Compound Interest

Percentage Accurate: 27.8% → 95.4%

Time: 28.6s

Alternatives: 15

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.8% 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: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n}\right)}^{n} - 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -50000000.0)
     (/ (+ (* t_0 100.0) -100.0) (/ i n))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_1 INFINITY)
         (/ (* n (- (* 100.0 (pow (/ i n) n)) 100.0)) i)
         (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (n * ((100.0 * pow((i / n), n)) - 100.0)) / i;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} 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 = (n * ((100.0 * Math.pow((i / n), n)) - 100.0)) / i;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -50000000.0:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = (n * ((100.0 * math.pow((i / n), n)) - 100.0)) / i
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(n * Float64(Float64(100.0 * (Float64(i / n) ^ n)) - 100.0)) / i);
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(n * N[(N[(100.0 * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]), $MachinePrecision] / i), $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)) < -5e7

   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/ 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. fma-def 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -5e7 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 26.3%

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

      1. *-un-lft-identity 26.3%

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

      2. add-exp-log 26.3%

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

      3. expm1-def 26.3%

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

      4. log-pow 40.5%

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

      5. log1p-udef 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   6. Simplified 99.8%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in i around inf 99.6%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in n around inf 99.8%

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

   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. Add Preprocessing

   3. Taylor expanded in i around 0 71.4%

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

      1. *-commutative 71.4%

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

   5. Simplified 71.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -50000000:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot {\left(\frac{i}{n}\right)}^{n} - 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq -1.72 \cdot 10^{-175}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{n}{\frac{i}{\log \left(\frac{i}{n}\right)}}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-212}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.3e-35)
   (/ (* (* n 100.0) (expm1 i)) i)
   (if (<= n -1.72e-175)
     (* (* n 100.0) (/ n (/ i (log (/ i n)))))
     (if (<= n 2.2e-212)
       (/ 0.0 (/ i n))
       (if (<= n 2.8e-37)
         (* 100.0 (/ i (/ i n)))
         (* 100.0 (/ n (/ i (expm1 i)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.3e-35) {
		tmp = ((n * 100.0) * expm1(i)) / i;
	} else if (n <= -1.72e-175) {
		tmp = (n * 100.0) * (n / (i / log((i / n))));
	} else if (n <= 2.2e-212) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n / (i / expm1(i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.3e-35) {
		tmp = ((n * 100.0) * Math.expm1(i)) / i;
	} else if (n <= -1.72e-175) {
		tmp = (n * 100.0) * (n / (i / Math.log((i / n))));
	} else if (n <= 2.2e-212) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.3e-35:
		tmp = ((n * 100.0) * math.expm1(i)) / i
	elif n <= -1.72e-175:
		tmp = (n * 100.0) * (n / (i / math.log((i / n))))
	elif n <= 2.2e-212:
		tmp = 0.0 / (i / n)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.3e-35)
		tmp = Float64(Float64(Float64(n * 100.0) * expm1(i)) / i);
	elseif (n <= -1.72e-175)
		tmp = Float64(Float64(n * 100.0) * Float64(n / Float64(i / log(Float64(i / n)))));
	elseif (n <= 2.2e-212)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.3e-35], N[(N[(N[(n * 100.0), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -1.72e-175], N[(N[(n * 100.0), $MachinePrecision] * N[(n / N[(i / N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.2e-212], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.30000000000000002e-35

   1. Initial program 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-/r/ 37.5%

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

      3. associate-*l* 37.5%

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

      4. sub-neg 37.5%

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

      5. metadata-eval 37.5%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in n around inf 41.1%

