Compound Interest

Percentage Accurate: 28.7% → 82.7%

Time: 25.0s

Alternatives: 14

Speedup: 8.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.7% 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: 82.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -4 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(\log \left(-i\right) - \log \left(-n\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{n}{\frac{i}{\log i - \log n}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -4e-128)
     t_0
     (if (<= n -5e-310)
       (/ (* 100.0 (* n (- (log (- i)) (log (- n))))) (/ i n))
       (if (<= n 1.8e-41)
         (* n (* 100.0 (/ n (/ i (- (log i) (log n))))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -4e-128) {
		tmp = t_0;
	} else if (n <= -5e-310) {
		tmp = (100.0 * (n * (log(-i) - log(-n)))) / (i / n);
	} else if (n <= 1.8e-41) {
		tmp = n * (100.0 * (n / (i / (log(i) - log(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 <= -4e-128) {
		tmp = t_0;
	} else if (n <= -5e-310) {
		tmp = (100.0 * (n * (Math.log(-i) - Math.log(-n)))) / (i / n);
	} else if (n <= 1.8e-41) {
		tmp = n * (100.0 * (n / (i / (Math.log(i) - Math.log(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 <= -4e-128:
		tmp = t_0
	elif n <= -5e-310:
		tmp = (100.0 * (n * (math.log(-i) - math.log(-n)))) / (i / n)
	elif n <= 1.8e-41:
		tmp = n * (100.0 * (n / (i / (math.log(i) - math.log(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 <= -4e-128)
		tmp = t_0;
	elseif (n <= -5e-310)
		tmp = Float64(Float64(100.0 * Float64(n * Float64(log(Float64(-i)) - log(Float64(-n))))) / Float64(i / n));
	elseif (n <= 1.8e-41)
		tmp = Float64(n * Float64(100.0 * Float64(n / Float64(i / Float64(log(i) - log(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, -4e-128], t$95$0, If[LessEqual[n, -5e-310], N[(N[(100.0 * N[(n * N[(N[Log[(-i)], $MachinePrecision] - N[Log[(-n)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-41], N[(n * N[(100.0 * N[(n / N[(i / N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.00000000000000022e-128 or 1.8e-41 < n

   1. Initial program 25.0%

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

      1. associate-/r/ 25.3%

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

      2. sub-neg 25.3%

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

      3. metadata-eval 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in n around inf 40.3%

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

      1. associate-/l* 40.3%

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

      2. expm1-def 92.4%

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

   7. Simplified 92.4%

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

   if -4.00000000000000022e-128 < n < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. associate-*r/ 76.3%

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

      2. sub-neg 76.3%

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

      3. distribute-lft-in 76.3%

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

      4. metadata-eval 76.3%

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

      5. metadata-eval 76.3%

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

      6. metadata-eval 76.3%

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

      7. fma-def 76.3%

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

      8. metadata-eval 76.3%

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

   3. Simplified 76.3%

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

      1. fma-udef 76.3%

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

      2. *-commutative 76.3%

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

   6. Applied egg-rr 76.3%

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

   7. Taylor expanded in n around 0 0.0%

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

      1. neg-mul-1 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 47.3%

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

   9. Simplified 47.3%

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

       1. frac-2neg 47.3%

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

       2. log-div 84.3%

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

   11. Applied egg-rr 84.3%

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

   if -4.999999999999985e-310 < n < 1.8e-41

   1. Initial program 25.0%

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

      1. associate-/r/ 25.4%

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

      2. associate-*r* 25.4%

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

      3. *-commutative 25.4%

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

      4. associate-*r/ 25.4%

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

      5. sub-neg 25.4%

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

      6. distribute-lft-in 25.4%

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

      7. metadata-eval 25.4%

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

      8. metadata-eval 25.4%

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

      9. metadata-eval 25.4%

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

      10. fma-def 25.4%

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

      11. metadata-eval 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in n around 0 79.5%

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

      1. associate-/l* 79.6%

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

      2. mul-1-neg 79.6%

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

      3. unsub-neg 79.6%

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

   7. Simplified 79.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-128}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(\log \left(-i\right) - \log \left(-n\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{n}{\frac{i}{\log i - \log n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-281}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n}{\frac{i}{\log i - \log n}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -9.5e-200)
     t_0
     (if (<= n 4e-281)
       0.0
       (if (<= n 4.2e-42)
         (* 100.0 (* n (/ n (/ i (- (log i) (log n))))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -9.5e-200) {
		tmp = t_0;
	} else if (n <= 4e-281) {
		tmp = 0.0;
	} else if (n <= 4.2e-42) {
		tmp = 100.0 * (n * (n / (i / (log(i) - log(n)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -9.5e-200) {
		tmp = t_0;
	} else if (n <= 4e-281) {
		tmp = 0.0;
	} else if (n <= 4.2e-42) {
		tmp = 100.0 * (n * (n / (i / (Math.log(i) - Math.log(n)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -9.5e-200:
		tmp = t_0
	elif n <= 4e-281:
		tmp = 0.0
	elif n <= 4.2e-42:
		tmp = 100.0 * (n * (n / (i / (math.log(i) - math.log(n)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -9.5e-200)
		tmp = t_0;
	elseif (n <= 4e-281)
		tmp = 0.0;
	elseif (n <= 4.2e-42)
		tmp = Float64(100.0 * Float64(n * Float64(n / Float64(i / Float64(log(i) - log(n))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.5e-200], t$95$0, If[LessEqual[n, 4e-281], 0.0, If[LessEqual[n, 4.2e-42], N[(100.0 * N[(n * N[(n / N[(i / N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.4999999999999995e-200 or 4.20000000000000013e-42 < n

