Compound Interest

Percentage Accurate: 28.8% → 77.5%

Time: 28.7s

Alternatives: 18

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: 28.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot e^{i} - 100\\ t_1 := \log i - \log n\\ \mathbf{if}\;i \leq -0.000102:\\ \;\;\;\;\frac{t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -3.05 \cdot 10^{-204}:\\ \;\;\;\;\frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 3.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{t\_0 \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.3 \cdot 10^{+118}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(n \cdot \frac{t\_1}{i} + {n}^{2} \cdot \left(\frac{1}{{i}^{2}} + {t\_1}^{2} \cdot \frac{0.5}{i}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (* 100.0 (exp i)) 100.0)) (t_1 (- (log i) (log n))))
   (if (<= i -0.000102)
     (/ t_0 (/ i n))
     (if (<= i -3.05e-204)
       (/ 1.0 (+ (* (* i (- (/ 0.5 (pow n 2.0)) (/ 0.5 n))) 0.01) (/ 0.01 n)))
       (if (<= i 2.5e-25)
         (* 100.0 n)
         (if (<= i 3.4e+61)
           (/ (* t_0 n) i)
           (if (<= i 1.3e+118)
             (*
              100.0
              (*
               n
               (+
                (* n (/ t_1 i))
                (*
                 (pow n 2.0)
                 (+ (/ 1.0 (pow i 2.0)) (* (pow t_1 2.0) (/ 0.5 i)))))))
             (* 100.0 (* n (/ (+ (pow (+ (/ i n) 1.0) n) -1.0) i))))))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * exp(i)) - 100.0;
	double t_1 = log(i) - log(n);
	double tmp;
	if (i <= -0.000102) {
		tmp = t_0 / (i / n);
	} else if (i <= -3.05e-204) {
		tmp = 1.0 / (((i * ((0.5 / pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 3.4e+61) {
		tmp = (t_0 * n) / i;
	} else if (i <= 1.3e+118) {
		tmp = 100.0 * (n * ((n * (t_1 / i)) + (pow(n, 2.0) * ((1.0 / pow(i, 2.0)) + (pow(t_1, 2.0) * (0.5 / i))))));
	} else {
		tmp = 100.0 * (n * ((pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (100.0d0 * exp(i)) - 100.0d0
    t_1 = log(i) - log(n)
    if (i <= (-0.000102d0)) then
        tmp = t_0 / (i / n)
    else if (i <= (-3.05d-204)) then
        tmp = 1.0d0 / (((i * ((0.5d0 / (n ** 2.0d0)) - (0.5d0 / n))) * 0.01d0) + (0.01d0 / n))
    else if (i <= 2.5d-25) then
        tmp = 100.0d0 * n
    else if (i <= 3.4d+61) then
        tmp = (t_0 * n) / i
    else if (i <= 1.3d+118) then
        tmp = 100.0d0 * (n * ((n * (t_1 / i)) + ((n ** 2.0d0) * ((1.0d0 / (i ** 2.0d0)) + ((t_1 ** 2.0d0) * (0.5d0 / i))))))
    else
        tmp = 100.0d0 * (n * (((((i / n) + 1.0d0) ** n) + (-1.0d0)) / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (100.0 * Math.exp(i)) - 100.0;
	double t_1 = Math.log(i) - Math.log(n);
	double tmp;
	if (i <= -0.000102) {
		tmp = t_0 / (i / n);
	} else if (i <= -3.05e-204) {
		tmp = 1.0 / (((i * ((0.5 / Math.pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 3.4e+61) {
		tmp = (t_0 * n) / i;
	} else if (i <= 1.3e+118) {
		tmp = 100.0 * (n * ((n * (t_1 / i)) + (Math.pow(n, 2.0) * ((1.0 / Math.pow(i, 2.0)) + (Math.pow(t_1, 2.0) * (0.5 / i))))));
	} else {
		tmp = 100.0 * (n * ((Math.pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * math.exp(i)) - 100.0
	t_1 = math.log(i) - math.log(n)
	tmp = 0
	if i <= -0.000102:
		tmp = t_0 / (i / n)
	elif i <= -3.05e-204:
		tmp = 1.0 / (((i * ((0.5 / math.pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n))
	elif i <= 2.5e-25:
		tmp = 100.0 * n
	elif i <= 3.4e+61:
		tmp = (t_0 * n) / i
	elif i <= 1.3e+118:
		tmp = 100.0 * (n * ((n * (t_1 / i)) + (math.pow(n, 2.0) * ((1.0 / math.pow(i, 2.0)) + (math.pow(t_1, 2.0) * (0.5 / i))))))
	else:
		tmp = 100.0 * (n * ((math.pow(((i / n) + 1.0), n) + -1.0) / i))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * exp(i)) - 100.0)
	t_1 = Float64(log(i) - log(n))
	tmp = 0.0
	if (i <= -0.000102)
		tmp = Float64(t_0 / Float64(i / n));
	elseif (i <= -3.05e-204)
		tmp = Float64(1.0 / Float64(Float64(Float64(i * Float64(Float64(0.5 / (n ^ 2.0)) - Float64(0.5 / n))) * 0.01) + Float64(0.01 / n)));
	elseif (i <= 2.5e-25)
		tmp = Float64(100.0 * n);
	elseif (i <= 3.4e+61)
		tmp = Float64(Float64(t_0 * n) / i);
	elseif (i <= 1.3e+118)
		tmp = Float64(100.0 * Float64(n * Float64(Float64(n * Float64(t_1 / i)) + Float64((n ^ 2.0) * Float64(Float64(1.0 / (i ^ 2.0)) + Float64((t_1 ^ 2.0) * Float64(0.5 / i)))))));
	else
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) + -1.0) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (100.0 * exp(i)) - 100.0;
	t_1 = log(i) - log(n);
	tmp = 0.0;
	if (i <= -0.000102)
		tmp = t_0 / (i / n);
	elseif (i <= -3.05e-204)
		tmp = 1.0 / (((i * ((0.5 / (n ^ 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	elseif (i <= 2.5e-25)
		tmp = 100.0 * n;
	elseif (i <= 3.4e+61)
		tmp = (t_0 * n) / i;
	elseif (i <= 1.3e+118)
		tmp = 100.0 * (n * ((n * (t_1 / i)) + ((n ^ 2.0) * ((1.0 / (i ^ 2.0)) + ((t_1 ^ 2.0) * (0.5 / i))))));
	else
		tmp = 100.0 * (n * (((((i / n) + 1.0) ^ n) + -1.0) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -0.000102], N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -3.05e-204], N[(1.0 / N[(N[(N[(i * N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.01), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e-25], N[(100.0 * n), $MachinePrecision], If[LessEqual[i, 3.4e+61], N[(N[(t$95$0 * n), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, 1.3e+118], N[(100.0 * N[(n * N[(N[(n * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision] + N[(N[Power[n, 2.0], $MachinePrecision] * N[(N[(1.0 / N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(0.5 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if i < -1.01999999999999999e-4

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in n around inf 84.7%

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

   if -1.01999999999999999e-4 < i < -3.04999999999999987e-204

   1. Initial program 12.3%

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

      1. associate-*r/ 12.3%

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

      2. sub-neg 12.3%

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

      3. distribute-rgt-in 12.3%

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

      4. metadata-eval 12.3%

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

      5. metadata-eval 12.3%

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

   3. Simplified 12.3%

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

   5. Taylor expanded in i around 0 64.4%

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

      1. clear-num 64.5%

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

      2. inv-pow 64.5%

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

      3. *-un-lft-identity 64.5%

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

      4. distribute-lft-out 64.5%

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

      5. times-frac 64.4%

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

      6. metadata-eval 64.4%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 64.4%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 64.4%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 64.4%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 64.4%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 91.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 91.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 91.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   if -3.04999999999999987e-204 < i < 2.49999999999999981e-25