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

      1. expm1-def 84.6%

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

   7. Simplified 84.6%

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

      1. associate-*l/ 84.7%

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

   9. Applied egg-rr 84.7%

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

   if -1.30000000000000002e-35 < n < -1.72000000000000005e-175

   1. Initial program 26.5%

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

      1. *-commutative 26.5%

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

      2. associate-/r/ 24.7%

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

      3. associate-*l* 24.7%

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

      4. sub-neg 24.7%

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

      5. metadata-eval 24.7%

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

   3. Simplified 24.7%

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

      1. expm1-log1p-u 24.7%

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

      2. expm1-udef 26.5%

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

      3. metadata-eval 26.5%

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

      4. sub-neg 26.5%

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

      5. add-exp-log 26.5%

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

      6. expm1-def 26.5%

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

      7. log-pow 26.5%

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

      8. log1p-udef 51.4%

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

   6. Applied egg-rr 51.4%

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

      1. expm1-def 97.9%

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

      2. expm1-log1p 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in n around 0 0.0%

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

       1. associate-/l* 0.0%

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

       2. mul-1-neg 0.0%

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

       3. sub-neg 0.0%

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

       4. log-div 66.8%

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

   11. Simplified 66.8%

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

   if -1.72000000000000005e-175 < n < 2.20000000000000003e-212

   1. Initial program 69.8%

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

      1. associate-*r/ 69.8%

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

      2. sub-neg 69.8%

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

      3. distribute-lft-in 69.8%

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

      4. metadata-eval 69.8%

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

      5. metadata-eval 69.8%

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

      6. metadata-eval 69.8%

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

      7. fma-def 69.8%

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

      8. metadata-eval 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in n around inf 59.8%

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

   6. Taylor expanded in i around 0 87.1%

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

   if 2.20000000000000003e-212 < n < 2.8000000000000001e-37

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 65.6%

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

   if 2.8000000000000001e-37 < n

   1. Initial program 20.2%

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

   3. Taylor expanded in n around inf 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-/l* 38.7%

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

      3. expm1-def 94.6%

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

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq -1.72 \cdot 10^{-175}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{n}{\frac{i}{\log \left(\frac{i}{n}\right)}}\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-212}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -5.4 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-210}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -5.4e-152)
     t_0
     (if (<= n 2.8e-210)
       (/ 0.0 (/ i n))
       (if (<= n 2.8e-37) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -5.4e-152) {
		tmp = t_0;
	} else if (n <= 2.8e-210) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		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 / (i / Math.expm1(i)));
	double tmp;
	if (n <= -5.4e-152) {
		tmp = t_0;
	} else if (n <= 2.8e-210) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / 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 <= -5.4e-152:
		tmp = t_0
	elif n <= 2.8e-210:
		tmp = 0.0 / (i / n)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / 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 <= -5.4e-152)
		tmp = t_0;
	elseif (n <= 2.8e-210)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.8e-37)
		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[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.4e-152], t$95$0, If[LessEqual[n, 2.8e-210], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.39999999999999997e-152 or 2.8000000000000001e-37 < n

   1. Initial program 28.4%

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

   3. Taylor expanded in n around inf 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-/l* 36.8%

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

      3. expm1-def 84.7%

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

   5. Simplified 84.7%

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

   if -5.39999999999999997e-152 < n < 2.8e-210

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. sub-neg 63.7%

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

      3. distribute-lft-in 63.7%

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

      4. metadata-eval 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. fma-def 63.7%

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

      8. metadata-eval 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in n around inf 50.8%

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

   6. Taylor expanded in i around 0 77.2%

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

   if 2.8e-210 < n < 2.8000000000000001e-37

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 65.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-210}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-152}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-211}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.4e-152)
   (* (* n 100.0) (/ (expm1 i) i))
   (if (<= n 8e-211)
     (/ 0.0 (/ i n))
     (if (<= n 1.75e-37)
       (* 100.0 (/ i (/ i n)))
       (* 100.0 (/ n (/ i (expm1 i))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.4e-152) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (n <= 8e-211) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.75e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n / (i / expm1(i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.4e-152) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else if (n <= 8e-211) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.75e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.4e-152:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	elif n <= 8e-211:
		tmp = 0.0 / (i / n)
	elif n <= 1.75e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.4e-152)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (n <= 8e-211)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.75e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.4e-152], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8e-211], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.39999999999999997e-152