   1. Initial program 25.5%

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

      1. associate-/r/ 25.9%

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

      2. sub-neg 25.9%

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

      3. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in n around inf 40.6%

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

      1. associate-/l* 40.6%

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

      2. expm1-def 91.9%

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

   7. Simplified 91.9%

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

   if -9.4999999999999995e-200 < n < 4.0000000000000001e-281

   1. Initial program 68.9%

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

      1. associate-*r/ 68.9%

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

      2. sub-neg 68.9%

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

      3. distribute-lft-in 68.9%

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

      4. metadata-eval 68.9%

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

      5. metadata-eval 68.9%

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

      6. metadata-eval 68.9%

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

      7. fma-def 68.9%

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

      8. metadata-eval 68.9%

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

   3. Simplified 68.9%

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

      1. fma-udef 68.9%

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

      2. *-commutative 68.9%

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

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in i around 0 83.1%

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

   8. Taylor expanded in i around 0 83.1%

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

   if 4.0000000000000001e-281 < n < 4.20000000000000013e-42

   1. Initial program 23.2%

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

      1. associate-/r/ 23.6%

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

      2. sub-neg 23.6%

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

      3. metadata-eval 23.6%

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

   3. Simplified 23.6%

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

   5. Taylor expanded in n around 0 78.3%

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

      1. associate-/l* 78.4%

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

      2. mul-1-neg 78.4%

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

      3. unsub-neg 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-200}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-281}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n}{\frac{i}{\log i - \log n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -4.3 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n}{\frac{i}{\log i - \log n}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -4.3e-129)
     t_0
     (if (<= n -5e-310)
       (* 100.0 (/ n (/ i (expm1 (* n (log (/ i n)))))))
       (if (<= n 1.45e-39)
         (* 100.0 (* n (/ n (/ i (- (log i) (log n))))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -4.3e-129) {
		tmp = t_0;
	} else if (n <= -5e-310) {
		tmp = 100.0 * (n / (i / expm1((n * log((i / n))))));
	} else if (n <= 1.45e-39) {
		tmp = 100.0 * (n * (n / (i / (log(i) - log(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 <= -4.3e-129) {
		tmp = t_0;
	} else if (n <= -5e-310) {
		tmp = 100.0 * (n / (i / Math.expm1((n * Math.log((i / n))))));
	} else if (n <= 1.45e-39) {
		tmp = 100.0 * (n * (n / (i / (Math.log(i) - Math.log(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 <= -4.3e-129:
		tmp = t_0
	elif n <= -5e-310:
		tmp = 100.0 * (n / (i / math.expm1((n * math.log((i / n))))))
	elif n <= 1.45e-39:
		tmp = 100.0 * (n * (n / (i / (math.log(i) - math.log(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 <= -4.3e-129)
		tmp = t_0;
	elseif (n <= -5e-310)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(Float64(n * log(Float64(i / n)))))));
	elseif (n <= 1.45e-39)
		tmp = Float64(100.0 * Float64(n * Float64(n / Float64(i / Float64(log(i) - log(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, -4.3e-129], t$95$0, If[LessEqual[n, -5e-310], N[(100.0 * N[(n / N[(i / N[(Exp[N[(n * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.45e-39], N[(100.0 * N[(n * N[(n / N[(i / N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.29999999999999981e-129 or 1.44999999999999994e-39 < n

   1. Initial program 25.0%

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

      1. associate-/r/ 25.3%

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

      2. sub-neg 25.3%

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

      3. metadata-eval 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in n around inf 40.3%

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

      1. associate-/l* 40.3%

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

      2. expm1-def 92.4%

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

   7. Simplified 92.4%

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

   if -4.29999999999999981e-129 < n < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-/r/ 76.3%

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

      3. associate-*l* 76.3%

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

      4. sub-neg 76.3%

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

      5. metadata-eval 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in i around inf 0.0%

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

      1. expm1-def 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. distribute-rgt-in 0.0%

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

      5. mul-1-neg 0.0%

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

      6. log-rec 0.0%

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

      7. remove-double-neg 0.0%

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

      8. distribute-rgt-in 0.0%

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

      9. +-commutative 0.0%

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

      10. mul-1-neg 0.0%

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

      11. unsub-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in n around inf 0.0%

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

      1. associate-/l* 0.0%

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

      2. expm1-def 0.0%

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

      3. log-div 84.2%

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

   10. Simplified 84.2%

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

   if -4.999999999999985e-310 < n < 1.44999999999999994e-39

   1. Initial program 25.0%

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

      1. associate-/r/ 25.4%

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

      2. sub-neg 25.4%

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

      3. metadata-eval 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in n around 0 79.1%

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

      1. associate-/l* 79.1%

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

      2. mul-1-neg 79.1%

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

      3. unsub-neg 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{-129}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n}{\frac{i}{\log i - \log n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -7 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{n}{\frac{i}{\log i - \log n}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -7e-128)
     t_0
     (if (<= n -5e-310)
       (* 100.0 (/ n (/ i (expm1 (* n (log (/ i n)))))))
       (if (<= n 4.2e-42)
         (* n (* 100.0 (/ n (/ i (- (log i) (log n))))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -7e-128) {
		tmp = t_0;
	} else if (n <= -5e-310) {
		tmp = 100.0 * (n / (i / expm1((n * log((i / n))))));
	} else if (n <= 4.2e-42) {
		tmp = n * (100.0 * (n / (i / (log(i) - log(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 <= -7e-128) {
		tmp = t_0;
	} else if (n <= -5e-310) {
		tmp = 100.0 * (n / (i / Math.expm1((n * Math.log((i / n))))));
	} else if (n <= 4.2e-42) {
		tmp = n * (100.0 * (n / (i / (Math.log(i) - Math.log(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 <= -7e-128:
		tmp = t_0
	elif n <= -5e-310:
		tmp = 100.0 * (n / (i / math.expm1((n * math.log((i / n))))))
	elif n <= 4.2e-42:
		tmp = n * (100.0 * (n / (i / (math.log(i) - math.log(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 <= -7e-128)
		tmp = t_0;
	elseif (n <= -5e-310)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(Float64(n * log(Float64(i / n)))))));
	elseif (n <= 4.2e-42)
		tmp = Float64(n * Float64(100.0 * Float64(n / Float64(i / Float64(log(i) - log(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, -7e-128], t$95$0, If[LessEqual[n, -5e-310], N[(100.0 * N[(n / N[(i / N[(Exp[N[(n * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.2e-42], N[(n * N[(100.0 * N[(n / N[(i / N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.99999999999999999e-128 or 4.20000000000000013e-42 < n

