Compound Interest

Percentage Accurate: 29.6% → 98.7%

Time: 29.6s

Alternatives: 25

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: 29.6% 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: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ t_2 := \frac{t_0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{if}\;t_1 \leq -5000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-214}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* (/ t_0 i) (* n 100.0))))
   (if (<= t_1 -5000000.0)
     t_2
     (if (<= t_1 5e-214)
       (/ 100.0 (/ (/ i n) (expm1 (* n (log1p (/ i n))))))
       (if (<= t_1 INFINITY) t_2 (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = (t_0 / i) * (n * 100.0);
	double tmp;
	if (t_1 <= -5000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-214) {
		tmp = 100.0 / ((i / n) / expm1((n * log1p((i / n)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = (t_0 / i) * (n * 100.0);
	double tmp;
	if (t_1 <= -5000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-214) {
		tmp = 100.0 / ((i / n) / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	t_2 = (t_0 / i) * (n * 100.0)
	tmp = 0
	if t_1 <= -5000000.0:
		tmp = t_2
	elif t_1 <= 5e-214:
		tmp = 100.0 / ((i / n) / math.expm1((n * math.log1p((i / n)))))
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(Float64(t_0 / i) * Float64(n * 100.0))
	tmp = 0.0
	if (t_1 <= -5000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-214)
		tmp = Float64(100.0 / Float64(Float64(i / n) / expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000.0], t$95$2, If[LessEqual[t$95$1, 5e-214], N[(100.0 / N[(N[(i / n), $MachinePrecision] / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $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)) < -5e6 or 4.9999999999999998e-214 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   if -5e6 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 4.9999999999999998e-214

   1. Initial program 23.4%

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

      1. clear-num 23.4%

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

      2. un-div-inv 23.4%

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

      3. pow-to-exp 23.4%

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

      4. expm1-def 37.3%

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

      5. *-commutative 37.3%

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

      6. log1p-udef 99.3%

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

   3. Applied egg-rr 99.3%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5000000:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-214}:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 2: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := t_0 \cdot 100\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))) (t_1 (* t_0 100.0)))
   (if (<= t_0 -2e-259)
     t_1
     (if (<= t_0 0.0)
       (* (* n 100.0) (/ (expm1 i) i))
       (if (<= t_0 INFINITY) t_1 (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -2e-259) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = t_0 * 100.0;
	double tmp;
	if (t_0 <= -2e-259) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	t_1 = t_0 * 100.0
	tmp = 0
	if t_0 <= -2e-259:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	t_1 = Float64(t_0 * 100.0)
	tmp = 0.0
	if (t_0 <= -2e-259)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-259], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $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)) < -2.0000000000000001e-259 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.7%

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

   if -2.0000000000000001e-259 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/r/ 19.2%

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

      3. associate-*l* 19.2%

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

      4. sub-neg 19.2%

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

      5. metadata-eval 19.2%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in n around inf 31.9%

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

      1. expm1-def 75.7%

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

   6. Simplified 75.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 3: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -2e-259)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* (* n 100.0) (/ (expm1 i) i))
       (if (<= t_1 INFINITY)
         (* n (/ (+ (* t_0 100.0) -100.0) i))
         (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -2e-259:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-259)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-259], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2.0000000000000001e-259

   1. Initial program 97.8%

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

   if -2.0000000000000001e-259 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/r/ 19.2%

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

      3. associate-*l* 19.2%

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

      4. sub-neg 19.2%

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

      5. metadata-eval 19.2%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in n around inf 31.9%

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

      1. expm1-def 75.7%

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

   6. Simplified 75.7%

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

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

   1. Initial program 99.4%

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

      1. associate-/r/ 99.4%

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

      2. associate-*r* 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r/ 99.5%

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

      5. sub-neg 99.5%

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

      6. distribute-lft-in 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. fma-udef 99.5%

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

      2. *-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 4: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{t_0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 -2e-259)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* (* n 100.0) (/ (expm1 i) i))
       (if (<= t_1 INFINITY)
         (* (/ t_0 i) (* n 100.0))
         (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 / i) * (n * 100.0);
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -2e-259) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / i) * (n * 100.0);
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= -2e-259:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	elif t_1 <= math.inf:
		tmp = (t_0 / i) * (n * 100.0)
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-259)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 / i) * Float64(n * 100.0));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-259], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -2.0000000000000001e-259

   1. Initial program 97.8%

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

   if -2.0000000000000001e-259 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/r/ 19.2%

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

      3. associate-*l* 19.2%

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

      4. sub-neg 19.2%

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

      5. metadata-eval 19.2%

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

   3. Simplified 19.2%

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

   4. Taylor expanded in n around inf 31.9%

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

      1. expm1-def 75.7%

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

   6. Simplified 75.7%

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

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

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-/r/ 99.4%

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

      3. associate-*l* 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 5: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := t_0 + -1\\ t_2 := \frac{t_1}{\frac{i}{n}}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-96}:\\ \;\;\;\;100 \cdot \left(t_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-214}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{t_1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (+ t_0 -1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 -5e-96)
     (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
     (if (<= t_2 5e-214)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_2 INFINITY)
         (* (/ t_1 i) (* n 100.0))
         (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -5e-96) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else if (t_2 <= 5e-214) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_1 / i) * (n * 100.0);
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = t_0 + -1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -5e-96) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else if (t_2 <= 5e-214) {
		tmp = n * ((100.0 * Math.expm1((n * Math.log1p((i / n))))) / i);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 / i) * (n * 100.0);
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = t_0 + -1.0
	t_2 = t_1 / (i / n)
	tmp = 0
	if t_2 <= -5e-96:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	elif t_2 <= 5e-214:
		tmp = n * ((100.0 * math.expm1((n * math.log1p((i / n))))) / i)
	elif t_2 <= math.inf:
		tmp = (t_1 / i) * (n * 100.0)
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -5e-96)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	elseif (t_2 <= 5e-214)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_1 / i) * Float64(n * 100.0));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-96], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-214], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(t$95$1 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.99999999999999995e-96

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

      2. clear-num 100.0%

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

      3. sub-neg 100.0%

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

      4. div-inv 100.0%

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

      5. clear-num 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.99999999999999995e-96 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 4.9999999999999998e-214