   1. Initial program 34.2%

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

      1. *-commutative 34.2%

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

      2. associate-/r/ 34.0%

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

      3. associate-*l* 34.1%

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

      4. sub-neg 34.1%

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

      5. metadata-eval 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in n around inf 35.4%

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

      1. expm1-def 77.5%

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

   7. Simplified 77.5%

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

   if -5.39999999999999997e-152 < n < 8.00000000000000069e-211

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. sub-neg 63.7%

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

      3. distribute-lft-in 63.7%

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

      4. metadata-eval 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. fma-def 63.7%

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

      8. metadata-eval 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in n around inf 50.8%

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

   6. Taylor expanded in i around 0 77.2%

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

   if 8.00000000000000069e-211 < n < 1.7500000000000001e-37

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 65.6%

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

   if 1.7500000000000001e-37 < n

   1. Initial program 20.2%

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

   3. Taylor expanded in n around inf 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-/l* 38.7%

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

      3. expm1-def 94.6%

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

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}} \cdot 100} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-152}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-211}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-210}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.9e-152)
   (* (/ n i) (* 100.0 (expm1 i)))
   (if (<= n 1.35e-210)
     (/ 0.0 (/ i n))
     (if (<= n 2.8e-37)
       (* 100.0 (/ i (/ i n)))
       (* (* n 100.0) (+ 1.0 (* i (- 0.5 (/ 0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.9e-152) {
		tmp = (n / i) * (100.0 * expm1(i));
	} else if (n <= 1.35e-210) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.9e-152) {
		tmp = (n / i) * (100.0 * Math.expm1(i));
	} else if (n <= 1.35e-210) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.9e-152:
		tmp = (n / i) * (100.0 * math.expm1(i))
	elif n <= 1.35e-210:
		tmp = 0.0 / (i / n)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.9e-152)
		tmp = Float64(Float64(n / i) * Float64(100.0 * expm1(i)));
	elseif (n <= 1.35e-210)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 - Float64(0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.9e-152], N[(N[(n / i), $MachinePrecision] * N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.35e-210], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.90000000000000011e-152

   1. Initial program 34.2%

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

      1. associate-*r/ 34.3%

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

      2. sub-neg 34.3%

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

      3. distribute-lft-in 34.3%

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

      4. metadata-eval 34.3%

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

      5. metadata-eval 34.3%

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

      6. metadata-eval 34.3%

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

      7. fma-def 34.3%

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

      8. metadata-eval 34.3%

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

   3. Simplified 34.3%

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

      1. fma-udef 34.3%

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

      2. metadata-eval 34.3%

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

      3. distribute-lft-in 34.3%

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

      4. metadata-eval 34.3%

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

      5. sub-neg 34.3%

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

      6. associate-*r/ 34.2%

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

      7. *-commutative 34.2%

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

      8. div-inv 34.2%

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

      9. clear-num 33.7%

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

      10. add-exp-log 33.7%

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

      11. expm1-def 33.7%

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

      12. log-pow 28.7%

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

      13. log1p-udef 71.2%

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

   6. Applied egg-rr 71.2%

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

   7. Taylor expanded in n around inf 35.4%

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

      1. expm1-def 73.9%

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

      2. *-commutative 73.9%

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

      3. associate-*r/ 63.0%

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

      4. associate-*r* 62.9%

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

   9. Simplified 62.9%

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

   if -5.90000000000000011e-152 < n < 1.34999999999999996e-210

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. sub-neg 63.7%

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

      3. distribute-lft-in 63.7%

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

      4. metadata-eval 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. fma-def 63.7%

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

      8. metadata-eval 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in n around inf 50.8%