   1. Initial program 25.0%

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

      1. associate-/r/ 25.3%

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

      2. sub-neg 25.3%

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

      3. metadata-eval 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in n around inf 40.3%

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

      1. associate-/l* 40.3%

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

      2. expm1-def 92.4%

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

   7. Simplified 92.4%

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

   if -6.99999999999999999e-128 < n < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-/r/ 76.3%

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

      3. associate-*l* 76.3%

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

      4. sub-neg 76.3%

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

      5. metadata-eval 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in i around inf 0.0%

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

      1. expm1-def 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. distribute-rgt-in 0.0%

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

      5. mul-1-neg 0.0%

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

      6. log-rec 0.0%

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

      7. remove-double-neg 0.0%

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

      8. distribute-rgt-in 0.0%

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

      9. +-commutative 0.0%

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

      10. mul-1-neg 0.0%

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

      11. unsub-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in n around inf 0.0%

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

      1. associate-/l* 0.0%

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

      2. expm1-def 0.0%

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

      3. log-div 84.2%

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

   10. Simplified 84.2%

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

   if -4.999999999999985e-310 < n < 4.20000000000000013e-42

   1. Initial program 25.0%

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

      1. associate-/r/ 25.4%

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

      2. associate-*r* 25.4%

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

      3. *-commutative 25.4%

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

      4. associate-*r/ 25.4%

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

      5. sub-neg 25.4%

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

      6. distribute-lft-in 25.4%

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

      7. metadata-eval 25.4%

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

      8. metadata-eval 25.4%

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

      9. metadata-eval 25.4%

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

      10. fma-def 25.4%

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

      11. metadata-eval 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in n around 0 79.5%

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

      1. associate-/l* 79.6%

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

      2. mul-1-neg 79.6%

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

      3. unsub-neg 79.6%

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

   7. Simplified 79.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-128}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \log \left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{n}{\frac{i}{\log i - \log n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-205}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.96 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{n}{i} \cdot \left(n \cdot \left(100 \cdot \log \left(\frac{i}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -2.5e-205)
     t_0
     (if (<= n 1.96e-230)
       0.0
       (if (<= n 2.6e-110)
         (* 100.0 (/ n (+ 1.0 (* i -0.5))))
         (if (<= n 1.8e-41) (* (/ n i) (* n (* 100.0 (log (/ i n))))) t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -2.5e-205) {
		tmp = t_0;
	} else if (n <= 1.96e-230) {
		tmp = 0.0;
	} else if (n <= 2.6e-110) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.8e-41) {
		tmp = (n / i) * (n * (100.0 * log((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 <= -2.5e-205) {
		tmp = t_0;
	} else if (n <= 1.96e-230) {
		tmp = 0.0;
	} else if (n <= 2.6e-110) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.8e-41) {
		tmp = (n / i) * (n * (100.0 * Math.log((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 <= -2.5e-205:
		tmp = t_0
	elif n <= 1.96e-230:
		tmp = 0.0
	elif n <= 2.6e-110:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1.8e-41:
		tmp = (n / i) * (n * (100.0 * math.log((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 <= -2.5e-205)
		tmp = t_0;
	elseif (n <= 1.96e-230)
		tmp = 0.0;
	elseif (n <= 2.6e-110)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1.8e-41)
		tmp = Float64(Float64(n / i) * Float64(n * Float64(100.0 * log(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, -2.5e-205], t$95$0, If[LessEqual[n, 1.96e-230], 0.0, If[LessEqual[n, 2.6e-110], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-41], N[(N[(n / i), $MachinePrecision] * N[(n * N[(100.0 * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.5e-205 or 1.8e-41 < n

   1. Initial program 25.5%

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

      1. associate-/r/ 25.9%

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

      2. sub-neg 25.9%

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

      3. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in n around inf 40.6%

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

      1. associate-/l* 40.6%

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

      2. expm1-def 91.9%

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

   7. Simplified 91.9%

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

   if -2.5e-205 < n < 1.96000000000000001e-230

   1. Initial program 57.7%

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

      1. associate-*r/ 57.7%

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

      2. sub-neg 57.7%

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

      3. distribute-lft-in 57.7%

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

      4. metadata-eval 57.7%

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

      5. metadata-eval 57.7%

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

      6. metadata-eval 57.7%

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

      7. fma-def 57.7%

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

      8. metadata-eval 57.7%

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

   3. Simplified 57.7%

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

      1. fma-udef 57.7%

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

      2. *-commutative 57.7%

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

   6. Applied egg-rr 57.7%

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

   7. Taylor expanded in i around 0 81.0%

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

   8. Taylor expanded in i around 0 81.0%

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

   if 1.96000000000000001e-230 < n < 2.5999999999999999e-110

   1. Initial program 3.7%

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

      1. associate-/r/ 4.2%

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

      2. sub-neg 4.2%

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

      3. metadata-eval 4.2%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in n around inf 4.2%