   1. Initial program 21.6%

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

      1. associate-/r/ 21.3%

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

      2. associate-*r* 21.3%

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

      3. *-commutative 21.3%

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

      4. associate-*r/ 21.3%

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

      5. sub-neg 21.3%

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

      6. distribute-lft-in 21.3%

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

      7. fma-def 21.3%

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

      8. metadata-eval 21.3%

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

      9. metadata-eval 21.3%

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

   3. Simplified 21.3%

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

      1. fma-udef 21.3%

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

      2. metadata-eval 21.3%

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

      3. distribute-lft-in 21.3%

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

      4. metadata-eval 21.3%

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

      5. sub-neg 21.3%

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

      6. *-commutative 21.3%

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

      7. pow-to-exp 21.3%

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

      8. expm1-def 35.8%

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

      9. *-commutative 35.8%

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

      10. log1p-udef 98.7%

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

   5. Applied egg-rr 98.7%

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

   if 4.9999999999999998e-214 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-96}:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-214}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 6: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2.2 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \frac{n}{i}\right) \cdot \log \left(\frac{i}{n}\right)\right)\\ \mathbf{elif}\;n \leq -2.35 \cdot 10^{-214}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-206}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{-1 + {\left(\frac{i}{n}\right)}^{n}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -9.5e-29)
     t_0
     (if (<= n -2.2e-59)
       (* 100.0 (* (* n (/ n i)) (log (/ i n))))
       (if (<= n -2.35e-214)
         (* 100.0 (/ (expm1 i) (/ i n)))
         (if (<= n 7.2e-206)
           (* (* n 100.0) (/ 0.0 i))
           (if (<= n 4.8e+14)
             (* 100.0 (/ i (/ i n)))
             (if (<= n 1.65e+31)
               (* 100.0 (/ n (/ i (+ -1.0 (pow (/ i n) n)))))
               t_0))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -9.5e-29) {
		tmp = t_0;
	} else if (n <= -2.2e-59) {
		tmp = 100.0 * ((n * (n / i)) * log((i / n)));
	} else if (n <= -2.35e-214) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 7.2e-206) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 4.8e+14) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.65e+31) {
		tmp = 100.0 * (n / (i / (-1.0 + pow((i / n), n))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -9.5e-29) {
		tmp = t_0;
	} else if (n <= -2.2e-59) {
		tmp = 100.0 * ((n * (n / i)) * Math.log((i / n)));
	} else if (n <= -2.35e-214) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 7.2e-206) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 4.8e+14) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.65e+31) {
		tmp = 100.0 * (n / (i / (-1.0 + Math.pow((i / n), n))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -9.5e-29:
		tmp = t_0
	elif n <= -2.2e-59:
		tmp = 100.0 * ((n * (n / i)) * math.log((i / n)))
	elif n <= -2.35e-214:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 7.2e-206:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 4.8e+14:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.65e+31:
		tmp = 100.0 * (n / (i / (-1.0 + math.pow((i / n), n))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -9.5e-29)
		tmp = t_0;
	elseif (n <= -2.2e-59)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(n / i)) * log(Float64(i / n))));
	elseif (n <= -2.35e-214)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 7.2e-206)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 4.8e+14)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.65e+31)
		tmp = Float64(100.0 * Float64(n / Float64(i / Float64(-1.0 + (Float64(i / n) ^ n)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.5e-29], t$95$0, If[LessEqual[n, -2.2e-59], N[(100.0 * N[(N[(n * N[(n / i), $MachinePrecision]), $MachinePrecision] * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.35e-214], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.2e-206], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.8e+14], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.65e+31], N[(100.0 * N[(n / N[(i / N[(-1.0 + N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -9.50000000000000023e-29 or 1.64999999999999996e31 < n

   1. Initial program 23.3%

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

   2. Taylor expanded in n around inf 36.1%

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

      1. *-commutative 36.1%

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

      2. associate-/l* 36.1%

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

      3. expm1-def 90.6%

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

   4. Simplified 90.6%

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

   if -9.50000000000000023e-29 < n < -2.1999999999999999e-59

   1. Initial program 6.6%

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

      1. *-commutative 6.6%

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

      2. associate-/r/ 6.6%

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

      3. sub-neg 6.6%

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

      4. metadata-eval 6.6%

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

      5. associate-*r* 6.6%

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

      6. metadata-eval 6.6%

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

      7. sub-neg 6.6%

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

      8. associate-*l/ 6.6%

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

      9. associate-/l* 6.6%

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

      10. pow-to-exp 6.6%

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

      11. expm1-def 86.8%

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

      12. *-commutative 86.8%

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

      13. log1p-udef 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in n around 0 0.0%

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

      1. associate-/l* 0.0%

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

      2. associate-/r/ 0.0%

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

      3. unpow2 0.0%

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

      4. mul-1-neg 0.0%

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

      5. unsub-neg 0.0%

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

      6. log-div 86.5%

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

   6. Simplified 86.5%

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

      1. associate-/l* 86.9%

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

      2. associate-/r/ 87.3%

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

   8. Applied egg-rr 87.3%

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

   if -2.1999999999999999e-59 < n < -2.3500000000000002e-214

   1. Initial program 39.4%

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

   2. Taylor expanded in n around inf 26.5%

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

      1. expm1-def 63.4%

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

   4. Simplified 63.4%

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

   if -2.3500000000000002e-214 < n < 7.19999999999999987e-206

   1. Initial program 51.4%

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

      1. *-commutative 51.4%

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

      2. associate-/r/ 52.0%

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

      3. associate-*l* 52.0%

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

      4. sub-neg 52.0%

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

      5. metadata-eval 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in i around 0 86.5%

