Compound Interest

Percentage Accurate: 28.3% → 95.7%

Time: 28.3s

Alternatives: 12

Speedup: 8.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot \frac{n}{i} + n \cdot \frac{-1}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-24)
     (* 100.0 (+ (* t_0 (/ n i)) (* n (/ -1.0 i))))
     (if (<= t_1 0.0)
       (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) (/ i n))
       (if (<= t_1 INFINITY)
         (/ (+ (* t_0 100.0) -100.0) (/ i n))
         (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = 100.0 * ((t_0 * (n / i)) + (n * (-1.0 / i)));
	} else if (t_1 <= 0.0) {
		tmp = (100.0 * expm1((n * log1p((i / n))))) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = 100.0 * ((t_0 * (n / i)) + (n * (-1.0 / i)));
	} else if (t_1 <= 0.0) {
		tmp = (100.0 * Math.expm1((n * Math.log1p((i / n))))) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -5e-24:
		tmp = 100.0 * ((t_0 * (n / i)) + (n * (-1.0 / i)))
	elif t_1 <= 0.0:
		tmp = (100.0 * math.expm1((n * math.log1p((i / n))))) / (i / n)
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-24)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) + Float64(n * Float64(-1.0 / i))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-24], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] + N[(n * N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.9999999999999998e-24

   1. Initial program 99.7%

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

      1. div-sub 99.7%

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

      2. clear-num 99.7%

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

      3. div-inv 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. div-inv 100.0%

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

      6. clear-num 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -4.9999999999999998e-24 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 26.2%

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

      1. associate-*r/ 26.2%

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

      2. *-commutative 26.2%

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

      3. add-exp-log 26.2%

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

      4. expm1-define 26.2%

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

      5. log-pow 36.0%

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

      6. log1p-define 99.6%

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

   4. Applied egg-rr 99.6%

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

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

   1. Initial program 99.7%

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

      1. associate-*r/ 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. metadata-eval 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   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. Add Preprocessing

   3. Taylor expanded in i around 0 78.0%

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

      1. *-commutative 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} + n \cdot \frac{-1}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot \frac{n}{i} + n \cdot \frac{-1}{i}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-24)
     (* 100.0 (+ (* t_0 (/ n i)) (* n (/ -1.0 i))))
     (if (<= t_1 0.0)
       (* n (* (expm1 (* n (log1p (/ i n)))) (/ 100.0 i)))
       (if (<= t_1 INFINITY)
         (/ (+ (* t_0 100.0) -100.0) (/ i n))
         (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = 100.0 * ((t_0 * (n / i)) + (n * (-1.0 / i)));
	} else if (t_1 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-24) {
		tmp = 100.0 * ((t_0 * (n / i)) + (n * (-1.0 / i)));
	} else if (t_1 <= 0.0) {
		tmp = n * (Math.expm1((n * Math.log1p((i / n)))) * (100.0 / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -5e-24:
		tmp = 100.0 * ((t_0 * (n / i)) + (n * (-1.0 / i)))
	elif t_1 <= 0.0:
		tmp = n * (math.expm1((n * math.log1p((i / n)))) * (100.0 / i))
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-24)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) + Float64(n * Float64(-1.0 / i))));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-24], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] + N[(n * N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.9999999999999998e-24

   1. Initial program 99.7%

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

      1. div-sub 99.7%

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

      2. clear-num 99.7%

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

      3. div-inv 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. div-inv 100.0%

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

      6. clear-num 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -4.9999999999999998e-24 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 26.2%

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

      1. associate-*r/ 26.2%

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

      2. clear-num 26.2%

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

      3. *-commutative 26.2%

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

      4. add-exp-log 26.2%

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

      5. expm1-define 26.2%

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

      6. log-pow 35.7%

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

      7. log1p-define 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. associate-/r/ 98.5%

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

      2. /-rgt-identity 98.5%

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

      3. times-frac 99.6%

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

      4. *-lft-identity 99.6%

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

      5. *-rgt-identity 99.6%

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

      6. associate-/r/ 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

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

   1. Initial program 99.7%

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

      1. associate-*r/ 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. metadata-eval 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   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. Add Preprocessing