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

      1. associate-/l* 4.2%

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

      2. expm1-def 38.1%

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

   7. Simplified 38.1%

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

   8. Taylor expanded in i around 0 73.3%

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

   if 2.5999999999999999e-110 < n < 1.8e-41

   1. Initial program 42.7%

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

      1. associate-*r/ 42.7%

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

      2. sub-neg 42.7%

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

      3. distribute-lft-in 42.7%

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

      4. metadata-eval 42.7%

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

      5. metadata-eval 42.7%

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

      6. metadata-eval 42.7%

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

      7. fma-def 42.7%

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

      8. metadata-eval 42.7%

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

   3. Simplified 42.7%

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

      1. fma-udef 42.7%

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

      2. *-commutative 42.7%

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

   6. Applied egg-rr 42.7%

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

   7. Taylor expanded in n around 0 89.9%

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

      1. neg-mul-1 89.9%

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

      2. sub-neg 89.9%

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

      3. log-div 90.2%

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

   9. Simplified 90.2%

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

       1. clear-num 79.8%

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

       2. associate-/r/ 90.0%

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

       3. clear-num 90.0%

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

       4. associate-*r* 89.9%

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

       5. *-commutative 89.9%

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

       6. associate-*l* 90.0%

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

   11. Applied egg-rr 90.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-205}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 1.96 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{n}{i} \cdot \left(n \cdot \left(100 \cdot \log \left(\frac{i}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -1.45 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \log \left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -1.45e-206)
     t_0
     (if (<= n 2.1e-230)
       0.0
       (if (<= n 2.6e-110)
         (* 100.0 (/ n (+ 1.0 (* i -0.5))))
         (if (<= n 4.2e-42) (/ (* 100.0 (* n (log (/ i n)))) (/ i n)) t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -1.45e-206) {
		tmp = t_0;
	} else if (n <= 2.1e-230) {
		tmp = 0.0;
	} else if (n <= 2.6e-110) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.2e-42) {
		tmp = (100.0 * (n * log((i / n)))) / (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 <= -1.45e-206) {
		tmp = t_0;
	} else if (n <= 2.1e-230) {
		tmp = 0.0;
	} else if (n <= 2.6e-110) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.2e-42) {
		tmp = (100.0 * (n * Math.log((i / n)))) / (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 <= -1.45e-206:
		tmp = t_0
	elif n <= 2.1e-230:
		tmp = 0.0
	elif n <= 2.6e-110:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 4.2e-42:
		tmp = (100.0 * (n * math.log((i / n)))) / (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 <= -1.45e-206)
		tmp = t_0;
	elseif (n <= 2.1e-230)
		tmp = 0.0;
	elseif (n <= 2.6e-110)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 4.2e-42)
		tmp = Float64(Float64(100.0 * Float64(n * log(Float64(i / n)))) / 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, -1.45e-206], t$95$0, If[LessEqual[n, 2.1e-230], 0.0, If[LessEqual[n, 2.6e-110], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.2e-42], N[(N[(100.0 * N[(n * N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.4500000000000001e-206 or 4.20000000000000013e-42 < n

   1. Initial program 25.5%

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

      1. associate-/r/ 25.9%

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

      2. sub-neg 25.9%

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

      3. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in n around inf 40.6%

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

      1. associate-/l* 40.6%

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

      2. expm1-def 91.9%

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

   7. Simplified 91.9%

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

   if -1.4500000000000001e-206 < n < 2.0999999999999998e-230

   1. Initial program 57.7%

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

      1. associate-*r/ 57.7%

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

      2. sub-neg 57.7%

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

      3. distribute-lft-in 57.7%

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

      4. metadata-eval 57.7%

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

      5. metadata-eval 57.7%

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

      6. metadata-eval 57.7%

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

      7. fma-def 57.7%

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

      8. metadata-eval 57.7%

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

   3. Simplified 57.7%

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

      1. fma-udef 57.7%

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

      2. *-commutative 57.7%

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

   6. Applied egg-rr 57.7%

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

   7. Taylor expanded in i around 0 81.0%

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

   8. Taylor expanded in i around 0 81.0%

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

   if 2.0999999999999998e-230 < n < 2.5999999999999999e-110

   1. Initial program 3.7%

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

      1. associate-/r/ 4.2%

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

      2. sub-neg 4.2%

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

      3. metadata-eval 4.2%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in n around inf 4.2%

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

      1. associate-/l* 4.2%

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

      2. expm1-def 38.1%

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

   7. Simplified 38.1%

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

   8. Taylor expanded in i around 0 73.3%

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

   if 2.5999999999999999e-110 < n < 4.20000000000000013e-42

   1. Initial program 42.7%

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

      1. associate-*r/ 42.7%

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

      2. sub-neg 42.7%

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

      3. distribute-lft-in 42.7%

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

      4. metadata-eval 42.7%

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

      5. metadata-eval 42.7%

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

      6. metadata-eval 42.7%

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

      7. fma-def 42.7%

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

      8. metadata-eval 42.7%

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

   3. Simplified 42.7%

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

      1. fma-udef 42.7%

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

      2. *-commutative 42.7%

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

   6. Applied egg-rr 42.7%

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

   7. Taylor expanded in n around 0 89.9%

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

      1. neg-mul-1 89.9%

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

      2. sub-neg 89.9%

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

      3. log-div 90.2%

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

   9. Simplified 90.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-206}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \log \left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -1.15e-200)
     t_0
     (if (<= n 1.85e-230)
       0.0
       (if (<= n 2.1e-105)
         (* 100.0 (/ n (+ 1.0 (* i -0.5))))
         (if (<= n 4.2e-42) 0.0 t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -1.15e-200) {
		tmp = t_0;
	} else if (n <= 1.85e-230) {
		tmp = 0.0;
	} else if (n <= 2.1e-105) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.2e-42) {
		tmp = 0.0;
	} 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 <= -1.15e-200) {
		tmp = t_0;
	} else if (n <= 1.85e-230) {
		tmp = 0.0;
	} else if (n <= 2.1e-105) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 4.2e-42) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -1.15e-200:
		tmp = t_0
	elif n <= 1.85e-230:
		tmp = 0.0
	elif n <= 2.1e-105:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 4.2e-42:
		tmp = 0.0
	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 <= -1.15e-200)
		tmp = t_0;
	elseif (n <= 1.85e-230)
		tmp = 0.0;
	elseif (n <= 2.1e-105)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 4.2e-42)
		tmp = 0.0;
	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, -1.15e-200], t$95$0, If[LessEqual[n, 1.85e-230], 0.0, If[LessEqual[n, 2.1e-105], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.2e-42], 0.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.15000000000000004e-200 or 4.20000000000000013e-42 < n