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

   if 7.19999999999999987e-206 < n < 4.8e14

   1. Initial program 9.0%

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

   2. Taylor expanded in i around 0 68.3%

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

   if 4.8e14 < n < 1.64999999999999996e31

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

      3. pow-to-exp 80.0%

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

      4. expm1-def 80.0%

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

      5. *-commutative 80.0%

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

      6. log1p-udef 80.0%

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

   3. Applied egg-rr 80.0%

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

      1. expm1-log1p-u 80.0%

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

      2. expm1-udef 80.0%

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

      3. associate-/r/ 80.0%

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

   5. Applied egg-rr 80.0%

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

      1. expm1-def 80.0%

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

      2. expm1-log1p 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

   8. Taylor expanded in i around inf 80.0%

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

      1. associate-/l* 80.0%

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

      2. sub-neg 80.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{100 \cdot \frac{n}{\frac{i}{{\left(\frac{i}{n}\right)}^{n} + -1}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -2.2 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \frac{n}{i}\right) \cdot \log \left(\frac{i}{n}\right)\right)\\ \mathbf{elif}\;n \leq -2.35 \cdot 10^{-214}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-206}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{-1 + {\left(\frac{i}{n}\right)}^{n}}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 7: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -2.9 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \frac{n}{i}\right) \cdot \log \left(\frac{i}{n}\right)\right)\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(i, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -9.5e-29)
     t_0
     (if (<= n -2.9e-59)
       (* 100.0 (* (* n (/ n i)) (log (/ i n))))
       (if (<= n 1.1e-5)
         (/ 100.0 (fma i (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n)))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -9.5e-29) {
		tmp = t_0;
	} else if (n <= -2.9e-59) {
		tmp = 100.0 * ((n * (n / i)) * log((i / n)));
	} else if (n <= 1.1e-5) {
		tmp = 100.0 / fma(i, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -9.5e-29)
		tmp = t_0;
	elseif (n <= -2.9e-59)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(n / i)) * log(Float64(i / n))));
	elseif (n <= 1.1e-5)
		tmp = Float64(100.0 / fma(i, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.5e-29], t$95$0, If[LessEqual[n, -2.9e-59], N[(100.0 * N[(N[(n * N[(n / i), $MachinePrecision]), $MachinePrecision] * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.1e-5], N[(100.0 / N[(i * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.50000000000000023e-29 or 1.1e-5 < n

   1. Initial program 25.5%

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

   2. Taylor expanded in n around inf 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-/l* 37.1%

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

      3. expm1-def 90.4%

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

   4. Simplified 90.4%

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

   if -9.50000000000000023e-29 < n < -2.90000000000000016e-59

   1. Initial program 6.6%

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

      1. *-commutative 6.6%

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

      2. associate-/r/ 6.6%

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

      3. sub-neg 6.6%

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

      4. metadata-eval 6.6%

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

      5. associate-*r* 6.6%

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

      6. metadata-eval 6.6%

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

      7. sub-neg 6.6%

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

      8. associate-*l/ 6.6%

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

      9. associate-/l* 6.6%

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

      10. pow-to-exp 6.6%

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

      11. expm1-def 86.8%

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

      12. *-commutative 86.8%

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

      13. log1p-udef 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in n around 0 0.0%

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

      1. associate-/l* 0.0%

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

      2. associate-/r/ 0.0%

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

      3. unpow2 0.0%

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

      4. mul-1-neg 0.0%

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

      5. unsub-neg 0.0%

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

      6. log-div 86.5%

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

   6. Simplified 86.5%

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

      1. associate-/l* 86.9%

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

      2. associate-/r/ 87.3%

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

   8. Applied egg-rr 87.3%

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

   if -2.90000000000000016e-59 < n < 1.1e-5

   1. Initial program 33.2%

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

      1. clear-num 33.2%

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

      2. un-div-inv 33.2%

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

      3. pow-to-exp 33.2%

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

      4. expm1-def 52.1%

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

      5. *-commutative 52.1%

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

      6. log1p-udef 81.0%

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

   3. Applied egg-rr 81.0%

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

   4. Taylor expanded in i around 0 75.9%

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

      1. fma-def 75.9%

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

      2. sub-neg 75.9%

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

      3. associate-*r/ 75.9%

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

      4. metadata-eval 75.9%

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

      5. unpow2 75.9%

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

      6. associate-*r/ 75.9%

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

      7. metadata-eval 75.9%

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

      8. distribute-neg-frac 75.9%

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

      9. metadata-eval 75.9%

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

   6. Simplified 75.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -2.9 \cdot 10^{-59}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \frac{n}{i}\right) \cdot \log \left(\frac{i}{n}\right)\right)\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(i, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 8: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-60}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \frac{n}{i}\right) \cdot \log \left(\frac{i}{n}\right)\right)\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-171}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -9.5e-29)
     t_0
     (if (<= n -8.5e-60)
       (* 100.0 (* (* n (/ n i)) (log (/ i n))))
       (if (<= n -6.4e-201)
         (* 100.0 (/ (expm1 i) (/ i n)))
         (if (<= n 4.1e-171) (* (* n 100.0) (/ 0.0 i)) t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -9.5e-29) {
		tmp = t_0;
	} else if (n <= -8.5e-60) {
		tmp = 100.0 * ((n * (n / i)) * log((i / n)));
	} else if (n <= -6.4e-201) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 4.1e-171) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -9.5e-29) {
		tmp = t_0;
	} else if (n <= -8.5e-60) {
		tmp = 100.0 * ((n * (n / i)) * Math.log((i / n)));
	} else if (n <= -6.4e-201) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 4.1e-171) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -9.5e-29:
		tmp = t_0
	elif n <= -8.5e-60:
		tmp = 100.0 * ((n * (n / i)) * math.log((i / n)))
	elif n <= -6.4e-201:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 4.1e-171:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -9.5e-29)
		tmp = t_0;
	elseif (n <= -8.5e-60)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(n / i)) * log(Float64(i / n))));
	elseif (n <= -6.4e-201)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 4.1e-171)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.5e-29], t$95$0, If[LessEqual[n, -8.5e-60], N[(100.0 * N[(N[(n * N[(n / i), $MachinePrecision]), $MachinePrecision] * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -6.4e-201], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.1e-171], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.50000000000000023e-29 or 4.1e-171 < n

   1. Initial program 22.5%

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

   2. Taylor expanded in n around inf 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-/l* 31.6%

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

      3. expm1-def 85.1%

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

   4. Simplified 85.1%

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

   if -9.50000000000000023e-29 < n < -8.50000000000000044e-60

   1. Initial program 6.6%

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

      1. *-commutative 6.6%

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

      2. associate-/r/ 6.6%

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

      3. sub-neg 6.6%

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

      4. metadata-eval 6.6%

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

      5. associate-*r* 6.6%

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

      6. metadata-eval 6.6%

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

      7. sub-neg 6.6%

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

      8. associate-*l/ 6.6%

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

      9. associate-/l* 6.6%

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

      10. pow-to-exp 6.6%

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

      11. expm1-def 86.8%

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

      12. *-commutative 86.8%

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

      13. log1p-udef 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in n around 0 0.0%