   3. Taylor expanded in i around 0 78.0%

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

      1. *-commutative 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} + n \cdot \frac{-1}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-185}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-308}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;n \cdot \left(100 \cdot \left(n \cdot \frac{\log i - \log n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* n (/ (expm1 i) i)))))
   (if (<= n -5.2e-185)
     t_0
     (if (<= n -1e-308)
       0.0
       (if (<= n 6.5e-161)
         (* n (* 100.0 (* n (/ (- (log i) (log n)) i))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (expm1(i) / i));
	double tmp;
	if (n <= -5.2e-185) {
		tmp = t_0;
	} else if (n <= -1e-308) {
		tmp = 0.0;
	} else if (n <= 6.5e-161) {
		tmp = n * (100.0 * (n * ((log(i) - log(n)) / i)));
	} 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 <= -5.2e-185) {
		tmp = t_0;
	} else if (n <= -1e-308) {
		tmp = 0.0;
	} else if (n <= 6.5e-161) {
		tmp = n * (100.0 * (n * ((Math.log(i) - Math.log(n)) / i)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (math.expm1(i) / i))
	tmp = 0
	if n <= -5.2e-185:
		tmp = t_0
	elif n <= -1e-308:
		tmp = 0.0
	elif n <= 6.5e-161:
		tmp = n * (100.0 * (n * ((math.log(i) - math.log(n)) / i)))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -5.2e-185)
		tmp = t_0;
	elseif (n <= -1e-308)
		tmp = 0.0;
	elseif (n <= 6.5e-161)
		tmp = Float64(n * Float64(100.0 * Float64(n * Float64(Float64(log(i) - log(n)) / i))));
	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] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.2e-185], t$95$0, If[LessEqual[n, -1e-308], 0.0, If[LessEqual[n, 6.5e-161], N[(n * N[(100.0 * N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999997e-185 or 6.50000000000000008e-161 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-/l* 37.4%

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

      3. expm1-define 85.3%

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

   5. Simplified 85.3%

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

   if -5.1999999999999997e-185 < n < -9.9999999999999991e-309

   1. Initial program 79.4%

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

      1. div-sub 79.4%

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

      2. clear-num 73.1%

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

      3. div-inv 78.1%

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

      4. cancel-sign-sub-inv 78.1%

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

      5. div-inv 78.1%

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

      6. clear-num 73.0%

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

   4. Applied egg-rr 73.0%

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

   5. Taylor expanded in i around 0 79.4%

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

      1. distribute-rgt1-in 79.4%

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

      2. metadata-eval 79.4%

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      3. mul0-lft 79.4%

         \[\leadsto 100 \cdot \frac{\color{blue}{0}}{i} \]

      4. div0 79.4%

         \[\leadsto 100 \cdot \color{blue}{0} \]

   7. Simplified 79.4%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if -9.9999999999999991e-309 < n < 6.50000000000000008e-161