   1. Initial program 25.5%

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

      1. associate-/r/ 25.9%

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

      2. sub-neg 25.9%

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

      3. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in n around inf 40.6%

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

      1. associate-/l* 40.6%

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

      2. expm1-def 91.9%

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

   7. Simplified 91.9%

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

   if -1.15000000000000004e-200 < n < 1.84999999999999991e-230 or 2.1e-105 < n < 4.20000000000000013e-42

   1. Initial program 55.2%

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

      1. associate-*r/ 55.2%

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

      2. sub-neg 55.2%

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

      3. distribute-lft-in 55.2%

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

      4. metadata-eval 55.2%

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

      5. metadata-eval 55.2%

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

      6. metadata-eval 55.2%

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

      7. fma-def 55.2%

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

      8. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

      1. fma-udef 55.2%

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

      2. *-commutative 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in i around 0 73.2%

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

   8. Taylor expanded in i around 0 73.2%

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

   if 1.84999999999999991e-230 < n < 2.1e-105

   1. Initial program 3.7%

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

      1. associate-/r/ 4.2%

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

      2. sub-neg 4.2%

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

      3. metadata-eval 4.2%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in n around inf 4.2%

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

      1. associate-/l* 4.2%

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

      2. expm1-def 36.2%

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

   7. Simplified 36.2%

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

   8. Taylor expanded in i around 0 69.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-200}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{if}\;n \leq -4.3 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (+ 1.0 (* i -0.5))))))
   (if (<= n -4.3e-190)
     t_0
     (if (<= n 2.3e-230)
       0.0
       (if (<= n 2.1e-105)
         t_0
         (if (<= n 4.2e-42)
           0.0
           (+ (* i -50.0) (* n (+ 100.0 (* i 50.0))))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (1.0 + (i * -0.5)));
	double tmp;
	if (n <= -4.3e-190) {
		tmp = t_0;
	} else if (n <= 2.3e-230) {
		tmp = 0.0;
	} else if (n <= 2.1e-105) {
		tmp = t_0;
	} else if (n <= 4.2e-42) {
		tmp = 0.0;
	} else {
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    if (n <= (-4.3d-190)) then
        tmp = t_0
    else if (n <= 2.3d-230) then
        tmp = 0.0d0
    else if (n <= 2.1d-105) then
        tmp = t_0
    else if (n <= 4.2d-42) then
        tmp = 0.0d0
    else
        tmp = (i * (-50.0d0)) + (n * (100.0d0 + (i * 50.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (1.0 + (i * -0.5)));
	double tmp;
	if (n <= -4.3e-190) {
		tmp = t_0;
	} else if (n <= 2.3e-230) {
		tmp = 0.0;
	} else if (n <= 2.1e-105) {
		tmp = t_0;
	} else if (n <= 4.2e-42) {
		tmp = 0.0;
	} else {
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (1.0 + (i * -0.5)))
	tmp = 0
	if n <= -4.3e-190:
		tmp = t_0
	elif n <= 2.3e-230:
		tmp = 0.0
	elif n <= 2.1e-105:
		tmp = t_0
	elif n <= 4.2e-42:
		tmp = 0.0
	else:
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))))
	tmp = 0.0
	if (n <= -4.3e-190)
		tmp = t_0;
	elseif (n <= 2.3e-230)
		tmp = 0.0;
	elseif (n <= 2.1e-105)
		tmp = t_0;
	elseif (n <= 4.2e-42)
		tmp = 0.0;
	else
		tmp = Float64(Float64(i * -50.0) + Float64(n * Float64(100.0 + Float64(i * 50.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n / (1.0 + (i * -0.5)));
	tmp = 0.0;
	if (n <= -4.3e-190)
		tmp = t_0;
	elseif (n <= 2.3e-230)
		tmp = 0.0;
	elseif (n <= 2.1e-105)
		tmp = t_0;
	elseif (n <= 4.2e-42)
		tmp = 0.0;
	else
		tmp = (i * -50.0) + (n * (100.0 + (i * 50.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.3e-190], t$95$0, If[LessEqual[n, 2.3e-230], 0.0, If[LessEqual[n, 2.1e-105], t$95$0, If[LessEqual[n, 4.2e-42], 0.0, N[(N[(i * -50.0), $MachinePrecision] + N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.3e-190 or 2.2999999999999998e-230 < n < 2.1e-105

   1. Initial program 26.7%

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

      1. associate-/r/ 27.0%

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

      2. sub-neg 27.0%

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

      3. metadata-eval 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in n around inf 38.2%

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

      1. associate-/l* 38.2%

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

      2. expm1-def 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in i around 0 58.0%

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

   if -4.3e-190 < n < 2.2999999999999998e-230 or 2.1e-105 < n < 4.20000000000000013e-42