   1. Initial program 5.4%

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

      1. associate-/r/ 6.1%

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

      2. *-commutative 6.1%

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

      3. sub-neg 6.1%

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

      4. metadata-eval 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in i around 0 89.5%

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

      1. *-commutative 89.5%

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

   7. Simplified 89.5%

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

   if 2.49999999999999981e-25 < i < 3.40000000000000026e61

   1. Initial program 28.2%

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

      1. associate-*r/ 28.2%

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

      2. sub-neg 28.2%

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

      3. distribute-rgt-in 28.2%

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

      4. metadata-eval 28.2%

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

      5. metadata-eval 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in n around inf 76.3%

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

   if 3.40000000000000026e61 < i < 1.30000000000000008e118

   1. Initial program 22.5%

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

      1. associate-/r/ 23.4%

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

      2. *-commutative 23.4%

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

      3. sub-neg 23.4%

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

      4. metadata-eval 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in n around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. mul-1-neg 100.0%

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

      10. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   if 1.30000000000000008e118 < i

   1. Initial program 69.1%

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

      1. associate-/r/ 69.3%

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

      2. *-commutative 69.3%

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

      3. sub-neg 69.3%

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

      4. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.000102:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -3.05 \cdot 10^{-204}:\\ \;\;\;\;\frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 3.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.3 \cdot 10^{+118}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(n \cdot \frac{\log i - \log n}{i} + {n}^{2} \cdot \left(\frac{1}{{i}^{2}} + {\left(\log i - \log n\right)}^{2} \cdot \frac{0.5}{i}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot e^{i} - 100\\ t_1 := \log i - \log n\\ \mathbf{if}\;i \leq -0.0018:\\ \;\;\;\;\frac{t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -1.85 \cdot 10^{-204}:\\ \;\;\;\;\frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 9.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{t\_0 \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.36 \cdot 10^{+118}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n \cdot t\_1 + {n}^{2} \cdot \left(\frac{1}{i} + 0.5 \cdot {t\_1}^{2}\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (* 100.0 (exp i)) 100.0)) (t_1 (- (log i) (log n))))
   (if (<= i -0.0018)
     (/ t_0 (/ i n))
     (if (<= i -1.85e-204)
       (/ 1.0 (+ (* (* i (- (/ 0.5 (pow n 2.0)) (/ 0.5 n))) 0.01) (/ 0.01 n)))
       (if (<= i 2.5e-25)
         (* 100.0 n)
         (if (<= i 9.6e+63)
           (/ (* t_0 n) i)
           (if (<= i 1.36e+118)
             (*
              100.0
              (*
               n
               (/
                (+
                 (* n t_1)
                 (* (pow n 2.0) (+ (/ 1.0 i) (* 0.5 (pow t_1 2.0)))))
                i)))
             (* 100.0 (* n (/ (+ (pow (+ (/ i n) 1.0) n) -1.0) i))))))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * exp(i)) - 100.0;
	double t_1 = log(i) - log(n);
	double tmp;
	if (i <= -0.0018) {
		tmp = t_0 / (i / n);
	} else if (i <= -1.85e-204) {
		tmp = 1.0 / (((i * ((0.5 / pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 9.6e+63) {
		tmp = (t_0 * n) / i;
	} else if (i <= 1.36e+118) {
		tmp = 100.0 * (n * (((n * t_1) + (pow(n, 2.0) * ((1.0 / i) + (0.5 * pow(t_1, 2.0))))) / i));
	} else {
		tmp = 100.0 * (n * ((pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (100.0d0 * exp(i)) - 100.0d0
    t_1 = log(i) - log(n)
    if (i <= (-0.0018d0)) then
        tmp = t_0 / (i / n)
    else if (i <= (-1.85d-204)) then
        tmp = 1.0d0 / (((i * ((0.5d0 / (n ** 2.0d0)) - (0.5d0 / n))) * 0.01d0) + (0.01d0 / n))
    else if (i <= 2.5d-25) then
        tmp = 100.0d0 * n
    else if (i <= 9.6d+63) then
        tmp = (t_0 * n) / i
    else if (i <= 1.36d+118) then
        tmp = 100.0d0 * (n * (((n * t_1) + ((n ** 2.0d0) * ((1.0d0 / i) + (0.5d0 * (t_1 ** 2.0d0))))) / i))
    else
        tmp = 100.0d0 * (n * (((((i / n) + 1.0d0) ** n) + (-1.0d0)) / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (100.0 * Math.exp(i)) - 100.0;
	double t_1 = Math.log(i) - Math.log(n);
	double tmp;
	if (i <= -0.0018) {
		tmp = t_0 / (i / n);
	} else if (i <= -1.85e-204) {
		tmp = 1.0 / (((i * ((0.5 / Math.pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 9.6e+63) {
		tmp = (t_0 * n) / i;
	} else if (i <= 1.36e+118) {
		tmp = 100.0 * (n * (((n * t_1) + (Math.pow(n, 2.0) * ((1.0 / i) + (0.5 * Math.pow(t_1, 2.0))))) / i));
	} else {
		tmp = 100.0 * (n * ((Math.pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * math.exp(i)) - 100.0
	t_1 = math.log(i) - math.log(n)
	tmp = 0
	if i <= -0.0018:
		tmp = t_0 / (i / n)
	elif i <= -1.85e-204:
		tmp = 1.0 / (((i * ((0.5 / math.pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n))
	elif i <= 2.5e-25:
		tmp = 100.0 * n
	elif i <= 9.6e+63:
		tmp = (t_0 * n) / i
	elif i <= 1.36e+118:
		tmp = 100.0 * (n * (((n * t_1) + (math.pow(n, 2.0) * ((1.0 / i) + (0.5 * math.pow(t_1, 2.0))))) / i))
	else:
		tmp = 100.0 * (n * ((math.pow(((i / n) + 1.0), n) + -1.0) / i))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * exp(i)) - 100.0)
	t_1 = Float64(log(i) - log(n))
	tmp = 0.0
	if (i <= -0.0018)
		tmp = Float64(t_0 / Float64(i / n));
	elseif (i <= -1.85e-204)
		tmp = Float64(1.0 / Float64(Float64(Float64(i * Float64(Float64(0.5 / (n ^ 2.0)) - Float64(0.5 / n))) * 0.01) + Float64(0.01 / n)));
	elseif (i <= 2.5e-25)
		tmp = Float64(100.0 * n);
	elseif (i <= 9.6e+63)
		tmp = Float64(Float64(t_0 * n) / i);
	elseif (i <= 1.36e+118)
		tmp = Float64(100.0 * Float64(n * Float64(Float64(Float64(n * t_1) + Float64((n ^ 2.0) * Float64(Float64(1.0 / i) + Float64(0.5 * (t_1 ^ 2.0))))) / i)));
	else
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) + -1.0) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (100.0 * exp(i)) - 100.0;
	t_1 = log(i) - log(n);
	tmp = 0.0;
	if (i <= -0.0018)
		tmp = t_0 / (i / n);
	elseif (i <= -1.85e-204)
		tmp = 1.0 / (((i * ((0.5 / (n ^ 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	elseif (i <= 2.5e-25)
		tmp = 100.0 * n;
	elseif (i <= 9.6e+63)
		tmp = (t_0 * n) / i;
	elseif (i <= 1.36e+118)
		tmp = 100.0 * (n * (((n * t_1) + ((n ^ 2.0) * ((1.0 / i) + (0.5 * (t_1 ^ 2.0))))) / i));
	else
		tmp = 100.0 * (n * (((((i / n) + 1.0) ^ n) + -1.0) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -0.0018], N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -1.85e-204], N[(1.0 / N[(N[(N[(i * N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.01), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e-25], N[(100.0 * n), $MachinePrecision], If[LessEqual[i, 9.6e+63], N[(N[(t$95$0 * n), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, 1.36e+118], N[(100.0 * N[(n * N[(N[(N[(n * t$95$1), $MachinePrecision] + N[(N[Power[n, 2.0], $MachinePrecision] * N[(N[(1.0 / i), $MachinePrecision] + N[(0.5 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if i < -0.0018