   1. Initial program 48.7%

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

      1. associate-*r/ 48.7%

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

      2. clear-num 48.7%

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

      3. *-commutative 48.7%

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

      4. add-exp-log 48.7%

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

      5. expm1-define 48.7%

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

      6. log-pow 64.2%

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

      7. log1p-define 68.2%

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

   4. Applied egg-rr 68.2%

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

      1. associate-/r/ 70.8%

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

      2. /-rgt-identity 70.8%

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

      3. times-frac 70.8%

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

      4. *-lft-identity 70.8%

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

      5. *-rgt-identity 70.8%

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

      6. associate-/r/ 71.2%

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

      7. *-commutative 71.2%

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

      8. associate-/l* 71.1%

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

   6. Simplified 71.1%

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

   7. Taylor expanded in n around 0 95.1%

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

      1. associate-/l* 95.2%

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

      2. mul-1-neg 95.2%

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

      3. unsub-neg 95.2%

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

   9. Simplified 95.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-185}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-308}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;n \cdot \left(100 \cdot \left(n \cdot \frac{\log i - \log n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -7.1 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{-54}:\\ \;\;\;\;n \cdot 100 + 100 \cdot \left(i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 6.4 \cdot 10^{+205}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(n + n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -7.1e-40)
     t_0
     (if (<= i 3.2e-54)
       (+ (* n 100.0) (* 100.0 (* i -0.5)))
       (if (<= i 6.4e+205) t_0 (* 100.0 (* (/ 1.0 i) (+ n n))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -7.1e-40) {
		tmp = t_0;
	} else if (i <= 3.2e-54) {
		tmp = (n * 100.0) + (100.0 * (i * -0.5));
	} else if (i <= 6.4e+205) {
		tmp = t_0;
	} else {
		tmp = 100.0 * ((1.0 / i) * (n + n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -7.1e-40) {
		tmp = t_0;
	} else if (i <= 3.2e-54) {
		tmp = (n * 100.0) + (100.0 * (i * -0.5));
	} else if (i <= 6.4e+205) {
		tmp = t_0;
	} else {
		tmp = 100.0 * ((1.0 / i) * (n + n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -7.1e-40:
		tmp = t_0
	elif i <= 3.2e-54:
		tmp = (n * 100.0) + (100.0 * (i * -0.5))
	elif i <= 6.4e+205:
		tmp = t_0
	else:
		tmp = 100.0 * ((1.0 / i) * (n + n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -7.1e-40)
		tmp = t_0;
	elseif (i <= 3.2e-54)
		tmp = Float64(Float64(n * 100.0) + Float64(100.0 * Float64(i * -0.5)));
	elseif (i <= 6.4e+205)
		tmp = t_0;
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 / i) * Float64(n + n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -7.1e-40], t$95$0, If[LessEqual[i, 3.2e-54], N[(N[(n * 100.0), $MachinePrecision] + N[(100.0 * N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 6.4e+205], t$95$0, N[(100.0 * N[(N[(1.0 / i), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -7.10000000000000023e-40 or 3.19999999999999998e-54 < i < 6.39999999999999993e205

   1. Initial program 44.7%

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

   3. Taylor expanded in n around inf 68.9%

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

      1. expm1-define 74.7%

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

   5. Simplified 74.7%

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

   if -7.10000000000000023e-40 < i < 3.19999999999999998e-54

   1. Initial program 9.5%

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

   3. Taylor expanded in i around 0 84.7%

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

   4. Taylor expanded in n around 0 84.7%

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

   if 6.39999999999999993e205 < i

   1. Initial program 42.9%

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

      1. div-sub 42.9%

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

      2. clear-num 36.6%

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

      3. div-inv 42.9%

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

      4. cancel-sign-sub-inv 42.9%

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

      5. div-inv 42.9%

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

      6. clear-num 36.3%

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

   4. Applied egg-rr 36.3%

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

   5. Taylor expanded in i around 0 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. div-inv 58.3%

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

      3. distribute-rgt-out 58.3%

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

      4. add-sqr-sqrt 1.4%

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

      5. sqrt-unprod 65.6%

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

      6. sqr-neg 65.6%

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

      7. sqrt-unprod 51.0%

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

      8. add-sqr-sqrt 58.6%

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

   7. Applied egg-rr 58.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.1 \cdot 10^{-40}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{-54}:\\ \;\;\;\;n \cdot 100 + 100 \cdot \left(i \cdot -0.5\right)\\ \mathbf{elif}\;i \leq 6.4 \cdot 10^{+205}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(n + n\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-190} \lor \neg \left(n \leq 7 \cdot 10^{-182}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.5e-190) (not (<= n 7e-182)))
   (* 100.0 (* n (/ (expm1 i) i)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.5e-190) || !(n <= 7e-182)) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.5e-190) || !(n <= 7e-182)) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.5e-190) or not (n <= 7e-182):
		tmp = 100.0 * (n * (math.expm1(i) / i))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.5e-190) || !(n <= 7e-182))
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.5e-190], N[Not[LessEqual[n, 7e-182]], $MachinePrecision]], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.50000000000000017e-190 or 6.99999999999999966e-182 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf 36.7%

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

      1. *-commutative 36.7%

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

      2. associate-/l* 36.7%

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

      3. expm1-define 84.0%

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

   5. Simplified 84.0%

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

   if -2.50000000000000017e-190 < n < 6.99999999999999966e-182

   1. Initial program 66.9%

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

      1. div-sub 66.9%

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

      2. clear-num 55.9%

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

      3. div-inv 55.9%

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

      4. cancel-sign-sub-inv 55.9%

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

      5. div-inv 55.9%

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

      6. clear-num 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in i around 0 85.8%

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

      1. distribute-rgt1-in 85.8%

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

      2. metadata-eval 85.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      3. mul0-lft 85.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{0}}{i} \]

      4. div0 85.8%

         \[\leadsto 100 \cdot \color{blue}{0} \]