   1. Initial program 55.2%

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

      1. associate-*r/ 55.2%

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

      2. sub-neg 55.2%

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

      3. distribute-lft-in 55.2%

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

      4. metadata-eval 55.2%

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

      5. metadata-eval 55.2%

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

      6. metadata-eval 55.2%

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

      7. fma-def 55.2%

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

      8. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

      1. fma-udef 55.2%

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

      2. *-commutative 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in i around 0 73.2%

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

   8. Taylor expanded in i around 0 73.2%

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

   if 4.20000000000000013e-42 < n

   1. Initial program 20.3%

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

      1. associate-/r/ 20.8%

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

      2. associate-*r* 20.8%

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

      3. *-commutative 20.8%

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

      4. associate-*r/ 20.8%

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

      5. sub-neg 20.8%

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

      6. distribute-lft-in 20.7%

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

      7. metadata-eval 20.7%

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

      8. metadata-eval 20.7%

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

      9. metadata-eval 20.7%

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

      10. fma-def 20.8%

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

      11. metadata-eval 20.8%

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

   3. Simplified 20.8%

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

   5. Taylor expanded in i around 0 73.0%

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

      1. associate-*r/ 73.0%

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

      2. metadata-eval 73.0%

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

   7. Simplified 73.0%

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

   8. Taylor expanded in n around 0 73.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{-190}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;i \cdot -50 + n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -5.2e-112)
     t_0
     (if (<= n 2.3e-230)
       0.0
       (if (<= n 2.1e-105)
         (/ (* 100.0 i) (/ i n))
         (if (<= n 4.2e-42) 0.0 t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -5.2e-112) {
		tmp = t_0;
	} else if (n <= 2.3e-230) {
		tmp = 0.0;
	} else if (n <= 2.1e-105) {
		tmp = (100.0 * i) / (i / n);
	} else if (n <= 4.2e-42) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-5.2d-112)) then
        tmp = t_0
    else if (n <= 2.3d-230) then
        tmp = 0.0d0
    else if (n <= 2.1d-105) then
        tmp = (100.0d0 * i) / (i / n)
    else if (n <= 4.2d-42) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -5.2e-112) {
		tmp = t_0;
	} else if (n <= 2.3e-230) {
		tmp = 0.0;
	} else if (n <= 2.1e-105) {
		tmp = (100.0 * i) / (i / n);
	} else if (n <= 4.2e-42) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -5.2e-112:
		tmp = t_0
	elif n <= 2.3e-230:
		tmp = 0.0
	elif n <= 2.1e-105:
		tmp = (100.0 * i) / (i / n)
	elif n <= 4.2e-42:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -5.2e-112)
		tmp = t_0;
	elseif (n <= 2.3e-230)
		tmp = 0.0;
	elseif (n <= 2.1e-105)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	elseif (n <= 4.2e-42)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -5.2e-112)
		tmp = t_0;
	elseif (n <= 2.3e-230)
		tmp = 0.0;
	elseif (n <= 2.1e-105)
		tmp = (100.0 * i) / (i / n);
	elseif (n <= 4.2e-42)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.2e-112], t$95$0, If[LessEqual[n, 2.3e-230], 0.0, If[LessEqual[n, 2.1e-105], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.2e-42], 0.0, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.19999999999999983e-112 or 4.20000000000000013e-42 < n

   1. Initial program 24.7%

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

      1. associate-/r/ 25.1%

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

      2. sub-neg 25.1%

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

      3. metadata-eval 25.1%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in n around inf 40.2%

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

      1. associate-/l* 40.2%

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

      2. expm1-def 92.4%

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

   7. Simplified 92.4%

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

   8. Taylor expanded in i around 0 62.7%

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

      1. +-commutative 62.7%

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

      2. associate-*r* 62.7%

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

      3. distribute-rgt-in 62.7%

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

      4. *-commutative 62.7%

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

   10. Simplified 62.7%

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

   if -5.19999999999999983e-112 < n < 2.2999999999999998e-230 or 2.1e-105 < n < 4.20000000000000013e-42

   1. Initial program 54.9%

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

      1. associate-*r/ 54.9%

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

      2. sub-neg 54.9%

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

      3. distribute-lft-in 54.9%

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

      4. metadata-eval 54.9%

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

      5. metadata-eval 54.9%

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

      6. metadata-eval 54.9%

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

      7. fma-def 54.9%

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

      8. metadata-eval 54.9%

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

   3. Simplified 54.9%

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

      1. fma-udef 54.9%

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

      2. *-commutative 54.9%

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

   6. Applied egg-rr 54.9%

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

   7. Taylor expanded in i around 0 70.5%

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

   8. Taylor expanded in i around 0 70.5%

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

   if 2.2999999999999998e-230 < n < 2.1e-105

   1. Initial program 3.7%

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

      1. associate-*r/ 3.7%

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

      2. sub-neg 3.7%

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

      3. distribute-lft-in 3.7%

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

      4. metadata-eval 3.7%

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

      5. metadata-eval 3.7%

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

      6. metadata-eval 3.7%

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

      7. fma-def 3.7%

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

      8. metadata-eval 3.7%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in i around 0 68.9%