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in n around inf 84.7%

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

   if -0.0018 < i < -1.8499999999999999e-204

   1. Initial program 12.3%

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

      1. associate-*r/ 12.3%

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

      2. sub-neg 12.3%

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

      3. distribute-rgt-in 12.3%

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

      4. metadata-eval 12.3%

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

      5. metadata-eval 12.3%

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

   3. Simplified 12.3%

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

   5. Taylor expanded in i around 0 64.4%

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

      1. clear-num 64.5%

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

      2. inv-pow 64.5%

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

      3. *-un-lft-identity 64.5%

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

      4. distribute-lft-out 64.5%

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

      5. times-frac 64.4%

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

      6. metadata-eval 64.4%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 64.4%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 64.4%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 64.4%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 64.4%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 91.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 91.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 91.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   if -1.8499999999999999e-204 < i < 2.49999999999999981e-25

   1. Initial program 5.4%

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

      1. associate-/r/ 6.1%

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

      2. *-commutative 6.1%

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

      3. sub-neg 6.1%

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

      4. metadata-eval 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in i around 0 89.5%

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

      1. *-commutative 89.5%

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

   7. Simplified 89.5%

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

   if 2.49999999999999981e-25 < i < 9.6e63

   1. Initial program 28.2%

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

      1. associate-*r/ 28.2%

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

      2. sub-neg 28.2%

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

      3. distribute-rgt-in 28.2%

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

      4. metadata-eval 28.2%

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

      5. metadata-eval 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in n around inf 76.3%

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

   if 9.6e63 < i < 1.36e118

   1. Initial program 22.5%

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

      1. associate-/r/ 23.4%

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

      2. *-commutative 23.4%

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

      3. sub-neg 23.4%

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

      4. metadata-eval 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in n around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. mul-1-neg 99.6%

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

      5. unsub-neg 99.6%

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

   7. Simplified 99.6%

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

   if 1.36e118 < i

   1. Initial program 69.1%

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

      1. associate-/r/ 69.3%

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

      2. *-commutative 69.3%

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

      3. sub-neg 69.3%

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

      4. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0018:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -1.85 \cdot 10^{-204}:\\ \;\;\;\;\frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 9.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.36 \cdot 10^{+118}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{n \cdot \left(\log i - \log n\right) + {n}^{2} \cdot \left(\frac{1}{i} + 0.5 \cdot {\left(\log i - \log n\right)}^{2}\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot e^{i} - 100\\ \mathbf{if}\;i \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -9.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{t\_0 \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.82 \cdot 10^{+118}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(n \cdot \frac{\log i - \log n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (* 100.0 (exp i)) 100.0)))
   (if (<= i -3.1e-5)
     (/ t_0 (/ i n))
     (if (<= i -9.6e-205)
       (/ 1.0 (+ (* (* i (- (/ 0.5 (pow n 2.0)) (/ 0.5 n))) 0.01) (/ 0.01 n)))
       (if (<= i 2.5e-25)
         (* 100.0 n)
         (if (<= i 2.05e+64)
           (/ (* t_0 n) i)
           (if (<= i 1.82e+118)
             (* 100.0 (* n (* n (/ (- (log i) (log n)) i))))
             (* 100.0 (* n (/ (+ (pow (+ (/ i n) 1.0) n) -1.0) i))))))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * exp(i)) - 100.0;
	double tmp;
	if (i <= -3.1e-5) {
		tmp = t_0 / (i / n);
	} else if (i <= -9.6e-205) {
		tmp = 1.0 / (((i * ((0.5 / pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 2.05e+64) {
		tmp = (t_0 * n) / i;
	} else if (i <= 1.82e+118) {
		tmp = 100.0 * (n * (n * ((log(i) - log(n)) / i)));
	} else {
		tmp = 100.0 * (n * ((pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	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 * exp(i)) - 100.0d0
    if (i <= (-3.1d-5)) then
        tmp = t_0 / (i / n)
    else if (i <= (-9.6d-205)) then
        tmp = 1.0d0 / (((i * ((0.5d0 / (n ** 2.0d0)) - (0.5d0 / n))) * 0.01d0) + (0.01d0 / n))
    else if (i <= 2.5d-25) then
        tmp = 100.0d0 * n
    else if (i <= 2.05d+64) then
        tmp = (t_0 * n) / i
    else if (i <= 1.82d+118) then
        tmp = 100.0d0 * (n * (n * ((log(i) - log(n)) / i)))
    else
        tmp = 100.0d0 * (n * (((((i / n) + 1.0d0) ** n) + (-1.0d0)) / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (100.0 * Math.exp(i)) - 100.0;
	double tmp;
	if (i <= -3.1e-5) {
		tmp = t_0 / (i / n);
	} else if (i <= -9.6e-205) {
		tmp = 1.0 / (((i * ((0.5 / Math.pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 2.05e+64) {
		tmp = (t_0 * n) / i;
	} else if (i <= 1.82e+118) {
		tmp = 100.0 * (n * (n * ((Math.log(i) - Math.log(n)) / i)));
	} else {
		tmp = 100.0 * (n * ((Math.pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * math.exp(i)) - 100.0
	tmp = 0
	if i <= -3.1e-5:
		tmp = t_0 / (i / n)
	elif i <= -9.6e-205:
		tmp = 1.0 / (((i * ((0.5 / math.pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n))
	elif i <= 2.5e-25:
		tmp = 100.0 * n
	elif i <= 2.05e+64:
		tmp = (t_0 * n) / i
	elif i <= 1.82e+118:
		tmp = 100.0 * (n * (n * ((math.log(i) - math.log(n)) / i)))
	else:
		tmp = 100.0 * (n * ((math.pow(((i / n) + 1.0), n) + -1.0) / i))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * exp(i)) - 100.0)
	tmp = 0.0
	if (i <= -3.1e-5)
		tmp = Float64(t_0 / Float64(i / n));
	elseif (i <= -9.6e-205)
		tmp = Float64(1.0 / Float64(Float64(Float64(i * Float64(Float64(0.5 / (n ^ 2.0)) - Float64(0.5 / n))) * 0.01) + Float64(0.01 / n)));
	elseif (i <= 2.5e-25)
		tmp = Float64(100.0 * n);
	elseif (i <= 2.05e+64)
		tmp = Float64(Float64(t_0 * n) / i);
	elseif (i <= 1.82e+118)
		tmp = Float64(100.0 * Float64(n * Float64(n * Float64(Float64(log(i) - log(n)) / i))));
	else
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) + -1.0) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (100.0 * exp(i)) - 100.0;
	tmp = 0.0;
	if (i <= -3.1e-5)
		tmp = t_0 / (i / n);
	elseif (i <= -9.6e-205)
		tmp = 1.0 / (((i * ((0.5 / (n ^ 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	elseif (i <= 2.5e-25)
		tmp = 100.0 * n;
	elseif (i <= 2.05e+64)
		tmp = (t_0 * n) / i;
	elseif (i <= 1.82e+118)
		tmp = 100.0 * (n * (n * ((log(i) - log(n)) / i)));
	else
		tmp = 100.0 * (n * (((((i / n) + 1.0) ^ n) + -1.0) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]}, If[LessEqual[i, -3.1e-5], N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -9.6e-205], N[(1.0 / N[(N[(N[(i * N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.01), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e-25], N[(100.0 * n), $MachinePrecision], If[LessEqual[i, 2.05e+64], N[(N[(t$95$0 * n), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, 1.82e+118], N[(100.0 * N[(n * N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if i < -3.10000000000000014e-5

