e+53) (not (<= i 6.8e-10)))
   (* 100.0 (/ i (/ i n)))
   (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1e+53) || !(i <= 6.8e-10)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-1d+53)) .or. (.not. (i <= 6.8d-10))) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1e+53) || !(i <= 6.8e-10)) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1e+53) or not (i <= 6.8e-10):
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1e+53) || !(i <= 6.8e-10))
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -1e+53) || ~((i <= 6.8e-10)))
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1e+53], N[Not[LessEqual[i, 6.8e-10]], $MachinePrecision]], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -9.9999999999999999e52 or 6.8000000000000003e-10 < i

   1. Initial program 53.4%

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

   2. Taylor expanded in i around 0 26.1%

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

   if -9.9999999999999999e52 < i < 6.8000000000000003e-10

   1. Initial program 7.4%

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

   2. Taylor expanded in i around 0 83.3%

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

      1. *-commutative 83.3%

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

   4. Simplified 83.3%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{+53} \lor \neg \left(i \leq 6.8 \cdot 10^{-10}\right):\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 14: 58.6% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \frac{n}{0.01 + i \cdot -0.005} \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (/ n (+ 0.01 (* i -0.005))))

C

double code(double i, double n) {
	return n / (0.01 + (i * -0.005));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n / (0.01d0 + (i * (-0.005d0)))
end function

Java

public static double code(double i, double n) {
	return n / (0.01 + (i * -0.005));
}

Python

def code(i, n):
	return n / (0.01 + (i * -0.005))

Julia

function code(i, n)
	return Float64(n / Float64(0.01 + Float64(i * -0.005)))
end

MATLAB

function tmp = code(i, n)
	tmp = n / (0.01 + (i * -0.005));
end

Wolfram

code[i_, n_] := N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.8%

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

2. Taylor expanded in n around inf 30.7%

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

   1. *-commutative 30.7%

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

   2. associate-/l* 30.7%

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

   3. expm1-def 77.6%

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

4. Simplified 77.6%

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

   1. associate-*l/ 77.2%

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

   2. associate-/l* 77.6%

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

6. Applied egg-rr 77.6%

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

7. Taylor expanded in i around 0 63.0%

   \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
8. Step-by-step derivation

   1. *-commutative 63.0%

      \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

9. Simplified 63.0%

   \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

10. Final simplification 63.0%

    \[\leadsto \frac{n}{0.01 + i \cdot -0.005} \]

Alternative 15: 49.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.8%

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

2. Taylor expanded in i around 0 54.0%

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

   1. *-commutative 54.0%

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

4. Simplified 54.0%

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

5. Final simplification 54.0%

   \[\leadsto n \cdot 100 \]

Developer target: 34.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(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))))