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

   6. Taylor expanded in i around 0 77.2%

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

   if 1.34999999999999996e-210 < n < 2.8000000000000001e-37

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 65.6%

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

   if 2.8000000000000001e-37 < n

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r/ 20.8%

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

      3. associate-*l* 20.8%

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

      4. sub-neg 20.8%

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

      5. metadata-eval 20.8%

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

   3. Simplified 20.8%

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

   5. Taylor expanded in i around 0 81.8%

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

      1. associate-*r/ 81.8%

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

      2. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-210}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-218}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.9e-152)
   (* 100.0 (* (expm1 i) (/ n i)))
   (if (<= n 3e-218)
     (/ 0.0 (/ i n))
     (if (<= n 2.8e-37)
       (* 100.0 (/ i (/ i n)))
       (* (* n 100.0) (+ 1.0 (* i (- 0.5 (/ 0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.9e-152) {
		tmp = 100.0 * (expm1(i) * (n / i));
	} else if (n <= 3e-218) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5.9e-152) {
		tmp = 100.0 * (Math.expm1(i) * (n / i));
	} else if (n <= 3e-218) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5.9e-152:
		tmp = 100.0 * (math.expm1(i) * (n / i))
	elif n <= 3e-218:
		tmp = 0.0 / (i / n)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.9e-152)
		tmp = Float64(100.0 * Float64(expm1(i) * Float64(n / i)));
	elseif (n <= 3e-218)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 - Float64(0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.9e-152], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3e-218], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.90000000000000011e-152

   1. Initial program 34.2%

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

      1. associate-*r/ 34.3%

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

      2. sub-neg 34.3%

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

      3. distribute-lft-in 34.3%

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

      4. metadata-eval 34.3%

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

      5. metadata-eval 34.3%

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

      6. metadata-eval 34.3%

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

      7. fma-def 34.3%

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

      8. metadata-eval 34.3%

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

   3. Simplified 34.3%

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

      1. fma-udef 34.3%

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

      2. metadata-eval 34.3%

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

      3. distribute-lft-in 34.3%

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

      4. metadata-eval 34.3%

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

      5. sub-neg 34.3%

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

      6. associate-*r/ 34.2%

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

      7. *-commutative 34.2%

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

      8. div-inv 34.2%

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

      9. clear-num 33.7%

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

      10. add-exp-log 33.7%

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

      11. expm1-def 33.7%

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

      12. log-pow 28.7%

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

      13. log1p-udef 71.2%

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

   6. Applied egg-rr 71.2%

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

   7. Taylor expanded in n around inf 35.4%

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

      1. expm1-def 73.9%

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

      2. associate-*l/ 63.0%

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

   9. Simplified 63.0%

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

   if -5.90000000000000011e-152 < n < 2.9999999999999998e-218

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. sub-neg 63.7%

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

      3. distribute-lft-in 63.7%

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

      4. metadata-eval 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. fma-def 63.7%

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

      8. metadata-eval 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in n around inf 50.8%

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

   6. Taylor expanded in i around 0 77.2%

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

   if 2.9999999999999998e-218 < n < 2.8000000000000001e-37

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 65.6%

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

   if 2.8000000000000001e-37 < n

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r/ 20.8%

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

      3. associate-*l* 20.8%

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

      4. sub-neg 20.8%

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

      5. metadata-eval 20.8%

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

   3. Simplified 20.8%

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

   5. Taylor expanded in i around 0 81.8%

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

      1. associate-*r/ 81.8%

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

      2. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-218}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-147}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.14 \cdot 10^{-210}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.8e-147)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n 1.14e-210)
     (/ 0.0 (/ i n))
     (if (<= n 2.65e-37)
       (* 100.0 (/ i (/ i n)))
       (* 100.0 (+ n (* (- 0.5 (/ 0.5 n)) (* i n))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.8e-147) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 1.14e-210) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.65e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.8d-147)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= 1.14d-210) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 2.65d-37) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * (n + ((0.5d0 - (0.5d0 / n)) * (i * n)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.8e-147) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 1.14e-210) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.65e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.8e-147:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= 1.14e-210:
		tmp = 0.0 / (i / n)
	elif n <= 2.65e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.8e-147)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= 1.14e-210)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.65e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.8e-147)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= 1.14e-210)
		tmp = 0.0 / (i / n);
	elseif (n <= 2.65e-37)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.8e-147], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.14e-210], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.65e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.8000000000000001e-147