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in n around inf 84.7%

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

   if -3.10000000000000014e-5 < i < -9.6000000000000007e-205

   1. Initial program 12.3%

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

      1. associate-*r/ 12.3%

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

      2. sub-neg 12.3%

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

      3. distribute-rgt-in 12.3%

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

      4. metadata-eval 12.3%

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

      5. metadata-eval 12.3%

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

   3. Simplified 12.3%

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

   5. Taylor expanded in i around 0 64.4%

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

      1. clear-num 64.5%

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

      2. inv-pow 64.5%

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

      3. *-un-lft-identity 64.5%

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

      4. distribute-lft-out 64.5%

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

      5. times-frac 64.4%

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

      6. metadata-eval 64.4%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 64.4%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 64.4%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 64.4%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 64.4%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 91.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 91.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 91.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   if -9.6000000000000007e-205 < i < 2.49999999999999981e-25

   1. Initial program 5.4%

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

      1. associate-/r/ 6.1%

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

      2. *-commutative 6.1%

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

      3. sub-neg 6.1%

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

      4. metadata-eval 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in i around 0 89.5%

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

      1. *-commutative 89.5%

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

   7. Simplified 89.5%

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

   if 2.49999999999999981e-25 < i < 2.04999999999999989e64

   1. Initial program 28.2%

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

      1. associate-*r/ 28.2%

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

      2. sub-neg 28.2%

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

      3. distribute-rgt-in 28.2%

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

      4. metadata-eval 28.2%

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

      5. metadata-eval 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in n around inf 76.3%

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

   if 2.04999999999999989e64 < i < 1.8200000000000001e118

   1. Initial program 22.5%

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

      1. associate-/r/ 23.4%

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

      2. *-commutative 23.4%

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

      3. sub-neg 23.4%

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

      4. metadata-eval 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in n around 0 80.5%

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

      1. associate-/l* 80.5%

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

      2. mul-1-neg 80.5%

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

      3. unsub-neg 80.5%

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

   7. Simplified 80.5%

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

   if 1.8200000000000001e118 < i

   1. Initial program 69.1%

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

      1. associate-/r/ 69.3%

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

      2. *-commutative 69.3%

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

      3. sub-neg 69.3%

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

      4. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -9.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.82 \cdot 10^{+118}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(n \cdot \frac{\log i - \log n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot e^{i} - 100\\ \mathbf{if}\;i \leq -0.000115:\\ \;\;\;\;\frac{t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -2.7 \cdot 10^{-204}:\\ \;\;\;\;\frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{t\_0 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (* 100.0 (exp i)) 100.0)))
   (if (<= i -0.000115)
     (/ t_0 (/ i n))
     (if (<= i -2.7e-204)
       (/ 1.0 (+ (* (* i (- (/ 0.5 (pow n 2.0)) (/ 0.5 n))) 0.01) (/ 0.01 n)))
       (if (<= i 2.5e-25)
         (* 100.0 n)
         (if (<= i 4.4e+43)
           (/ (* t_0 n) i)
           (* 100.0 (* n (/ (+ (pow (+ (/ i n) 1.0) n) -1.0) i)))))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * exp(i)) - 100.0;
	double tmp;
	if (i <= -0.000115) {
		tmp = t_0 / (i / n);
	} else if (i <= -2.7e-204) {
		tmp = 1.0 / (((i * ((0.5 / pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 4.4e+43) {
		tmp = (t_0 * n) / i;
	} else {
		tmp = 100.0 * (n * ((pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	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 * exp(i)) - 100.0d0
    if (i <= (-0.000115d0)) then
        tmp = t_0 / (i / n)
    else if (i <= (-2.7d-204)) then
        tmp = 1.0d0 / (((i * ((0.5d0 / (n ** 2.0d0)) - (0.5d0 / n))) * 0.01d0) + (0.01d0 / n))
    else if (i <= 2.5d-25) then
        tmp = 100.0d0 * n
    else if (i <= 4.4d+43) then
        tmp = (t_0 * n) / i
    else
        tmp = 100.0d0 * (n * (((((i / n) + 1.0d0) ** n) + (-1.0d0)) / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (100.0 * Math.exp(i)) - 100.0;
	double tmp;
	if (i <= -0.000115) {
		tmp = t_0 / (i / n);
	} else if (i <= -2.7e-204) {
		tmp = 1.0 / (((i * ((0.5 / Math.pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 4.4e+43) {
		tmp = (t_0 * n) / i;
	} else {
		tmp = 100.0 * (n * ((Math.pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * math.exp(i)) - 100.0
	tmp = 0
	if i <= -0.000115:
		tmp = t_0 / (i / n)
	elif i <= -2.7e-204:
		tmp = 1.0 / (((i * ((0.5 / math.pow(n, 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n))
	elif i <= 2.5e-25:
		tmp = 100.0 * n
	elif i <= 4.4e+43:
		tmp = (t_0 * n) / i
	else:
		tmp = 100.0 * (n * ((math.pow(((i / n) + 1.0), n) + -1.0) / i))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * exp(i)) - 100.0)
	tmp = 0.0
	if (i <= -0.000115)
		tmp = Float64(t_0 / Float64(i / n));
	elseif (i <= -2.7e-204)
		tmp = Float64(1.0 / Float64(Float64(Float64(i * Float64(Float64(0.5 / (n ^ 2.0)) - Float64(0.5 / n))) * 0.01) + Float64(0.01 / n)));
	elseif (i <= 2.5e-25)
		tmp = Float64(100.0 * n);
	elseif (i <= 4.4e+43)
		tmp = Float64(Float64(t_0 * n) / i);
	else
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) + -1.0) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (100.0 * exp(i)) - 100.0;
	tmp = 0.0;
	if (i <= -0.000115)
		tmp = t_0 / (i / n);
	elseif (i <= -2.7e-204)
		tmp = 1.0 / (((i * ((0.5 / (n ^ 2.0)) - (0.5 / n))) * 0.01) + (0.01 / n));
	elseif (i <= 2.5e-25)
		tmp = 100.0 * n;
	elseif (i <= 4.4e+43)
		tmp = (t_0 * n) / i;
	else
		tmp = 100.0 * (n * (((((i / n) + 1.0) ^ n) + -1.0) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]}, If[LessEqual[i, -0.000115], N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, -2.7e-204], N[(1.0 / N[(N[(N[(i * N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.01), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e-25], N[(100.0 * n), $MachinePrecision], If[LessEqual[i, 4.4e+43], N[(N[(t$95$0 * n), $MachinePrecision] / i), $MachinePrecision], N[(100.0 * N[(n * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -1.15e-4