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

      1. associate-/l* 0.0%

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

      2. associate-/r/ 0.0%

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

      3. unpow2 0.0%

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

      4. mul-1-neg 0.0%

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

      5. unsub-neg 0.0%

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

      6. log-div 86.5%

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

   6. Simplified 86.5%

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

      1. associate-/l* 86.9%

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

      2. associate-/r/ 87.3%

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

   8. Applied egg-rr 87.3%

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

   if -8.50000000000000044e-60 < n < -6.4000000000000002e-201

   1. Initial program 39.4%

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

   2. Taylor expanded in n around inf 26.5%

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

      1. expm1-def 63.4%

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

   4. Simplified 63.4%

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

   if -6.4000000000000002e-201 < n < 4.1e-171

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-60}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \frac{n}{i}\right) \cdot \log \left(\frac{i}{n}\right)\right)\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-171}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 9: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -6.8 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-203}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 210000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -6.8e-124)
     t_0
     (if (<= n 8e-203)
       (* (* n 100.0) (/ 0.0 i))
       (if (<= n 210000.0) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -6.8e-124) {
		tmp = t_0;
	} else if (n <= 8e-203) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 210000.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -6.8e-124) {
		tmp = t_0;
	} else if (n <= 8e-203) {
		tmp = (n * 100.0) * (0.0 / i);
	} else if (n <= 210000.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -6.8e-124:
		tmp = t_0
	elif n <= 8e-203:
		tmp = (n * 100.0) * (0.0 / i)
	elif n <= 210000.0:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -6.8e-124)
		tmp = t_0;
	elseif (n <= 8e-203)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	elseif (n <= 210000.0)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.8e-124], t$95$0, If[LessEqual[n, 8e-203], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 210000.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.8000000000000001e-124 or 2.1e5 < n

   1. Initial program 24.5%

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

      1. clear-num 24.5%

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

      2. un-div-inv 24.5%

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

      3. pow-to-exp 16.2%

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

      4. expm1-def 22.4%

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

      5. *-commutative 22.4%

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

      6. log1p-udef 71.3%

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

   3. Applied egg-rr 71.3%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 25.0%

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

      3. associate-/r/ 23.6%

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

   5. Applied egg-rr 23.6%

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

      1. expm1-def 38.0%

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

      2. expm1-log1p 69.2%

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

      3. *-commutative 69.2%

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

      4. associate-/r/ 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in n around inf 35.1%

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

      1. expm1-def 84.6%

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

   10. Simplified 84.6%

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

   if -6.8000000000000001e-124 < n < 8.0000000000000003e-203

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. associate-/r/ 51.0%

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

      3. associate-*l* 51.0%

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

      4. sub-neg 51.0%

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

      5. metadata-eval 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in i around 0 77.2%

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

   if 8.0000000000000003e-203 < n < 2.1e5

   1. Initial program 9.0%

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

   2. Taylor expanded in i around 0 68.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.8 \cdot 10^{-124}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-203}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{elif}\;n \leq 210000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]

Alternative 10: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-213} \lor \neg \left(n \leq 3.9 \cdot 10^{-173}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9.8e-213) (not (<= n 3.9e-173)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (* (* n 100.0) (/ 0.0 i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9.8e-213) || !(n <= 3.9e-173)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = (n * 100.0) * (0.0 / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9.8e-213) || !(n <= 3.9e-173)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = (n * 100.0) * (0.0 / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9.8e-213) or not (n <= 3.9e-173):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = (n * 100.0) * (0.0 / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9.8e-213) || !(n <= 3.9e-173))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9.8e-213], N[Not[LessEqual[n, 3.9e-173]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.7999999999999997e-213 or 3.89999999999999987e-173 < n

   1. Initial program 23.6%

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

   2. Taylor expanded in n around inf 30.0%

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

      1. *-commutative 30.0%

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

      2. associate-/l* 30.0%

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

      3. expm1-def 80.2%

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

   4. Simplified 80.2%

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

   if -9.7999999999999997e-213 < n < 3.89999999999999987e-173

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-213} \lor \neg \left(n \leq 3.9 \cdot 10^{-173}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \end{array} \]

Alternative 11: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{+257}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(0.16666666666666666 \cdot \left(i \cdot i\right) + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq -3.4 \cdot 10^{-215}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-169}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.2e+257)
   (* 100.0 (+ n (* n (+ (* 0.16666666666666666 (* i i)) (* i 0.5)))))
   (if (<= n -3.4e-215)
     (* 100.0 (/ (expm1 i) (/ i n)))
     (if (<= n 5.3e-169)
       (* (* n 100.0) (/ 0.0 i))
       (* n (/ (* 100.0 (+ i (* (* i i) (- 0.5 (/ 0.5 n))))) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.2e+257) {
		tmp = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))));
	} else if (n <= -3.4e-215) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 5.3e-169) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -6.2e+257) {
		tmp = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))));
	} else if (n <= -3.4e-215) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 5.3e-169) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -6.2e+257:
		tmp = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))))
	elif n <= -3.4e-215:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 5.3e-169:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.2e+257)
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(0.16666666666666666 * Float64(i * i)) + Float64(i * 0.5)))));
	elseif (n <= -3.4e-215)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 5.3e-169)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = Float64(n * Float64(Float64(100.0 * Float64(i + Float64(Float64(i * i) * Float64(0.5 - Float64(0.5 / n))))) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.2e+257], N[(100.0 * N[(n + N[(n * N[(N[(0.16666666666666666 * N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -3.4e-215], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.3e-169], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(i + N[(N[(i * i), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.2000000000000001e257

   1. Initial program 0.8%

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

   2. Taylor expanded in n around inf 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-/l* 37.1%

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

      3. expm1-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in i around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-*r* 76.9%

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

      3. associate-*r* 76.9%

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

      4. distribute-rgt-out 78.1%

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

      5. *-commutative 78.1%

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

      6. unpow2 78.1%

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

   7. Simplified 78.1%

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

   if -6.2000000000000001e257 < n < -3.40000000000000001e-215

   1. Initial program 30.1%

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

   2. Taylor expanded in n around inf 32.9%

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

      1. expm1-def 69.7%

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

   4. Simplified 69.7%

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

   if -3.40000000000000001e-215 < n < 5.3e-169

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

   if 5.3e-169 < n

   1. Initial program 19.5%

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

      1. associate-/r/ 20.0%

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

      2. associate-*r* 20.0%

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

      3. *-commutative 20.0%

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

      4. associate-*r/ 20.0%

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

      5. sub-neg 20.0%

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

      6. distribute-lft-in 20.0%

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

      7. fma-def 20.0%

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

      8. metadata-eval 20.0%

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

      9. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

   4. Taylor expanded in i around 0 74.0%

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

      1. distribute-lft-out 74.0%

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

      2. unpow2 74.0%

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

      3. associate-*r/ 74.0%

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

      4. metadata-eval 74.0%

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

   6. Simplified 74.0%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{+257}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(0.16666666666666666 \cdot \left(i \cdot i\right) + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq -3.4 \cdot 10^{-215}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-169}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \]