   1. Initial program 34.2%

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

   3. Taylor expanded in i around 0 50.6%

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

      1. associate-*r* 50.6%

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

      2. associate-*r/ 50.6%

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

      3. metadata-eval 50.6%

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

   5. Simplified 50.6%

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

   6. Taylor expanded in n around inf 50.8%

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

      1. *-commutative 50.8%

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

      2. *-commutative 50.8%

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

      3. associate-*l* 50.8%

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

   8. Simplified 50.8%

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

   9. Taylor expanded in i around 0 50.8%

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

       1. +-commutative 50.8%

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

       2. *-commutative 50.8%

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

       3. *-commutative 50.8%

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

       4. *-commutative 50.8%

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

       5. associate-*l* 50.8%

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

       6. distribute-lft-out 50.9%

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

   11. Simplified 50.9%

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

   if -9.8000000000000001e-147 < n < 1.14e-210

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. sub-neg 63.7%

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

      3. distribute-lft-in 63.7%

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

      4. metadata-eval 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. fma-def 63.7%

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

      8. metadata-eval 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in n around inf 50.8%

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

   6. Taylor expanded in i around 0 77.2%

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

   if 1.14e-210 < n < 2.64999999999999998e-37

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 65.6%

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

   if 2.64999999999999998e-37 < n

   1. Initial program 20.2%

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

   3. Taylor expanded in i around 0 81.7%

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

      1. associate-*r* 81.7%

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

      2. associate-*r/ 81.7%

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

      3. metadata-eval 81.7%

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

   5. Simplified 81.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-147}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.14 \cdot 10^{-210}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.4 \cdot 10^{-152}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 7.4 \cdot 10^{-211}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.4e-152)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n 7.4e-211)
     (/ 0.0 (/ i n))
     (if (<= n 1.9e-37)
       (* 100.0 (/ i (/ i n)))
       (* (* n 100.0) (+ 1.0 (* i (- 0.5 (/ 0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.4e-152) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 7.4e-211) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.9e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-7.4d-152)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= 7.4d-211) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 1.9d-37) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * (0.5d0 - (0.5d0 / n))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.4e-152) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 7.4e-211) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.9e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.4e-152:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= 7.4e-211:
		tmp = 0.0 / (i / n)
	elif n <= 1.9e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.4e-152)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= 7.4e-211)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.9e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * Float64(0.5 - Float64(0.5 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -7.4e-152)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= 7.4e-211)
		tmp = 0.0 / (i / n);
	elseif (n <= 1.9e-37)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) * (1.0 + (i * (0.5 - (0.5 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.4e-152], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.4e-211], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.9e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -7.3999999999999997e-152

   1. Initial program 34.2%

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

   3. Taylor expanded in i around 0 50.6%

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

      1. associate-*r* 50.6%

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

      2. associate-*r/ 50.6%

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

      3. metadata-eval 50.6%

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

   5. Simplified 50.6%

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

   6. Taylor expanded in n around inf 50.8%

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

      1. *-commutative 50.8%

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

      2. *-commutative 50.8%

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

      3. associate-*l* 50.8%

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

   8. Simplified 50.8%

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

   9. Taylor expanded in i around 0 50.8%

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

       1. +-commutative 50.8%

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

       2. *-commutative 50.8%

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

       3. *-commutative 50.8%

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

       4. *-commutative 50.8%

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

       5. associate-*l* 50.8%

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

       6. distribute-lft-out 50.9%

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

   11. Simplified 50.9%

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

   if -7.3999999999999997e-152 < n < 7.3999999999999996e-211

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. sub-neg 63.7%

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

      3. distribute-lft-in 63.7%

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

      4. metadata-eval 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. fma-def 63.7%

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

      8. metadata-eval 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in n around inf 50.8%