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in n around inf 84.7%

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

   if -1.15e-4 < i < -2.69999999999999991e-204

   1. Initial program 12.3%

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

      1. associate-*r/ 12.3%

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

      2. sub-neg 12.3%

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

      3. distribute-rgt-in 12.3%

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

      4. metadata-eval 12.3%

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

      5. metadata-eval 12.3%

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

   3. Simplified 12.3%

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

   5. Taylor expanded in i around 0 64.4%

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

      1. clear-num 64.5%

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

      2. inv-pow 64.5%

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

      3. *-un-lft-identity 64.5%

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

      4. distribute-lft-out 64.5%

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

      5. times-frac 64.4%

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

      6. metadata-eval 64.4%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 64.4%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 64.4%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 64.4%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 64.4%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 91.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 91.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 91.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 91.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   if -2.69999999999999991e-204 < i < 2.49999999999999981e-25

   1. Initial program 5.4%

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

      1. associate-/r/ 6.1%

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

      2. *-commutative 6.1%

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

      3. sub-neg 6.1%

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

      4. metadata-eval 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in i around 0 89.5%

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

      1. *-commutative 89.5%

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

   7. Simplified 89.5%

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

   if 2.49999999999999981e-25 < i < 4.40000000000000001e43

   1. Initial program 18.4%

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

      1. associate-*r/ 18.4%

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

      2. sub-neg 18.4%

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

      3. distribute-rgt-in 18.4%

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

      4. metadata-eval 18.4%

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

      5. metadata-eval 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in n around inf 78.6%

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

   if 4.40000000000000001e43 < i

   1. Initial program 60.6%

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

      1. associate-/r/ 60.9%

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

      2. *-commutative 60.9%

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

      3. sub-neg 60.9%

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

      4. metadata-eval 60.9%

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

   3. Simplified 60.9%

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)} \]
   4. Add Preprocessing
3. Recombined 5 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.000115:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -2.7 \cdot 10^{-204}:\\ \;\;\;\;\frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot e^{i} + -100}{i}\\ \mathbf{if}\;i \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{+64} \lor \neg \left(i \leq 2.5 \cdot 10^{+118}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{{n}^{2}}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (+ (* 100.0 (exp i)) -100.0) i))))
   (if (<= i -7.4e-6)
     t_0
     (if (<= i 2.5e-25)
       (+ (* 100.0 n) (* 50.0 (* i n)))
       (if (or (<= i 2.05e+64) (not (<= i 2.5e+118)))
         t_0
         (* 200.0 (/ (pow n 2.0) i)))))))

C

double code(double i, double n) {
	double t_0 = n * (((100.0 * exp(i)) + -100.0) / i);
	double tmp;
	if (i <= -7.4e-6) {
		tmp = t_0;
	} else if (i <= 2.5e-25) {
		tmp = (100.0 * n) + (50.0 * (i * n));
	} else if ((i <= 2.05e+64) || !(i <= 2.5e+118)) {
		tmp = t_0;
	} else {
		tmp = 200.0 * (pow(n, 2.0) / i);
	}
	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 * exp(i)) + (-100.0d0)) / i)
    if (i <= (-7.4d-6)) then
        tmp = t_0
    else if (i <= 2.5d-25) then
        tmp = (100.0d0 * n) + (50.0d0 * (i * n))
    else if ((i <= 2.05d+64) .or. (.not. (i <= 2.5d+118))) then
        tmp = t_0
    else
        tmp = 200.0d0 * ((n ** 2.0d0) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (((100.0 * Math.exp(i)) + -100.0) / i);
	double tmp;
	if (i <= -7.4e-6) {
		tmp = t_0;
	} else if (i <= 2.5e-25) {
		tmp = (100.0 * n) + (50.0 * (i * n));
	} else if ((i <= 2.05e+64) || !(i <= 2.5e+118)) {
		tmp = t_0;
	} else {
		tmp = 200.0 * (Math.pow(n, 2.0) / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (((100.0 * math.exp(i)) + -100.0) / i)
	tmp = 0
	if i <= -7.4e-6:
		tmp = t_0
	elif i <= 2.5e-25:
		tmp = (100.0 * n) + (50.0 * (i * n))
	elif (i <= 2.05e+64) or not (i <= 2.5e+118):
		tmp = t_0
	else:
		tmp = 200.0 * (math.pow(n, 2.0) / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(Float64(100.0 * exp(i)) + -100.0) / i))
	tmp = 0.0
	if (i <= -7.4e-6)
		tmp = t_0;
	elseif (i <= 2.5e-25)
		tmp = Float64(Float64(100.0 * n) + Float64(50.0 * Float64(i * n)));
	elseif ((i <= 2.05e+64) || !(i <= 2.5e+118))
		tmp = t_0;
	else
		tmp = Float64(200.0 * Float64((n ^ 2.0) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (((100.0 * exp(i)) + -100.0) / i);
	tmp = 0.0;
	if (i <= -7.4e-6)
		tmp = t_0;
	elseif (i <= 2.5e-25)
		tmp = (100.0 * n) + (50.0 * (i * n));
	elseif ((i <= 2.05e+64) || ~((i <= 2.5e+118)))
		tmp = t_0;
	else
		tmp = 200.0 * ((n ^ 2.0) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -7.4e-6], t$95$0, If[LessEqual[i, 2.5e-25], N[(N[(100.0 * n), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[i, 2.05e+64], N[Not[LessEqual[i, 2.5e+118]], $MachinePrecision]], t$95$0, N[(200.0 * N[(N[Power[n, 2.0], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -7.4000000000000003e-6 or 2.49999999999999981e-25 < i < 2.04999999999999989e64 or 2.49999999999999986e118 < i

   1. Initial program 51.7%

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

      1. associate-*r/ 51.8%

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

      2. sub-neg 51.8%

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

      3. distribute-rgt-in 51.8%

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

      4. metadata-eval 51.8%

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

      5. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in n around inf 71.6%

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

      1. associate-/l* 71.6%

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

      2. sub-neg 71.6%

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

      3. *-commutative 71.6%

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

      4. metadata-eval 71.6%

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

   7. Simplified 71.6%

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

   if -7.4000000000000003e-6 < i < 2.49999999999999981e-25

   1. Initial program 7.5%

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

      1. associate-/r/ 8.1%

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

      2. *-commutative 8.1%

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

      3. sub-neg 8.1%

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

      4. metadata-eval 8.1%

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

   3. Simplified 8.1%

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

   5. Taylor expanded in i around 0 86.9%

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

   6. Taylor expanded in n around inf 87.1%

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

   if 2.04999999999999989e64 < i < 2.49999999999999986e118

   1. Initial program 32.2%

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

      1. associate-*r/ 32.0%

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

      2. sub-neg 32.0%

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

      3. distribute-rgt-in 32.3%

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

      4. metadata-eval 32.3%

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

      5. metadata-eval 32.3%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in i around 0 1.6%

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

      1. clear-num 1.6%

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

      2. inv-pow 1.6%

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

      3. *-un-lft-identity 1.6%

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

      4. distribute-lft-out 1.6%

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

      5. times-frac 1.6%

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

      6. metadata-eval 1.6%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 1.6%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 1.6%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 1.6%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 66.7%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 66.7%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 66.7%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   13. Taylor expanded in n around 0 65.8%