Alternative 12: 65.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+117}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(0.16666666666666666 \cdot \left(i \cdot i\right) + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq -2.76 \cdot 10^{-207}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-174}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.3e+117)
   (* 100.0 (+ n (* n (+ (* 0.16666666666666666 (* i i)) (* i 0.5)))))
   (if (<= n -2.76e-207)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 8e-174)
       (* (* n 100.0) (/ 0.0 i))
       (* n (/ (* 100.0 (+ i (* (* i i) (- 0.5 (/ 0.5 n))))) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.3e+117) {
		tmp = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))));
	} else if (n <= -2.76e-207) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 8e-174) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.3d+117)) then
        tmp = 100.0d0 * (n + (n * ((0.16666666666666666d0 * (i * i)) + (i * 0.5d0))))
    else if (n <= (-2.76d-207)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 8d-174) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = n * ((100.0d0 * (i + ((i * i) * (0.5d0 - (0.5d0 / n))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.3e+117) {
		tmp = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))));
	} else if (n <= -2.76e-207) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 8e-174) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.3e+117:
		tmp = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))))
	elif n <= -2.76e-207:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 8e-174:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.3e+117)
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(0.16666666666666666 * Float64(i * i)) + Float64(i * 0.5)))));
	elseif (n <= -2.76e-207)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 8e-174)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = Float64(n * Float64(Float64(100.0 * Float64(i + Float64(Float64(i * i) * Float64(0.5 - Float64(0.5 / n))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.3e+117)
		tmp = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))));
	elseif (n <= -2.76e-207)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 8e-174)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.3e+117], N[(100.0 * N[(n + N[(n * N[(N[(0.16666666666666666 * N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.76e-207], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8e-174], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(i + N[(N[(i * i), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.29999999999999988e117

   1. Initial program 21.5%

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

   2. Taylor expanded in n around inf 49.4%

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

      1. *-commutative 49.4%

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

      2. associate-/l* 49.4%

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

      3. expm1-def 90.6%

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

   4. Simplified 90.6%

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

   5. Taylor expanded in i around 0 60.1%

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

      1. +-commutative 60.1%

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

      2. associate-*r* 60.1%

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

      3. associate-*r* 60.1%

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

      4. distribute-rgt-out 60.9%

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

      5. *-commutative 60.9%

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

      6. unpow2 60.9%

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

   7. Simplified 60.9%

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

   if -2.29999999999999988e117 < n < -2.76000000000000013e-207

   1. Initial program 31.5%

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

   2. Taylor expanded in n around inf 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/l* 19.5%

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

      3. expm1-def 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in i around 0 61.5%

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

      1. *-commutative 61.5%

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

   7. Simplified 61.5%

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

   if -2.76000000000000013e-207 < n < 8e-174

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

   if 8e-174 < n

   1. Initial program 19.5%

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

      1. associate-/r/ 20.0%

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

      2. associate-*r* 20.0%

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

      3. *-commutative 20.0%

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

      4. associate-*r/ 20.0%

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

      5. sub-neg 20.0%

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

      6. distribute-lft-in 20.0%

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

      7. fma-def 20.0%

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

      8. metadata-eval 20.0%

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

      9. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

   4. Taylor expanded in i around 0 74.0%

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

      1. distribute-lft-out 74.0%

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

      2. unpow2 74.0%

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

      3. associate-*r/ 74.0%

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

      4. metadata-eval 74.0%

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

   6. Simplified 74.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+117}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(0.16666666666666666 \cdot \left(i \cdot i\right) + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq -2.76 \cdot 10^{-207}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-174}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \]

Alternative 13: 66.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.25\right)\right)\\ \mathbf{if}\;n \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-212}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (+ n (* n (+ (* i 0.5) (* (* i i) 0.25)))))))
   (if (<= n -1.2e+117)
     t_0
     (if (<= n -4.8e-212)
       (* 100.0 (/ n (+ 1.0 (* i -0.5))))
       (if (<= n 2.8e-172) (* (* n 100.0) (/ 0.0 i)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * ((i * 0.5) + ((i * i) * 0.25))));
	double tmp;
	if (n <= -1.2e+117) {
		tmp = t_0;
	} else if (n <= -4.8e-212) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 2.8e-172) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n + (n * ((i * 0.5d0) + ((i * i) * 0.25d0))))
    if (n <= (-1.2d+117)) then
        tmp = t_0
    else if (n <= (-4.8d-212)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 2.8d-172) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * ((i * 0.5) + ((i * i) * 0.25))));
	double tmp;
	if (n <= -1.2e+117) {
		tmp = t_0;
	} else if (n <= -4.8e-212) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 2.8e-172) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n + (n * ((i * 0.5) + ((i * i) * 0.25))))
	tmp = 0
	if n <= -1.2e+117:
		tmp = t_0
	elif n <= -4.8e-212:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 2.8e-172:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(i * 0.5) + Float64(Float64(i * i) * 0.25)))))
	tmp = 0.0
	if (n <= -1.2e+117)
		tmp = t_0;
	elseif (n <= -4.8e-212)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 2.8e-172)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n + (n * ((i * 0.5) + ((i * i) * 0.25))));
	tmp = 0.0;
	if (n <= -1.2e+117)
		tmp = t_0;
	elseif (n <= -4.8e-212)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 2.8e-172)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n + N[(n * N[(N[(i * 0.5), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.2e+117], t$95$0, If[LessEqual[n, -4.8e-212], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-172], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1999999999999999e117 or 2.80000000000000011e-172 < n

   1. Initial program 20.3%

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

   2. Taylor expanded in n around inf 34.5%

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

      1. *-commutative 34.5%

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

      2. associate-/l* 34.5%

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

      3. expm1-def 85.5%

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

   4. Simplified 85.5%

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

   5. Taylor expanded in i around 0 59.5%

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

      1. *-commutative 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in i around 0 67.8%