   7. Simplified 85.8%

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-190} \lor \neg \left(n \leq 7 \cdot 10^{-182}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-179} \lor \neg \left(n \leq 4.2 \cdot 10^{-162}\right):\\ \;\;\;\;n \cdot 100 + \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right) \cdot \left(i \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3.4e-179) (not (<= n 4.2e-162)))
   (+ (* n 100.0) (* (* n (- 0.5 (/ 0.5 n))) (* i 100.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3.4e-179) || !(n <= 4.2e-162)) {
		tmp = (n * 100.0) + ((n * (0.5 - (0.5 / n))) * (i * 100.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-3.4d-179)) .or. (.not. (n <= 4.2d-162))) then
        tmp = (n * 100.0d0) + ((n * (0.5d0 - (0.5d0 / n))) * (i * 100.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -3.4e-179) || !(n <= 4.2e-162)) {
		tmp = (n * 100.0) + ((n * (0.5 - (0.5 / n))) * (i * 100.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -3.4e-179) or not (n <= 4.2e-162):
		tmp = (n * 100.0) + ((n * (0.5 - (0.5 / n))) * (i * 100.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3.4e-179) || !(n <= 4.2e-162))
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(n * Float64(0.5 - Float64(0.5 / n))) * Float64(i * 100.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -3.4e-179) || ~((n <= 4.2e-162)))
		tmp = (n * 100.0) + ((n * (0.5 - (0.5 / n))) * (i * 100.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -3.4e-179], N[Not[LessEqual[n, 4.2e-162]], $MachinePrecision]], N[(N[(n * 100.0), $MachinePrecision] + N[(N[(n * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -3.3999999999999997e-179 or 4.2e-162 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in i around 0 58.0%

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

      1. add-log-exp 49.5%

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

      2. associate-*r* 49.5%

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

      3. *-commutative 49.5%

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

      4. exp-prod 63.0%

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

      5. *-commutative 63.0%

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

      6. exp-prod 63.0%

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

      7. cancel-sign-sub-inv 63.0%

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

      8. un-div-inv 63.0%

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

      9. metadata-eval 63.0%

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

   5. Applied egg-rr 63.0%

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

      1. log-pow 65.6%

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

      2. metadata-eval 65.6%

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

      3. distribute-neg-frac 65.6%

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

      4. metadata-eval 65.6%

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

      5. associate-*r/ 65.6%

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

      6. sub-neg 65.6%

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

      7. associate-*r/ 65.6%

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

      8. metadata-eval 65.6%

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

      9. log-pow 58.0%

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

      10. rem-log-exp 58.0%

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

      11. *-commutative 58.0%

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

   7. Simplified 58.0%

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

   if -3.3999999999999997e-179 < n < 4.2e-162

   1. Initial program 61.9%

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

      1. div-sub 61.9%

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

      2. clear-num 51.0%

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

      3. div-inv 52.1%

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

      4. cancel-sign-sub-inv 52.1%

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

      5. div-inv 52.1%

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

      6. clear-num 55.6%

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in i around 0 78.5%

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

      1. distribute-rgt1-in 78.5%

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

      2. metadata-eval 78.5%

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      3. mul0-lft 78.5%

         \[\leadsto 100 \cdot \frac{\color{blue}{0}}{i} \]

      4. div0 78.5%

         \[\leadsto 100 \cdot \color{blue}{0} \]

   7. Simplified 78.5%

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-179} \lor \neg \left(n \leq 4.2 \cdot 10^{-162}\right):\\ \;\;\;\;n \cdot 100 + \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right) \cdot \left(i \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-175} \lor \neg \left(n \leq 7 \cdot 10^{-162}\right):\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.5e-175) (not (<= n 7e-162)))
   (* 100.0 (+ n (* i (* n (- 0.5 (/ 0.5 n))))))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.5e-175) || !(n <= 7e-162)) {
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-5.5d-175)) .or. (.not. (n <= 7d-162))) then
        tmp = 100.0d0 * (n + (i * (n * (0.5d0 - (0.5d0 / n)))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5.5e-175) || !(n <= 7e-162)) {
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5.5e-175) or not (n <= 7e-162):
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.5e-175) || !(n <= 7e-162))
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(0.5 - Float64(0.5 / n))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -5.5e-175) || ~((n <= 7e-162)))
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.5e-175], N[Not[LessEqual[n, 7e-162]], $MachinePrecision]], N[(100.0 * N[(n + N[(i * N[(n * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -5.50000000000000054e-175 or 6.9999999999999998e-162 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in i around 0 58.0%