       \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;n \cdot \frac{100 \cdot e^{i} + -100}{i}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{+64} \lor \neg \left(i \leq 2.5 \cdot 10^{+118}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot e^{i} + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{{n}^{2}}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot e^{i}\\ t_1 := \frac{\left(t\_0 - 100\right) \cdot n}{i}\\ \mathbf{if}\;i \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{+119}:\\ \;\;\;\;200 \cdot \frac{{n}^{2}}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{t\_0 + -100}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (exp i))) (t_1 (/ (* (- t_0 100.0) n) i)))
   (if (<= i -7.4e-6)
     t_1
     (if (<= i 2.5e-25)
       (+ (* 100.0 n) (* 50.0 (* i n)))
       (if (<= i 2.05e+64)
         t_1
         (if (<= i 1.9e+119)
           (* 200.0 (/ (pow n 2.0) i))
           (* n (/ (+ t_0 -100.0) i))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * exp(i);
	double t_1 = ((t_0 - 100.0) * n) / i;
	double tmp;
	if (i <= -7.4e-6) {
		tmp = t_1;
	} else if (i <= 2.5e-25) {
		tmp = (100.0 * n) + (50.0 * (i * n));
	} else if (i <= 2.05e+64) {
		tmp = t_1;
	} else if (i <= 1.9e+119) {
		tmp = 200.0 * (pow(n, 2.0) / i);
	} else {
		tmp = n * ((t_0 + -100.0) / i);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 100.0d0 * exp(i)
    t_1 = ((t_0 - 100.0d0) * n) / i
    if (i <= (-7.4d-6)) then
        tmp = t_1
    else if (i <= 2.5d-25) then
        tmp = (100.0d0 * n) + (50.0d0 * (i * n))
    else if (i <= 2.05d+64) then
        tmp = t_1
    else if (i <= 1.9d+119) then
        tmp = 200.0d0 * ((n ** 2.0d0) / i)
    else
        tmp = n * ((t_0 + (-100.0d0)) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * Math.exp(i);
	double t_1 = ((t_0 - 100.0) * n) / i;
	double tmp;
	if (i <= -7.4e-6) {
		tmp = t_1;
	} else if (i <= 2.5e-25) {
		tmp = (100.0 * n) + (50.0 * (i * n));
	} else if (i <= 2.05e+64) {
		tmp = t_1;
	} else if (i <= 1.9e+119) {
		tmp = 200.0 * (Math.pow(n, 2.0) / i);
	} else {
		tmp = n * ((t_0 + -100.0) / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * math.exp(i)
	t_1 = ((t_0 - 100.0) * n) / i
	tmp = 0
	if i <= -7.4e-6:
		tmp = t_1
	elif i <= 2.5e-25:
		tmp = (100.0 * n) + (50.0 * (i * n))
	elif i <= 2.05e+64:
		tmp = t_1
	elif i <= 1.9e+119:
		tmp = 200.0 * (math.pow(n, 2.0) / i)
	else:
		tmp = n * ((t_0 + -100.0) / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * exp(i))
	t_1 = Float64(Float64(Float64(t_0 - 100.0) * n) / i)
	tmp = 0.0
	if (i <= -7.4e-6)
		tmp = t_1;
	elseif (i <= 2.5e-25)
		tmp = Float64(Float64(100.0 * n) + Float64(50.0 * Float64(i * n)));
	elseif (i <= 2.05e+64)
		tmp = t_1;
	elseif (i <= 1.9e+119)
		tmp = Float64(200.0 * Float64((n ^ 2.0) / i));
	else
		tmp = Float64(n * Float64(Float64(t_0 + -100.0) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * exp(i);
	t_1 = ((t_0 - 100.0) * n) / i;
	tmp = 0.0;
	if (i <= -7.4e-6)
		tmp = t_1;
	elseif (i <= 2.5e-25)
		tmp = (100.0 * n) + (50.0 * (i * n));
	elseif (i <= 2.05e+64)
		tmp = t_1;
	elseif (i <= 1.9e+119)
		tmp = 200.0 * ((n ^ 2.0) / i);
	else
		tmp = n * ((t_0 + -100.0) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 - 100.0), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[i, -7.4e-6], t$95$1, If[LessEqual[i, 2.5e-25], N[(N[(100.0 * n), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.05e+64], t$95$1, If[LessEqual[i, 1.9e+119], N[(200.0 * N[(N[Power[n, 2.0], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(t$95$0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -7.4000000000000003e-6 or 2.49999999999999981e-25 < i < 2.04999999999999989e64

   1. Initial program 44.6%

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

      1. associate-*r/ 44.7%

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

      2. sub-neg 44.7%

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

      3. distribute-rgt-in 44.7%

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

      4. metadata-eval 44.7%

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

      5. metadata-eval 44.7%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in n around inf 82.3%

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

   if -7.4000000000000003e-6 < i < 2.49999999999999981e-25

   1. Initial program 7.5%

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

      1. associate-/r/ 8.1%

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

      2. *-commutative 8.1%

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

      3. sub-neg 8.1%

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

      4. metadata-eval 8.1%

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

   3. Simplified 8.1%

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

   5. Taylor expanded in i around 0 86.9%

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

   6. Taylor expanded in n around inf 87.1%

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

   if 2.04999999999999989e64 < i < 1.89999999999999995e119

   1. Initial program 32.2%

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

      1. associate-*r/ 32.0%

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

      2. sub-neg 32.0%

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

      3. distribute-rgt-in 32.3%

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

      4. metadata-eval 32.3%

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

      5. metadata-eval 32.3%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in i around 0 1.6%

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

      1. clear-num 1.6%

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

      2. inv-pow 1.6%

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

      3. *-un-lft-identity 1.6%

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

      4. distribute-lft-out 1.6%

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

      5. times-frac 1.6%

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

      6. metadata-eval 1.6%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 1.6%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 1.6%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 1.6%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 66.7%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 66.7%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 66.7%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   13. Taylor expanded in n around 0 65.8%

       \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]

   if 1.89999999999999995e119 < i

   1. Initial program 68.0%

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

      1. associate-*r/ 68.0%

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

      2. sub-neg 68.0%

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

      3. distribute-rgt-in 68.0%

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

      4. metadata-eval 68.0%

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

      5. metadata-eval 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in n around inf 47.1%