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

      1. +-commutative 67.8%

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

      2. associate-*r* 67.8%

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

      3. associate-*r* 67.8%

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

      4. distribute-rgt-out 68.2%

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

      5. unpow2 68.2%

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

   10. Simplified 68.2%

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

   if -1.1999999999999999e117 < n < -4.79999999999999978e-212

   1. Initial program 31.5%

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

   2. Taylor expanded in n around inf 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/l* 19.5%

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

      3. expm1-def 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in i around 0 61.5%

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

      1. *-commutative 61.5%

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

   7. Simplified 61.5%

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

   if -4.79999999999999978e-212 < n < 2.80000000000000011e-172

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.25\right)\right)\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-212}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + \left(i \cdot i\right) \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 14: 66.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n + n \cdot \left(0.16666666666666666 \cdot \left(i \cdot i\right) + i \cdot 0.5\right)\right)\\ \mathbf{if}\;n \leq -2.6 \cdot 10^{+116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-208}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 8.8 \cdot 10^{-172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (* 100.0 (+ n (* n (+ (* 0.16666666666666666 (* i i)) (* i 0.5)))))))
   (if (<= n -2.6e+116)
     t_0
     (if (<= n -3.6e-208)
       (* 100.0 (/ n (+ 1.0 (* i -0.5))))
       (if (<= n 8.8e-172) (* (* n 100.0) (/ 0.0 i)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))));
	double tmp;
	if (n <= -2.6e+116) {
		tmp = t_0;
	} else if (n <= -3.6e-208) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 8.8e-172) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n + (n * ((0.16666666666666666d0 * (i * i)) + (i * 0.5d0))))
    if (n <= (-2.6d+116)) then
        tmp = t_0
    else if (n <= (-3.6d-208)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 8.8d-172) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))));
	double tmp;
	if (n <= -2.6e+116) {
		tmp = t_0;
	} else if (n <= -3.6e-208) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 8.8e-172) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))))
	tmp = 0
	if n <= -2.6e+116:
		tmp = t_0
	elif n <= -3.6e-208:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 8.8e-172:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(0.16666666666666666 * Float64(i * i)) + Float64(i * 0.5)))))
	tmp = 0.0
	if (n <= -2.6e+116)
		tmp = t_0;
	elseif (n <= -3.6e-208)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 8.8e-172)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n + (n * ((0.16666666666666666 * (i * i)) + (i * 0.5))));
	tmp = 0.0;
	if (n <= -2.6e+116)
		tmp = t_0;
	elseif (n <= -3.6e-208)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 8.8e-172)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n + N[(n * N[(N[(0.16666666666666666 * N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.6e+116], t$95$0, If[LessEqual[n, -3.6e-208], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.8e-172], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.59999999999999987e116 or 8.80000000000000036e-172 < n

   1. Initial program 20.3%

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

   2. Taylor expanded in n around inf 34.5%

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

      1. *-commutative 34.5%

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

      2. associate-/l* 34.5%

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

      3. expm1-def 85.5%

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

   4. Simplified 85.5%

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

   5. Taylor expanded in i around 0 67.9%

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

      1. +-commutative 67.9%

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

      2. associate-*r* 67.9%

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

      3. associate-*r* 67.9%

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

      4. distribute-rgt-out 68.3%

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

      5. *-commutative 68.3%

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

      6. unpow2 68.3%

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

   7. Simplified 68.3%

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

   if -2.59999999999999987e116 < n < -3.5999999999999998e-208

   1. Initial program 31.5%

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

   2. Taylor expanded in n around inf 19.5%

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

      1. *-commutative 19.5%

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

      2. associate-/l* 19.5%

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

      3. expm1-def 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in i around 0 61.5%

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

      1. *-commutative 61.5%

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

   7. Simplified 61.5%

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

   if -3.5999999999999998e-208 < n < 8.80000000000000036e-172

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{+116}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(0.16666666666666666 \cdot \left(i \cdot i\right) + i \cdot 0.5\right)\right)\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-208}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 8.8 \cdot 10^{-172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(0.16666666666666666 \cdot \left(i \cdot i\right) + i \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 15: 63.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-206}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 8.8 \cdot 10^{-172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-206)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 8.8e-172)
     (* (* n 100.0) (/ 0.0 i))
     (* 100.0 (+ n (* (- 0.5 (/ 0.5 n)) (* i n)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-206) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 8.8e-172) {
		tmp = (n * 100.0) * (0.0 / i);
	} 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 <= (-1.12d-206)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 8.8d-172) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    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 <= -1.12e-206) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 8.8e-172) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.12e-206:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 8.8e-172:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-206)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 8.8e-172)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	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 <= -1.12e-206)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 8.8e-172)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-206], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.8e-172], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $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 3 regimes

2. if n < -1.11999999999999997e-206

   1. Initial program 26.9%

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

   2. Taylor expanded in n around inf 33.1%

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

      1. *-commutative 33.1%

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

      2. associate-/l* 33.1%

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

      3. expm1-def 78.2%

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

   4. Simplified 78.2%

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

   5. Taylor expanded in i around 0 57.1%

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

      1. *-commutative 57.1%

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

   7. Simplified 57.1%

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

   if -1.11999999999999997e-206 < n < 8.80000000000000036e-172

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

   if 8.80000000000000036e-172 < n

   1. Initial program 19.5%

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

   2. Taylor expanded in i around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. associate-*r/ 71.2%

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

      3. metadata-eval 71.2%

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

   4. Simplified 71.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-206}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 8.8 \cdot 10^{-172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 16: 63.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;i \cdot -50 + 100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.4e-213)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 4.9e-172)
     (* (* n 100.0) (/ 0.0 i))
     (+ (* i -50.0) (* 100.0 (* n (+ 1.0 (* i 0.5))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.4e-213) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.9e-172) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = (i * -50.0) + (100.0 * (n * (1.0 + (i * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.4d-213)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 4.9d-172) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = (i * (-50.0d0)) + (100.0d0 * (n * (1.0d0 + (i * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.4e-213) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.9e-172) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = (i * -50.0) + (100.0 * (n * (1.0 + (i * 0.5))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.4e-213:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 4.9e-172:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = (i * -50.0) + (100.0 * (n * (1.0 + (i * 0.5))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.4e-213)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 4.9e-172)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = Float64(Float64(i * -50.0) + Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -5.4e-213)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 4.9e-172)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = (i * -50.0) + (100.0 * (n * (1.0 + (i * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.4e-213], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.9e-172], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], N[(N[(i * -50.0), $MachinePrecision] + N[(100.0 * N[(n * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.4000000000000001e-213