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

      1. *-commutative 68.9%

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

   7. Simplified 68.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-112}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{if}\;n \leq -3 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.96 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (+ 1.0 (* i -0.5))))))
   (if (<= n -3e-209)
     t_0
     (if (<= n 1.96e-230)
       0.0
       (if (<= n 2.1e-105)
         t_0
         (if (<= n 4.2e-42) 0.0 (* n (+ 100.0 (* i 50.0)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (1.0 + (i * -0.5)));
	double tmp;
	if (n <= -3e-209) {
		tmp = t_0;
	} else if (n <= 1.96e-230) {
		tmp = 0.0;
	} else if (n <= 2.1e-105) {
		tmp = t_0;
	} else if (n <= 4.2e-42) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    if (n <= (-3d-209)) then
        tmp = t_0
    else if (n <= 1.96d-230) then
        tmp = 0.0d0
    else if (n <= 2.1d-105) then
        tmp = t_0
    else if (n <= 4.2d-42) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (1.0 + (i * -0.5)));
	double tmp;
	if (n <= -3e-209) {
		tmp = t_0;
	} else if (n <= 1.96e-230) {
		tmp = 0.0;
	} else if (n <= 2.1e-105) {
		tmp = t_0;
	} else if (n <= 4.2e-42) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (1.0 + (i * -0.5)))
	tmp = 0
	if n <= -3e-209:
		tmp = t_0
	elif n <= 1.96e-230:
		tmp = 0.0
	elif n <= 2.1e-105:
		tmp = t_0
	elif n <= 4.2e-42:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))))
	tmp = 0.0
	if (n <= -3e-209)
		tmp = t_0;
	elseif (n <= 1.96e-230)
		tmp = 0.0;
	elseif (n <= 2.1e-105)
		tmp = t_0;
	elseif (n <= 4.2e-42)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (n / (1.0 + (i * -0.5)));
	tmp = 0.0;
	if (n <= -3e-209)
		tmp = t_0;
	elseif (n <= 1.96e-230)
		tmp = 0.0;
	elseif (n <= 2.1e-105)
		tmp = t_0;
	elseif (n <= 4.2e-42)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3e-209], t$95$0, If[LessEqual[n, 1.96e-230], 0.0, If[LessEqual[n, 2.1e-105], t$95$0, If[LessEqual[n, 4.2e-42], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.9999999999999999e-209 or 1.96000000000000001e-230 < n < 2.1e-105

   1. Initial program 26.7%

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

      1. associate-/r/ 27.0%

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

      2. sub-neg 27.0%

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

      3. metadata-eval 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in n around inf 38.2%

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

      1. associate-/l* 38.2%

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

      2. expm1-def 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in i around 0 58.0%

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

   if -2.9999999999999999e-209 < n < 1.96000000000000001e-230 or 2.1e-105 < n < 4.20000000000000013e-42

   1. Initial program 55.2%

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

      1. associate-*r/ 55.2%

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

      2. sub-neg 55.2%

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

      3. distribute-lft-in 55.2%

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

      4. metadata-eval 55.2%

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

      5. metadata-eval 55.2%

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

      6. metadata-eval 55.2%

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

      7. fma-def 55.2%

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

      8. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

      1. fma-udef 55.2%

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

      2. *-commutative 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in i around 0 73.2%

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

   8. Taylor expanded in i around 0 73.2%

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

   if 4.20000000000000013e-42 < n

   1. Initial program 20.3%

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

      1. associate-/r/ 20.8%

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

      2. sub-neg 20.8%

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

      3. metadata-eval 20.8%

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

   3. Simplified 20.8%

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

   5. Taylor expanded in n around inf 37.9%

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

      1. associate-/l* 37.9%

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

      2. expm1-def 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in i around 0 72.6%

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

      1. +-commutative 72.6%

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

      2. associate-*r* 72.6%

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

      3. distribute-rgt-in 72.6%

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

      4. *-commutative 72.6%

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

   10. Simplified 72.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-209}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.96 \cdot 10^{-230}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{+74}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{+110}:\\ \;\;\;\;50 \cdot \left(n \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -3.2e+74)
   0.0
   (if (<= i 2.0) (* n 100.0) (if (<= i 4.5e+110) (* 50.0 (* n i)) 0.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -3.2e+74) {
		tmp = 0.0;
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else if (i <= 4.5e+110) {
		tmp = 50.0 * (n * i);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-3.2d+74)) then
        tmp = 0.0d0
    else if (i <= 2.0d0) then
        tmp = n * 100.0d0
    else if (i <= 4.5d+110) then
        tmp = 50.0d0 * (n * i)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -3.2e+74) {
		tmp = 0.0;
	} else if (i <= 2.0) {
		tmp = n * 100.0;
	} else if (i <= 4.5e+110) {
		tmp = 50.0 * (n * i);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -3.2e+74:
		tmp = 0.0
	elif i <= 2.0:
		tmp = n * 100.0
	elif i <= 4.5e+110:
		tmp = 50.0 * (n * i)
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -3.2e+74)
		tmp = 0.0;
	elseif (i <= 2.0)
		tmp = Float64(n * 100.0);
	elseif (i <= 4.5e+110)
		tmp = Float64(50.0 * Float64(n * i));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -3.2e+74)
		tmp = 0.0;
	elseif (i <= 2.0)
		tmp = n * 100.0;
	elseif (i <= 4.5e+110)
		tmp = 50.0 * (n * i);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -3.2e+74], 0.0, If[LessEqual[i, 2.0], N[(n * 100.0), $MachinePrecision], If[LessEqual[i, 4.5e+110], N[(50.0 * N[(n * i), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if i < -3.19999999999999995e74 or 4.5000000000000003e110 < i

   1. Initial program 64.7%

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

      1. associate-*r/ 64.7%

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

      2. sub-neg 64.7%

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

      3. distribute-lft-in 64.7%

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

      4. metadata-eval 64.7%

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

      5. metadata-eval 64.7%

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

      6. metadata-eval 64.7%

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

      7. fma-def 64.7%

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

      8. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

      1. fma-udef 64.7%

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

      2. *-commutative 64.7%

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

   6. Applied egg-rr 64.7%

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

   7. Taylor expanded in i around 0 35.5%

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

   8. Taylor expanded in i around 0 35.5%

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

   if -3.19999999999999995e74 < i < 2

   1. Initial program 10.8%

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

      1. associate-/r/ 11.3%

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

      2. sub-neg 11.3%

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

      3. metadata-eval 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in i around 0 76.8%