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

   6. Taylor expanded in i around 0 77.2%

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

   if 7.3999999999999996e-211 < n < 1.9000000000000002e-37

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 65.6%

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

   if 1.9000000000000002e-37 < n

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r/ 20.8%

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

      3. associate-*l* 20.8%

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

      4. sub-neg 20.8%

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

      5. metadata-eval 20.8%

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

   3. Simplified 20.8%

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

   5. Taylor expanded in i around 0 81.8%

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

      1. associate-*r/ 81.8%

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

      2. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.4 \cdot 10^{-152}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 7.4 \cdot 10^{-211}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{if}\;i \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{+32}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 3.6 \cdot 10^{+244} \lor \neg \left(i \leq 1.45 \cdot 10^{+300}\right):\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* i (/ n i)))))
   (if (<= i -1e+16)
     t_0
     (if (<= i 2.6e+32)
       (* n 100.0)
       (if (or (<= i 3.6e+244) (not (<= i 1.45e+300)))
         (* 50.0 (* i n))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i * (n / i));
	double tmp;
	if (i <= -1e+16) {
		tmp = t_0;
	} else if (i <= 2.6e+32) {
		tmp = n * 100.0;
	} else if ((i <= 3.6e+244) || !(i <= 1.45e+300)) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (i * (n / i))
    if (i <= (-1d+16)) then
        tmp = t_0
    else if (i <= 2.6d+32) then
        tmp = n * 100.0d0
    else if ((i <= 3.6d+244) .or. (.not. (i <= 1.45d+300))) then
        tmp = 50.0d0 * (i * n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (i * (n / i));
	double tmp;
	if (i <= -1e+16) {
		tmp = t_0;
	} else if (i <= 2.6e+32) {
		tmp = n * 100.0;
	} else if ((i <= 3.6e+244) || !(i <= 1.45e+300)) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i * (n / i))
	tmp = 0
	if i <= -1e+16:
		tmp = t_0
	elif i <= 2.6e+32:
		tmp = n * 100.0
	elif (i <= 3.6e+244) or not (i <= 1.45e+300):
		tmp = 50.0 * (i * n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i * Float64(n / i)))
	tmp = 0.0
	if (i <= -1e+16)
		tmp = t_0;
	elseif (i <= 2.6e+32)
		tmp = Float64(n * 100.0);
	elseif ((i <= 3.6e+244) || !(i <= 1.45e+300))
		tmp = Float64(50.0 * Float64(i * n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (i * (n / i));
	tmp = 0.0;
	if (i <= -1e+16)
		tmp = t_0;
	elseif (i <= 2.6e+32)
		tmp = n * 100.0;
	elseif ((i <= 3.6e+244) || ~((i <= 1.45e+300)))
		tmp = 50.0 * (i * n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1e+16], t$95$0, If[LessEqual[i, 2.6e+32], N[(n * 100.0), $MachinePrecision], If[Or[LessEqual[i, 3.6e+244], N[Not[LessEqual[i, 1.45e+300]], $MachinePrecision]], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1e16 or 3.6e244 < i < 1.44999999999999993e300