   1. Initial program 26.9%

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

   2. Taylor expanded in n around inf 33.1%

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

      1. *-commutative 33.1%

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

      2. associate-/l* 33.1%

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

      3. expm1-def 78.2%

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

   4. Simplified 78.2%

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

   5. Taylor expanded in i around 0 57.1%

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

      1. *-commutative 57.1%

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

   7. Simplified 57.1%

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

   if -5.4000000000000001e-213 < n < 4.9000000000000001e-172

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

   if 4.9000000000000001e-172 < n

   1. Initial program 19.5%

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

   2. Taylor expanded in i around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. associate-*r/ 71.2%

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

      3. metadata-eval 71.2%

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

   4. Simplified 71.2%

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

   5. Taylor expanded in n around 0 71.2%

      \[\leadsto \color{blue}{-50 \cdot i + 100 \cdot \left(n \cdot \left(1 + 0.5 \cdot i\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-172}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;i \cdot -50 + 100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 17: 63.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.18 \cdot 10^{-210}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-173}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.18e-210)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 2.9e-173)
     (* (* n 100.0) (/ 0.0 i))
     (+ (* n 100.0) (* (* i n) 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.18e-210) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 2.9e-173) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.18d-210)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 2.9d-173) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = (n * 100.0d0) + ((i * n) * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.18e-210) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 2.9e-173) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.18e-210:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 2.9e-173:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = (n * 100.0) + ((i * n) * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.18e-210)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 2.9e-173)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(i * n) * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.18e-210)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 2.9e-173)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = (n * 100.0) + ((i * n) * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.18e-210], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.9e-173], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.18e-210

   1. Initial program 26.9%

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

   2. Taylor expanded in n around inf 33.1%

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

      1. *-commutative 33.1%

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

      2. associate-/l* 33.1%

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

      3. expm1-def 78.2%

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

   4. Simplified 78.2%

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

   5. Taylor expanded in i around 0 57.1%

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

      1. *-commutative 57.1%

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

   7. Simplified 57.1%

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

   if -1.18e-210 < n < 2.8999999999999998e-173

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

   if 2.8999999999999998e-173 < n

   1. Initial program 19.5%

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

   2. Taylor expanded in n around inf 26.1%

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

      1. *-commutative 26.1%

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

      2. associate-/l* 26.1%

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

      3. expm1-def 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in i around 0 70.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.18 \cdot 10^{-210}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-173}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \]

Alternative 18: 60.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+78} \lor \neg \left(n \leq 1.95 \cdot 10^{-40}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.9e+78) (not (<= n 1.95e-40)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e+78) || !(n <= 1.95e-40)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.9d+78)) .or. (.not. (n <= 1.95d-40))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e+78) || !(n <= 1.95e-40)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.9e+78) or not (n <= 1.95e-40):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.9e+78) || !(n <= 1.95e-40))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.9e+78) || ~((n <= 1.95e-40)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.9e+78], N[Not[LessEqual[n, 1.95e-40]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.9e78 or 1.9499999999999999e-40 < n

   1. Initial program 23.2%

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

   2. Taylor expanded in n around inf 38.3%

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

      1. *-commutative 38.3%

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

      2. associate-/l* 38.3%

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

      3. expm1-def 91.9%

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

   4. Simplified 91.9%

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

   5. Taylor expanded in i around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. associate-*r* 68.2%

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

      3. distribute-rgt-out 68.2%

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

   7. Simplified 68.2%

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

   if -1.9e78 < n < 1.9499999999999999e-40

   1. Initial program 32.8%

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

   2. Taylor expanded in i around 0 58.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+78} \lor \neg \left(n \leq 1.95 \cdot 10^{-40}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 19: 57.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.000118:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{+105}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.000118)
   (/ -200.0 (/ i n))
   (if (<= i 1.7e+105) (* 100.0 (+ n (* i -0.5))) (* (/ n i) -200.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.000118) {
		tmp = -200.0 / (i / n);
	} else if (i <= 1.7e+105) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-0.000118d0)) then
        tmp = (-200.0d0) / (i / n)
    else if (i <= 1.7d+105) then
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    else
        tmp = (n / i) * (-200.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.000118) {
		tmp = -200.0 / (i / n);
	} else if (i <= 1.7e+105) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.000118:
		tmp = -200.0 / (i / n)
	elif i <= 1.7e+105:
		tmp = 100.0 * (n + (i * -0.5))
	else:
		tmp = (n / i) * -200.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.000118)
		tmp = Float64(-200.0 / Float64(i / n));
	elseif (i <= 1.7e+105)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	else
		tmp = Float64(Float64(n / i) * -200.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.000118)
		tmp = -200.0 / (i / n);
	elseif (i <= 1.7e+105)
		tmp = 100.0 * (n + (i * -0.5));
	else
		tmp = (n / i) * -200.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.000118], N[(-200.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.7e+105], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.18e-4

   1. Initial program 54.0%

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

   2. Taylor expanded in n around inf 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-/l* 68.2%

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

      3. expm1-def 68.2%

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

   4. Simplified 68.2%

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

   5. Taylor expanded in i around 0 28.4%

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

      1. *-commutative 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in i around inf 28.4%

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

      1. clear-num 29.5%

         \[\leadsto -200 \cdot \color{blue}{\frac{1}{\frac{i}{n}}} \]

      2. un-div-inv 29.5%

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

   10. Applied egg-rr 29.5%

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

   if -1.18e-4 < i < 1.7e105

   1. Initial program 11.7%

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

   2. Taylor expanded in i around 0 78.9%

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

      1. associate-*r* 78.7%

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

      2. associate-*r/ 78.7%

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

      3. metadata-eval 78.7%

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

   4. Simplified 78.7%

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

   5. Taylor expanded in n around 0 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   if 1.7e105 < i

   1. Initial program 50.1%

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

   2. Taylor expanded in n around inf 35.7%

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

      1. *-commutative 35.7%

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

      2. associate-/l* 35.7%

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

      3. expm1-def 35.7%

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

   4. Simplified 35.7%

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

   5. Taylor expanded in i around 0 41.6%

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

      1. *-commutative 41.6%

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

   7. Simplified 41.6%

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

   8. Taylor expanded in i around inf 41.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.000118:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{+105}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \]

Alternative 20: 63.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-170}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-213)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 4.4e-170) (* (* n 100.0) (/ 0.0 i)) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-213) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.4e-170) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.12d-213)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 4.4d-170) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-213) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.4e-170) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.12e-213:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 4.4e-170:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-213)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 4.4e-170)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.12e-213)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 4.4e-170)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-213], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.4e-170], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1200000000000001e-213