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

      1. *-commutative 76.8%

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

   7. Simplified 76.8%

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

   if 2 < i < 4.5000000000000003e110

   1. Initial program 32.1%

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

      1. associate-/r/ 32.1%

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

      2. associate-*r* 32.1%

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

      3. *-commutative 32.1%

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

      4. associate-*r/ 32.1%

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

      5. sub-neg 32.1%

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

      6. distribute-lft-in 32.1%

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

      7. metadata-eval 32.1%

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

      8. metadata-eval 32.1%

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

      9. metadata-eval 32.1%

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

      10. fma-def 32.1%

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

      11. metadata-eval 32.1%

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

   3. Simplified 32.1%

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

   5. Taylor expanded in i around 0 27.6%

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

      1. associate-*r/ 27.6%

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

      2. metadata-eval 27.6%

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

   7. Simplified 27.6%

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

   8. Taylor expanded in n around 0 27.6%

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

   9. Taylor expanded in i around inf 27.6%

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

   10. Taylor expanded in n around inf 27.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{+74}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{+110}:\\ \;\;\;\;50 \cdot \left(n \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-112} \lor \neg \left(n \leq 4.2 \cdot 10^{-42}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.2e-112) (not (<= n 4.2e-42)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.2e-112) || !(n <= 4.2e-42)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-5.2d-112)) .or. (.not. (n <= 4.2d-42))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5.2e-112) || !(n <= 4.2e-42)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5.2e-112) or not (n <= 4.2e-42):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.2e-112) || !(n <= 4.2e-42))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -5.2e-112) || ~((n <= 4.2e-42)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.2e-112], N[Not[LessEqual[n, 4.2e-42]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -5.19999999999999983e-112 or 4.20000000000000013e-42 < n

   1. Initial program 24.7%

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

      1. associate-/r/ 25.1%

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

      2. sub-neg 25.1%

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

      3. metadata-eval 25.1%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in n around inf 40.2%

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

      1. associate-/l* 40.2%

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

      2. expm1-def 92.4%

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

   7. Simplified 92.4%

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

   8. Taylor expanded in i around 0 62.7%

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

      1. +-commutative 62.7%

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

      2. associate-*r* 62.7%

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

      3. distribute-rgt-in 62.7%

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

      4. *-commutative 62.7%

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

   10. Simplified 62.7%

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

   if -5.19999999999999983e-112 < n < 4.20000000000000013e-42

   1. Initial program 42.1%

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

      1. associate-*r/ 42.1%

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

      2. sub-neg 42.1%

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

      3. distribute-lft-in 42.1%

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

      4. metadata-eval 42.1%

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

      5. metadata-eval 42.1%

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

      6. metadata-eval 42.1%

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

      7. fma-def 42.1%

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

      8. metadata-eval 42.1%

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

   3. Simplified 42.1%

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

      1. fma-udef 42.1%

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

      2. *-commutative 42.1%

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

   6. Applied egg-rr 42.1%

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

   7. Taylor expanded in i around 0 62.1%

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

   8. Taylor expanded in i around 0 62.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{-112} \lor \neg \left(n \leq 4.2 \cdot 10^{-42}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-112} \lor \neg \left(n \leq 4.2 \cdot 10^{-42}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -6e-112) (not (<= n 4.2e-42))) (* n 100.0) 0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -6e-112) || !(n <= 4.2e-42)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-6d-112)) .or. (.not. (n <= 4.2d-42))) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -6e-112) || !(n <= 4.2e-42)) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -6e-112) or not (n <= 4.2e-42):
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -6e-112) || !(n <= 4.2e-42))
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -6e-112) || ~((n <= 4.2e-42)))
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -6e-112], N[Not[LessEqual[n, 4.2e-42]], $MachinePrecision]], N[(n * 100.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -6.0000000000000002e-112 or 4.20000000000000013e-42 < n

   1. Initial program 24.7%

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

      1. associate-/r/ 25.1%

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

      2. sub-neg 25.1%

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

      3. metadata-eval 25.1%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in i around 0 56.9%

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

      1. *-commutative 56.9%

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

   7. Simplified 56.9%

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

   if -6.0000000000000002e-112 < n < 4.20000000000000013e-42

   1. Initial program 42.1%

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

      1. associate-*r/ 42.1%

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

      2. sub-neg 42.1%

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

      3. distribute-lft-in 42.1%

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

      4. metadata-eval 42.1%

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

      5. metadata-eval 42.1%

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

      6. metadata-eval 42.1%

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

      7. fma-def 42.1%

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

      8. metadata-eval 42.1%

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

   3. Simplified 42.1%

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

      1. fma-udef 42.1%

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

      2. *-commutative 42.1%

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

   6. Applied egg-rr 42.1%

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

   7. Taylor expanded in i around 0 62.1%

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

   8. Taylor expanded in i around 0 62.1%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-112} \lor \neg \left(n \leq 4.2 \cdot 10^{-42}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 18.7% accurate, 114.0× speedup

Math

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

FPCore

(FPCore (i n) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

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

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 28.8%

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

   1. associate-*r/ 28.8%

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

   2. sub-neg 28.8%

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

   3. distribute-lft-in 28.7%

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

   4. metadata-eval 28.7%

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

   5. metadata-eval 28.7%

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

   6. metadata-eval 28.7%

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

   7. fma-def 28.8%

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

   8. metadata-eval 28.8%

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

3. Simplified 28.8%

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

   1. fma-udef 28.7%

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

   2. *-commutative 28.7%

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

6. Applied egg-rr 28.7%

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

7. Taylor expanded in i around 0 16.6%

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

8. Taylor expanded in i around 0 16.9%

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

9. Final simplification 16.9%

   \[\leadsto 0 \]
10. Add Preprocessing

Developer target: 35.0% 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 2024023 
(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))))