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

      1. associate-*r/ 58.0%

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

      2. metadata-eval 58.0%

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

   5. Simplified 58.0%

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

   if -5.50000000000000054e-175 < n < 6.9999999999999998e-162

   1. Initial program 61.9%

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

      1. div-sub 61.9%

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

      2. clear-num 51.0%

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

      3. div-inv 52.1%

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

      4. cancel-sign-sub-inv 52.1%

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

      5. div-inv 52.1%

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

      6. clear-num 55.6%

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in i around 0 78.5%

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

      1. distribute-rgt1-in 78.5%

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

      2. metadata-eval 78.5%

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      3. mul0-lft 78.5%

         \[\leadsto 100 \cdot \frac{\color{blue}{0}}{i} \]

      4. div0 78.5%

         \[\leadsto 100 \cdot \color{blue}{0} \]

   7. Simplified 78.5%

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-175} \lor \neg \left(n \leq 7 \cdot 10^{-162}\right):\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-181} \lor \neg \left(n \leq 1.45 \cdot 10^{-182}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.4e-181) (not (<= n 1.45e-182)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e-181) || !(n <= 1.45e-182)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.4d-181)) .or. (.not. (n <= 1.45d-182))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e-181) || !(n <= 1.45e-182)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.4e-181) or not (n <= 1.45e-182):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.4e-181) || !(n <= 1.45e-182))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.4e-181) || ~((n <= 1.45e-182)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.4e-181], N[Not[LessEqual[n, 1.45e-182]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.39999999999999993e-181 or 1.44999999999999993e-182 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf 36.7%

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

      1. *-commutative 36.7%

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

      2. associate-/l* 36.7%

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

      3. expm1-define 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in i around 0 57.2%

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

      1. associate-*r* 57.2%

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

      2. distribute-rgt-out 57.2%

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

   8. Applied egg-rr 57.2%

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

   if -1.39999999999999993e-181 < n < 1.44999999999999993e-182

   1. Initial program 66.9%

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

      1. div-sub 66.9%

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

      2. clear-num 55.9%

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

      3. div-inv 55.9%

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

      4. cancel-sign-sub-inv 55.9%

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

      5. div-inv 55.9%

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

      6. clear-num 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in i around 0 85.8%

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

      1. distribute-rgt1-in 85.8%

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

      2. metadata-eval 85.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      3. mul0-lft 85.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{0}}{i} \]

      4. div0 85.8%

         \[\leadsto 100 \cdot \color{blue}{0} \]

   7. Simplified 85.8%

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-181} \lor \neg \left(n \leq 1.45 \cdot 10^{-182}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.16 \cdot 10^{-175}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-182}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.16e-175)
   (+ (* n 100.0) (* 50.0 (* i n)))
   (if (<= n 2.6e-182) 0.0 (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.16e-175) {
		tmp = (n * 100.0) + (50.0 * (i * n));
	} else if (n <= 2.6e-182) {
		tmp = 0.0;
	} 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 <= (-1.16d-175)) then
        tmp = (n * 100.0d0) + (50.0d0 * (i * n))
    else if (n <= 2.6d-182) then
        tmp = 0.0d0
    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 <= -1.16e-175) {
		tmp = (n * 100.0) + (50.0 * (i * n));
	} else if (n <= 2.6e-182) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.16e-175:
		tmp = (n * 100.0) + (50.0 * (i * n))
	elif n <= 2.6e-182:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.16e-175)
		tmp = Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)));
	elseif (n <= 2.6e-182)
		tmp = 0.0;
	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 <= -1.16e-175)
		tmp = (n * 100.0) + (50.0 * (i * n));
	elseif (n <= 2.6e-182)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.16e-175], N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.6e-182], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.16000000000000002e-175

   1. Initial program 27.7%

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

   3. Taylor expanded in n around inf 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-/l* 42.1%

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

      3. expm1-define 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in i around 0 50.1%

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

   if -1.16000000000000002e-175 < n < 2.60000000000000006e-182

   1. Initial program 66.9%

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

      1. div-sub 66.9%

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

      2. clear-num 55.9%

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

      3. div-inv 55.9%

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

      4. cancel-sign-sub-inv 55.9%

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

      5. div-inv 55.9%

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

      6. clear-num 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in i around 0 85.8%

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

      1. distribute-rgt1-in 85.8%

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

      2. metadata-eval 85.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      3. mul0-lft 85.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{0}}{i} \]

      4. div0 85.8%

         \[\leadsto 100 \cdot \color{blue}{0} \]