   1. Initial program 26.9%

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

   2. Taylor expanded in n around inf 33.1%

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

      1. *-commutative 33.1%

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

      2. associate-/l* 33.1%

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

      3. expm1-def 78.2%

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

   4. Simplified 78.2%

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

   5. Taylor expanded in i around 0 57.1%

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

      1. *-commutative 57.1%

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

   7. Simplified 57.1%

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

   if -1.1200000000000001e-213 < n < 4.40000000000000029e-170

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.6%

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

      3. associate-*l* 49.6%

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

      4. sub-neg 49.6%

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

      5. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in i around 0 83.2%

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

   if 4.40000000000000029e-170 < n

   1. Initial program 19.5%

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

   2. Taylor expanded in n around inf 26.1%

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

      1. *-commutative 26.1%

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

      2. associate-/l* 26.1%

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

      3. expm1-def 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in i around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. associate-*r* 70.4%

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

      3. distribute-rgt-out 70.4%

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

   7. Simplified 70.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-170}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 21: 61.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 7 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 7e-6)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 7e-6) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 7d-6) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 7e-6) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 7e-6:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 7e-6)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 7e-6)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 7e-6], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6.99999999999999989e-6

   1. Initial program 28.6%

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

   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 63.7%

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

   4. Simplified 63.7%

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

   5. Taylor expanded in i around 0 59.5%

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

      1. *-commutative 59.5%

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

   7. Simplified 59.5%

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

   if 6.99999999999999989e-6 < n

   1. Initial program 25.3%

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

   2. Taylor expanded in n around inf 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-/l* 36.4%

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

      3. expm1-def 93.9%

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

   4. Simplified 93.9%

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

   5. Taylor expanded in i around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. associate-*r* 76.0%

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

      3. distribute-rgt-out 76.0%

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

   7. Simplified 76.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 7 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 22: 57.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 9.2 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2.0) (not (<= i 9.2e+107))) (* (/ n i) -200.0) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 9.2e+107)) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-2.0d0)) .or. (.not. (i <= 9.2d+107))) then
        tmp = (n / i) * (-200.0d0)
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 9.2e+107)) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2.0) or not (i <= 9.2e+107):
		tmp = (n / i) * -200.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2.0) || !(i <= 9.2e+107))
		tmp = Float64(Float64(n / i) * -200.0);
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -2.0) || ~((i <= 9.2e+107)))
		tmp = (n / i) * -200.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2.0], N[Not[LessEqual[i, 9.2e+107]], $MachinePrecision]], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2 or 9.2000000000000001e107 < i

   1. Initial program 51.7%

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

   2. Taylor expanded in n around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-/l* 53.9%

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

      3. expm1-def 53.9%

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

   4. Simplified 53.9%

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

   5. Taylor expanded in i around 0 35.3%

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

      1. *-commutative 35.3%

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

   7. Simplified 35.3%

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

   8. Taylor expanded in i around inf 35.3%

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

   if -2 < i < 9.2000000000000001e107

   1. Initial program 12.8%

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

   2. Taylor expanded in i around 0 75.2%

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

      1. *-commutative 75.2%

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

   4. Simplified 75.2%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 9.2 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 23: 57.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0)
   (/ -200.0 (/ i n))
   (if (<= i 9.5e+108) (* n 100.0) (* (/ n i) -200.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = -200.0 / (i / n);
	} else if (i <= 9.5e+108) {
		tmp = n * 100.0;
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2.0d0)) then
        tmp = (-200.0d0) / (i / n)
    else if (i <= 9.5d+108) then
        tmp = n * 100.0d0
    else
        tmp = (n / i) * (-200.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = -200.0 / (i / n);
	} else if (i <= 9.5e+108) {
		tmp = n * 100.0;
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = -200.0 / (i / n)
	elif i <= 9.5e+108:
		tmp = n * 100.0
	else:
		tmp = (n / i) * -200.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(-200.0 / Float64(i / n));
	elseif (i <= 9.5e+108)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(n / i) * -200.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = -200.0 / (i / n);
	elseif (i <= 9.5e+108)
		tmp = n * 100.0;
	else
		tmp = (n / i) * -200.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(-200.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 9.5e+108], N[(n * 100.0), $MachinePrecision], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 53.1%

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

   2. Taylor expanded in n around inf 69.4%

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

      1. *-commutative 69.4%

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

      2. associate-/l* 69.4%

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

      3. expm1-def 69.4%

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

   4. Simplified 69.4%

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

   5. Taylor expanded in i around 0 28.8%

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

      1. *-commutative 28.8%

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

   7. Simplified 28.8%

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

   8. Taylor expanded in i around inf 28.8%

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

      1. clear-num 29.9%

         \[\leadsto -200 \cdot \color{blue}{\frac{1}{\frac{i}{n}}} \]

      2. un-div-inv 29.9%

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

   10. Applied egg-rr 29.9%

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

   if -2 < i < 9.50000000000000097e108

   1. Initial program 12.8%

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

   2. Taylor expanded in i around 0 75.2%

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

      1. *-commutative 75.2%

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

   4. Simplified 75.2%

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

   if 9.50000000000000097e108 < i

   1. Initial program 50.0%

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

   2. Taylor expanded in n around inf 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-/l* 35.0%

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

      3. expm1-def 35.0%

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

   4. Simplified 35.0%

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

   5. Taylor expanded in i around 0 43.4%

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

      1. *-commutative 43.4%

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

   7. Simplified 43.4%

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

   8. Taylor expanded in i around inf 43.4%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \]

Alternative 24: 2.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.7%

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

2. Taylor expanded in i around 0 53.5%

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

   1. associate-*r* 53.4%

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

   2. associate-*r/ 53.4%

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

   3. metadata-eval 53.4%

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

4. Simplified 53.4%

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

5. Taylor expanded in n around 0 2.7%

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

   1. *-commutative 2.7%

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

7. Simplified 2.7%

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

8. Final simplification 2.7%

   \[\leadsto i \cdot -50 \]

Alternative 25: 48.5% 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 27.7%

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

2. Taylor expanded in i around 0 48.3%

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

   1. *-commutative 48.3%

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

4. Simplified 48.3%

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

5. Final simplification 48.3%

   \[\leadsto n \cdot 100 \]

Developer target: 35.1% 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 2023285 
(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))))