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

      1. associate-/l* 47.1%

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

      2. sub-neg 47.1%

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

      3. *-commutative 47.1%

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

      4. metadata-eval 47.1%

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

   7. Simplified 47.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{+119}:\\ \;\;\;\;200 \cdot \frac{{n}^{2}}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot e^{i} + -100}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot e^{i}\\ t_1 := t\_0 - 100\\ \mathbf{if}\;i \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{t\_1 \cdot n}{i}\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{+118}:\\ \;\;\;\;200 \cdot \frac{{n}^{2}}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{t\_0 + -100}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (exp i))) (t_1 (- t_0 100.0)))
   (if (<= i -7.4e-6)
     (/ t_1 (/ i n))
     (if (<= i 2.5e-25)
       (+ (* 100.0 n) (* 50.0 (* i n)))
       (if (<= i 2.05e+64)
         (/ (* t_1 n) i)
         (if (<= i 2.4e+118)
           (* 200.0 (/ (pow n 2.0) i))
           (* n (/ (+ t_0 -100.0) i))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * exp(i);
	double t_1 = t_0 - 100.0;
	double tmp;
	if (i <= -7.4e-6) {
		tmp = t_1 / (i / n);
	} else if (i <= 2.5e-25) {
		tmp = (100.0 * n) + (50.0 * (i * n));
	} else if (i <= 2.05e+64) {
		tmp = (t_1 * n) / i;
	} else if (i <= 2.4e+118) {
		tmp = 200.0 * (pow(n, 2.0) / i);
	} else {
		tmp = n * ((t_0 + -100.0) / i);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 100.0d0 * exp(i)
    t_1 = t_0 - 100.0d0
    if (i <= (-7.4d-6)) then
        tmp = t_1 / (i / n)
    else if (i <= 2.5d-25) then
        tmp = (100.0d0 * n) + (50.0d0 * (i * n))
    else if (i <= 2.05d+64) then
        tmp = (t_1 * n) / i
    else if (i <= 2.4d+118) then
        tmp = 200.0d0 * ((n ** 2.0d0) / i)
    else
        tmp = n * ((t_0 + (-100.0d0)) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * Math.exp(i);
	double t_1 = t_0 - 100.0;
	double tmp;
	if (i <= -7.4e-6) {
		tmp = t_1 / (i / n);
	} else if (i <= 2.5e-25) {
		tmp = (100.0 * n) + (50.0 * (i * n));
	} else if (i <= 2.05e+64) {
		tmp = (t_1 * n) / i;
	} else if (i <= 2.4e+118) {
		tmp = 200.0 * (Math.pow(n, 2.0) / i);
	} else {
		tmp = n * ((t_0 + -100.0) / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * math.exp(i)
	t_1 = t_0 - 100.0
	tmp = 0
	if i <= -7.4e-6:
		tmp = t_1 / (i / n)
	elif i <= 2.5e-25:
		tmp = (100.0 * n) + (50.0 * (i * n))
	elif i <= 2.05e+64:
		tmp = (t_1 * n) / i
	elif i <= 2.4e+118:
		tmp = 200.0 * (math.pow(n, 2.0) / i)
	else:
		tmp = n * ((t_0 + -100.0) / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * exp(i))
	t_1 = Float64(t_0 - 100.0)
	tmp = 0.0
	if (i <= -7.4e-6)
		tmp = Float64(t_1 / Float64(i / n));
	elseif (i <= 2.5e-25)
		tmp = Float64(Float64(100.0 * n) + Float64(50.0 * Float64(i * n)));
	elseif (i <= 2.05e+64)
		tmp = Float64(Float64(t_1 * n) / i);
	elseif (i <= 2.4e+118)
		tmp = Float64(200.0 * Float64((n ^ 2.0) / i));
	else
		tmp = Float64(n * Float64(Float64(t_0 + -100.0) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * exp(i);
	t_1 = t_0 - 100.0;
	tmp = 0.0;
	if (i <= -7.4e-6)
		tmp = t_1 / (i / n);
	elseif (i <= 2.5e-25)
		tmp = (100.0 * n) + (50.0 * (i * n));
	elseif (i <= 2.05e+64)
		tmp = (t_1 * n) / i;
	elseif (i <= 2.4e+118)
		tmp = 200.0 * ((n ^ 2.0) / i);
	else
		tmp = n * ((t_0 + -100.0) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 100.0), $MachinePrecision]}, If[LessEqual[i, -7.4e-6], N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e-25], N[(N[(100.0 * n), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.05e+64], N[(N[(t$95$1 * n), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, 2.4e+118], N[(200.0 * N[(N[Power[n, 2.0], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(t$95$0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -7.4000000000000003e-6

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in n around inf 84.7%

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

   if -7.4000000000000003e-6 < i < 2.49999999999999981e-25

   1. Initial program 7.5%

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

      1. associate-/r/ 8.1%

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

      2. *-commutative 8.1%

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

      3. sub-neg 8.1%

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

      4. metadata-eval 8.1%

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

   3. Simplified 8.1%

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

   5. Taylor expanded in i around 0 86.9%

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

   6. Taylor expanded in n around inf 87.1%

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

   if 2.49999999999999981e-25 < i < 2.04999999999999989e64

   1. Initial program 28.2%

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

      1. associate-*r/ 28.2%

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

      2. sub-neg 28.2%

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

      3. distribute-rgt-in 28.2%

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

      4. metadata-eval 28.2%

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

      5. metadata-eval 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in n around inf 76.3%

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

   if 2.04999999999999989e64 < i < 2.4e118

   1. Initial program 32.2%

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

      1. associate-*r/ 32.0%

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

      2. sub-neg 32.0%

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

      3. distribute-rgt-in 32.3%

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

      4. metadata-eval 32.3%

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

      5. metadata-eval 32.3%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in i around 0 1.6%

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

      1. clear-num 1.6%

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

      2. inv-pow 1.6%

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

      3. *-un-lft-identity 1.6%

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

      4. distribute-lft-out 1.6%

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

      5. times-frac 1.6%

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

      6. metadata-eval 1.6%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 1.6%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 1.6%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 1.6%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 66.7%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 66.7%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 66.7%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 66.7%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   13. Taylor expanded in n around 0 65.8%

       \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]

   if 2.4e118 < i

   1. Initial program 68.0%

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

      1. associate-*r/ 68.0%

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

      2. sub-neg 68.0%

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

      3. distribute-rgt-in 68.0%

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

      4. metadata-eval 68.0%

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

      5. metadata-eval 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in n around inf 47.1%

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

      1. associate-/l* 47.1%

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

      2. sub-neg 47.1%

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

      3. *-commutative 47.1%

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

      4. metadata-eval 47.1%

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

   7. Simplified 47.1%

      \[\leadsto \color{blue}{n \cdot \frac{e^{i} \cdot 100 + -100}{i}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{+118}:\\ \;\;\;\;200 \cdot \frac{{n}^{2}}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot e^{i} + -100}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot e^{i} - 100\\ \mathbf{if}\;i \leq -0.000165:\\ \;\;\;\;\frac{t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;n \cdot \left({i}^{2} \cdot 16.666666666666668 + \left(100 + i \cdot 50\right)\right)\\ \mathbf{elif}\;i \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{t\_0 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (* 100.0 (exp i)) 100.0)))
   (if (<= i -0.000165)
     (/ t_0 (/ i n))
     (if (<= i 2.5e-25)
       (* n (+ (* (pow i 2.0) 16.666666666666668) (+ 100.0 (* i 50.0))))
       (if (<= i 4.4e+43)
         (/ (* t_0 n) i)
         (* 100.0 (* n (/ (+ (pow (+ (/ i n) 1.0) n) -1.0) i))))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * exp(i)) - 100.0;
	double tmp;
	if (i <= -0.000165) {
		tmp = t_0 / (i / n);
	} else if (i <= 2.5e-25) {
		tmp = n * ((pow(i, 2.0) * 16.666666666666668) + (100.0 + (i * 50.0)));
	} else if (i <= 4.4e+43) {
		tmp = (t_0 * n) / i;
	} else {
		tmp = 100.0 * (n * ((pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	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 * exp(i)) - 100.0d0
    if (i <= (-0.000165d0)) then
        tmp = t_0 / (i / n)
    else if (i <= 2.5d-25) then
        tmp = n * (((i ** 2.0d0) * 16.666666666666668d0) + (100.0d0 + (i * 50.0d0)))
    else if (i <= 4.4d+43) then
        tmp = (t_0 * n) / i
    else
        tmp = 100.0d0 * (n * (((((i / n) + 1.0d0) ** n) + (-1.0d0)) / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (100.0 * Math.exp(i)) - 100.0;
	double tmp;
	if (i <= -0.000165) {
		tmp = t_0 / (i / n);
	} else if (i <= 2.5e-25) {
		tmp = n * ((Math.pow(i, 2.0) * 16.666666666666668) + (100.0 + (i * 50.0)));
	} else if (i <= 4.4e+43) {
		tmp = (t_0 * n) / i;
	} else {
		tmp = 100.0 * (n * ((Math.pow(((i / n) + 1.0), n) + -1.0) / i));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * math.exp(i)) - 100.0
	tmp = 0
	if i <= -0.000165:
		tmp = t_0 / (i / n)
	elif i <= 2.5e-25:
		tmp = n * ((math.pow(i, 2.0) * 16.666666666666668) + (100.0 + (i * 50.0)))
	elif i <= 4.4e+43:
		tmp = (t_0 * n) / i
	else:
		tmp = 100.0 * (n * ((math.pow(((i / n) + 1.0), n) + -1.0) / i))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * exp(i)) - 100.0)
	tmp = 0.0
	if (i <= -0.000165)
		tmp = Float64(t_0 / Float64(i / n));
	elseif (i <= 2.5e-25)
		tmp = Float64(n * Float64(Float64((i ^ 2.0) * 16.666666666666668) + Float64(100.0 + Float64(i * 50.0))));
	elseif (i <= 4.4e+43)
		tmp = Float64(Float64(t_0 * n) / i);
	else
		tmp = Float64(100.0 * Float64(n * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) + -1.0) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (100.0 * exp(i)) - 100.0;
	tmp = 0.0;
	if (i <= -0.000165)
		tmp = t_0 / (i / n);
	elseif (i <= 2.5e-25)
		tmp = n * (((i ^ 2.0) * 16.666666666666668) + (100.0 + (i * 50.0)));
	elseif (i <= 4.4e+43)
		tmp = (t_0 * n) / i;
	else
		tmp = 100.0 * (n * (((((i / n) + 1.0) ^ n) + -1.0) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]}, If[LessEqual[i, -0.000165], N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e-25], N[(n * N[(N[(N[Power[i, 2.0], $MachinePrecision] * 16.666666666666668), $MachinePrecision] + N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.4e+43], N[(N[(t$95$0 * n), $MachinePrecision] / i), $MachinePrecision], N[(100.0 * N[(n * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.65e-4