   7. Simplified 85.8%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if 2.60000000000000006e-182 < n

   1. Initial program 15.6%

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

   3. Taylor expanded in n around inf 31.4%

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

      1. *-commutative 31.4%

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

      2. associate-/l* 31.4%

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

      3. expm1-define 83.9%

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

   5. Simplified 83.9%

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

   6. Taylor expanded in i around 0 64.1%

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

      1. associate-*r* 64.1%

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

      2. distribute-rgt-out 64.1%

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

   8. Applied egg-rr 64.1%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.16 \cdot 10^{-175}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-182}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-189} \lor \neg \left(n \leq 2 \cdot 10^{-181}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.5e-189) (not (<= n 2e-181))) (* n 100.0) 0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.5e-189) || !(n <= 2e-181)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.5d-189)) .or. (.not. (n <= 2d-181))) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.5e-189) || !(n <= 2e-181)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.5e-189) or not (n <= 2e-181):
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.5e-189) || !(n <= 2e-181))
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2.5e-189) || ~((n <= 2e-181)))
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.5e-189], N[Not[LessEqual[n, 2e-181]], $MachinePrecision]], N[(n * 100.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.4999999999999999e-189 or 2.00000000000000009e-181 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in i around 0 52.1%

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

      1. *-commutative 52.1%

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

   5. Simplified 52.1%

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

   if -2.4999999999999999e-189 < n < 2.00000000000000009e-181

   1. Initial program 66.9%

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

      1. div-sub 66.9%

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

      2. clear-num 55.9%

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

      3. div-inv 55.9%

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

      4. cancel-sign-sub-inv 55.9%

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

      5. div-inv 55.9%

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

      6. clear-num 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in i around 0 85.8%

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

      1. distribute-rgt1-in 85.8%

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

      2. metadata-eval 85.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      3. mul0-lft 85.8%

         \[\leadsto 100 \cdot \frac{\color{blue}{0}}{i} \]

      4. div0 85.8%

         \[\leadsto 100 \cdot \color{blue}{0} \]

   7. Simplified 85.8%

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-189} \lor \neg \left(n \leq 2 \cdot 10^{-181}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 2.3% 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 28.1%

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

   1. associate-*r/ 28.1%

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

   2. sub-neg 28.1%

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

   3. distribute-lft-in 28.1%

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

   4. metadata-eval 28.1%

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

   5. metadata-eval 28.1%

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

3. Simplified 28.1%

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

5. Taylor expanded in i around 0 16.3%

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

   1. *-commutative 16.3%

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

7. Simplified 16.3%

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

   1. frac-2neg 16.3%

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

   2. div-inv 16.2%

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

   3. +-commutative 16.2%

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

   4. associate-+l+ 41.4%

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

   5. metadata-eval 41.4%

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

   6. distribute-neg-frac2 41.4%

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

   7. clear-num 40.3%

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

   8. add-sqr-sqrt 19.4%

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

   9. sqrt-unprod 32.1%

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

   10. sqr-neg 32.1%

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

   11. sqrt-unprod 4.5%

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

   12. add-sqr-sqrt 8.5%

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

9. Applied egg-rr 8.5%

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

    1. *-commutative 8.5%

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

    2. +-rgt-identity 8.5%

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

    3. distribute-rgt-neg-in 8.5%

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

    4. metadata-eval 8.5%

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

11. Simplified 8.5%

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

12. Taylor expanded in n around 0 2.3%

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

13. Final simplification 2.3%

    \[\leadsto n \cdot -100 \]
14. Add Preprocessing

Alternative 12: 18.2% accurate, 114.0× speedup

Math

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

FPCore

(FPCore (i n) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

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

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 28.1%

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

   1. div-sub 28.1%

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

   2. clear-num 25.6%

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

   3. div-inv 26.3%

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

   4. cancel-sign-sub-inv 26.3%

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

   5. div-inv 26.3%

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

   6. clear-num 26.5%

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

4. Applied egg-rr 26.5%

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

5. Taylor expanded in i around 0 19.1%

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

   1. distribute-rgt1-in 19.1%

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

   2. metadata-eval 19.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

   3. mul0-lft 19.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{0}}{i} \]

   4. div0 19.1%

      \[\leadsto 100 \cdot \color{blue}{0} \]

7. Simplified 19.1%

   \[\leadsto 100 \cdot \color{blue}{0} \]

8. Final simplification 19.1%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer target: 34.1% 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 2024053 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (* 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))))