   1. Initial program 65.3%

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

      1. associate-*r/ 65.4%

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

      2. sub-neg 65.4%

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

      3. distribute-lft-in 65.4%

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

      4. metadata-eval 65.4%

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

      5. metadata-eval 65.4%

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

      6. metadata-eval 65.4%

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

      7. fma-def 65.4%

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

      8. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in i around 0 30.8%

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

      1. *-commutative 30.8%

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

   7. Simplified 30.8%

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

      1. div-inv 30.8%

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

      2. *-commutative 30.8%

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

      3. clear-num 26.5%

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

      4. associate-*l* 26.5%

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

   9. Applied egg-rr 26.5%

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

   if -1e16 < i < 2.6000000000000002e32

   1. Initial program 10.6%

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

   3. Taylor expanded in i around 0 75.4%

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

      1. *-commutative 75.4%

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

   5. Simplified 75.4%

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

   if 2.6000000000000002e32 < i < 3.6e244 or 1.44999999999999993e300 < i

   1. Initial program 44.6%

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

   3. Taylor expanded in i around 0 35.6%

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

      1. associate-*r* 35.6%

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

      2. associate-*r/ 35.6%

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

      3. metadata-eval 35.6%

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

   5. Simplified 35.6%

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

   6. Taylor expanded in n around inf 35.9%

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

      1. *-commutative 35.9%

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

      2. *-commutative 35.9%

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

      3. associate-*l* 35.9%

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

   8. Simplified 35.9%

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

   9. Taylor expanded in i around inf 35.9%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{+16}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq 2.6 \cdot 10^{+32}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 3.6 \cdot 10^{+244} \lor \neg \left(i \leq 1.45 \cdot 10^{+300}\right):\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -4.9 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -4.9e-151)
     t_0
     (if (<= n 3.6e-212)
       (/ 0.0 (/ i n))
       (if (<= n 2.8e-37) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -4.9e-151) {
		tmp = t_0;
	} else if (n <= 3.6e-212) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-4.9d-151)) then
        tmp = t_0
    else if (n <= 3.6d-212) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 2.8d-37) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -4.9e-151) {
		tmp = t_0;
	} else if (n <= 3.6e-212) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8e-37) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -4.9e-151:
		tmp = t_0
	elif n <= 3.6e-212:
		tmp = 0.0 / (i / n)
	elif n <= 2.8e-37:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -4.9e-151)
		tmp = t_0;
	elseif (n <= 3.6e-212)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.8e-37)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -4.9e-151)
		tmp = t_0;
	elseif (n <= 3.6e-212)
		tmp = 0.0 / (i / n);
	elseif (n <= 2.8e-37)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.9e-151], t$95$0, If[LessEqual[n, 3.6e-212], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-37], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.89999999999999966e-151 or 2.8000000000000001e-37 < n

   1. Initial program 28.4%

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

   3. Taylor expanded in i around 0 63.6%

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

      1. associate-*r* 63.6%

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

      2. associate-*r/ 63.6%

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

      3. metadata-eval 63.6%

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

   5. Simplified 63.6%

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

   6. Taylor expanded in n around inf 63.6%

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

      1. *-commutative 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-*l* 63.6%

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

   8. Simplified 63.6%

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

   9. Taylor expanded in i around 0 63.6%

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

       1. +-commutative 63.6%

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

       2. *-commutative 63.6%

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

       3. *-commutative 63.6%

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

       4. *-commutative 63.6%

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

       5. associate-*l* 63.6%

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

       6. distribute-lft-out 63.6%

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

   11. Simplified 63.6%

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

   if -4.89999999999999966e-151 < n < 3.6000000000000001e-212

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. sub-neg 63.7%

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

      3. distribute-lft-in 63.7%

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

      4. metadata-eval 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. fma-def 63.7%

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

      8. metadata-eval 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in n around inf 50.8%

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

   6. Taylor expanded in i around 0 77.2%

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

   if 3.6000000000000001e-212 < n < 2.8000000000000001e-37

   1. Initial program 6.3%

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

   3. Taylor expanded in i around 0 65.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{-151}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+56} \lor \neg \left(n \leq 10^{-37}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.6e+56) (not (<= n 1e-37)))
   (* n 100.0)
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.6e+56) || !(n <= 1e-37)) {
		tmp = n * 100.0;
	} 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 <= (-1.6d+56)) .or. (.not. (n <= 1d-37))) then
        tmp = n * 100.0d0
    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 <= -1.6e+56) || !(n <= 1e-37)) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.6e+56) or not (n <= 1e-37):
		tmp = n * 100.0
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.6e+56) || !(n <= 1e-37))
		tmp = Float64(n * 100.0);
	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 <= -1.6e+56) || ~((n <= 1e-37)))
		tmp = n * 100.0;
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.6e+56], N[Not[LessEqual[n, 1e-37]], $MachinePrecision]], N[(n * 100.0), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.60000000000000002e56 or 1.00000000000000007e-37 < n

   1. Initial program 26.2%

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

   3. Taylor expanded in i around 0 54.5%

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

      1. *-commutative 54.5%

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

   5. Simplified 54.5%

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

   if -1.60000000000000002e56 < n < 1.00000000000000007e-37

   1. Initial program 38.0%

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

   3. Taylor expanded in i around 0 52.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+56} \lor \neg \left(n \leq 10^{-37}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.86 \cdot 10^{+56} \lor \neg \left(n \leq 1.5\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.86e+56) (not (<= n 1.5)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.86e+56) || !(n <= 1.5)) {
		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 <= (-1.86d+56)) .or. (.not. (n <= 1.5d0))) 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 <= -1.86e+56) || !(n <= 1.5)) {
		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 <= -1.86e+56) or not (n <= 1.5):
		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 <= -1.86e+56) || !(n <= 1.5))
		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 <= -1.86e+56) || ~((n <= 1.5)))
		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, -1.86e+56], N[Not[LessEqual[n, 1.5]], $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 < -1.86000000000000007e56 or 1.5 < n

   1. Initial program 26.8%

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

   3. Taylor expanded in i around 0 68.2%

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

      1. associate-*r* 68.2%

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

      2. associate-*r/ 68.2%

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

      3. metadata-eval 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in n around inf 68.2%

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

      1. *-commutative 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*l* 68.2%

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

   8. Simplified 68.2%

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

   9. Taylor expanded in i around 0 68.2%

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

       1. +-commutative 68.2%

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

       2. *-commutative 68.2%

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

       3. *-commutative 68.2%

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

       4. *-commutative 68.2%

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

       5. associate-*l* 68.2%

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

       6. distribute-lft-out 68.2%

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

   11. Simplified 68.2%

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

   if -1.86000000000000007e56 < n < 1.5

   1. Initial program 36.7%

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

   3. Taylor expanded in i around 0 54.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.86 \cdot 10^{+56} \lor \neg \left(n \leq 1.5\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.6 \cdot 10^{+32}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.6e+32) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.6e+32) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (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 (i <= 2.6d+32) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.6e+32) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.6e+32:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.6e+32)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 2.6e+32)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.6e+32], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2.6000000000000002e32

   1. Initial program 25.9%

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

   3. Taylor expanded in i around 0 57.3%

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

      1. *-commutative 57.3%

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

   5. Simplified 57.3%

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

   if 2.6000000000000002e32 < i

   1. Initial program 45.9%

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

   3. Taylor expanded in i around 0 33.2%

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

      1. associate-*r* 33.2%

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

      2. associate-*r/ 33.2%

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

      3. metadata-eval 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

      2. *-commutative 33.5%

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

      3. associate-*l* 33.5%

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

   8. Simplified 33.5%

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

   9. Taylor expanded in i around inf 33.5%

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.6 \cdot 10^{+32}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 2.8% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

3. Taylor expanded in i around 0 51.8%

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

   1. associate-*r* 52.0%

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

   2. associate-*r/ 52.0%

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

   3. metadata-eval 52.0%

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

5. Simplified 52.0%

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

6. Taylor expanded in n around 0 2.3%

   \[\leadsto \color{blue}{-50 \cdot i} \]
7. Step-by-step derivation

   1. *-commutative 2.3%

      \[\leadsto \color{blue}{i \cdot -50} \]

8. Simplified 2.3%

   \[\leadsto \color{blue}{i \cdot -50} \]

9. Final simplification 2.3%

   \[\leadsto i \cdot -50 \]
10. Add Preprocessing

Alternative 15: 49.4% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Taylor expanded in i around 0 44.2%

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

   1. *-commutative 44.2%

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

5. Simplified 44.2%

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

6. Final simplification 44.2%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer target: 34.2% 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 2024031 
(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))))