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in n around inf 84.7%

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

   if -1.65e-4 < i < 2.49999999999999981e-25

   1. Initial program 7.5%

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

      1. associate-*r/ 7.5%

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

      2. sub-neg 7.5%

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

      3. distribute-rgt-in 7.5%

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

      4. metadata-eval 7.5%

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

      5. metadata-eval 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in n around inf 9.0%

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

   6. Taylor expanded in i around 0 87.2%

      \[\leadsto \color{blue}{16.666666666666668 \cdot \left({i}^{2} \cdot n\right) + \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-commutative 87.2%

         \[\leadsto \color{blue}{\left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right) + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right)} \]

      2. associate-*r* 87.2%

         \[\leadsto \left(\color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n\right) + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right) \]

      3. +-commutative 87.2%

         \[\leadsto \color{blue}{\left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)} + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right) \]

      4. distribute-rgt-in 87.2%

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right) \]

      5. *-commutative 87.2%

         \[\leadsto \color{blue}{\left(100 + 50 \cdot i\right) \cdot n} + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right) \]

      6. associate-*r* 87.2%

         \[\leadsto \left(100 + 50 \cdot i\right) \cdot n + \color{blue}{\left(16.666666666666668 \cdot {i}^{2}\right) \cdot n} \]

      7. distribute-rgt-out 87.2%

         \[\leadsto \color{blue}{n \cdot \left(\left(100 + 50 \cdot i\right) + 16.666666666666668 \cdot {i}^{2}\right)} \]

      8. *-commutative 87.2%

         \[\leadsto n \cdot \left(\left(100 + 50 \cdot i\right) + \color{blue}{{i}^{2} \cdot 16.666666666666668}\right) \]

   8. Simplified 87.2%

      \[\leadsto \color{blue}{n \cdot \left(\left(100 + 50 \cdot i\right) + {i}^{2} \cdot 16.666666666666668\right)} \]

   if 2.49999999999999981e-25 < i < 4.40000000000000001e43

   1. Initial program 18.4%

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

      1. associate-*r/ 18.4%

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

      2. sub-neg 18.4%

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

      3. distribute-rgt-in 18.4%

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

      4. metadata-eval 18.4%

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

      5. metadata-eval 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in n around inf 78.6%

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

   if 4.40000000000000001e43 < i

   1. Initial program 60.6%

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

      1. associate-/r/ 60.9%

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

      2. *-commutative 60.9%

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

      3. sub-neg 60.9%

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

      4. metadata-eval 60.9%

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

   3. Simplified 60.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.000165:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;n \cdot \left({i}^{2} \cdot 16.666666666666668 + \left(100 + i \cdot 50\right)\right)\\ \mathbf{elif}\;i \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} + -1}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot e^{i} - 100\\ \mathbf{if}\;i \leq -0.000165:\\ \;\;\;\;\frac{t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;n \cdot \left({i}^{2} \cdot 16.666666666666668 + \left(100 + i \cdot 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (* 100.0 (exp i)) 100.0)))
   (if (<= i -0.000165)
     (/ t_0 (/ i n))
     (if (<= i 2.5e-25)
       (* n (+ (* (pow i 2.0) 16.666666666666668) (+ 100.0 (* i 50.0))))
       (/ (* t_0 n) i)))))

C

double code(double i, double n) {
	double t_0 = (100.0 * exp(i)) - 100.0;
	double tmp;
	if (i <= -0.000165) {
		tmp = t_0 / (i / n);
	} else if (i <= 2.5e-25) {
		tmp = n * ((pow(i, 2.0) * 16.666666666666668) + (100.0 + (i * 50.0)));
	} else {
		tmp = (t_0 * n) / i;
	}
	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 * exp(i)) - 100.0d0
    if (i <= (-0.000165d0)) then
        tmp = t_0 / (i / n)
    else if (i <= 2.5d-25) then
        tmp = n * (((i ** 2.0d0) * 16.666666666666668d0) + (100.0d0 + (i * 50.0d0)))
    else
        tmp = (t_0 * n) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (100.0 * Math.exp(i)) - 100.0;
	double tmp;
	if (i <= -0.000165) {
		tmp = t_0 / (i / n);
	} else if (i <= 2.5e-25) {
		tmp = n * ((Math.pow(i, 2.0) * 16.666666666666668) + (100.0 + (i * 50.0)));
	} else {
		tmp = (t_0 * n) / i;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * math.exp(i)) - 100.0
	tmp = 0
	if i <= -0.000165:
		tmp = t_0 / (i / n)
	elif i <= 2.5e-25:
		tmp = n * ((math.pow(i, 2.0) * 16.666666666666668) + (100.0 + (i * 50.0)))
	else:
		tmp = (t_0 * n) / i
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * exp(i)) - 100.0)
	tmp = 0.0
	if (i <= -0.000165)
		tmp = Float64(t_0 / Float64(i / n));
	elseif (i <= 2.5e-25)
		tmp = Float64(n * Float64(Float64((i ^ 2.0) * 16.666666666666668) + Float64(100.0 + Float64(i * 50.0))));
	else
		tmp = Float64(Float64(t_0 * n) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (100.0 * exp(i)) - 100.0;
	tmp = 0.0;
	if (i <= -0.000165)
		tmp = t_0 / (i / n);
	elseif (i <= 2.5e-25)
		tmp = n * (((i ^ 2.0) * 16.666666666666668) + (100.0 + (i * 50.0)));
	else
		tmp = (t_0 * n) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[Exp[i], $MachinePrecision]), $MachinePrecision] - 100.0), $MachinePrecision]}, If[LessEqual[i, -0.000165], N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e-25], N[(n * N[(N[(N[Power[i, 2.0], $MachinePrecision] * 16.666666666666668), $MachinePrecision] + N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * n), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.65e-4

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in n around inf 84.7%

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

   if -1.65e-4 < i < 2.49999999999999981e-25

   1. Initial program 7.5%

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

      1. associate-*r/ 7.5%

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

      2. sub-neg 7.5%

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

      3. distribute-rgt-in 7.5%

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

      4. metadata-eval 7.5%

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

      5. metadata-eval 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in n around inf 9.0%

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

   6. Taylor expanded in i around 0 87.2%

      \[\leadsto \color{blue}{16.666666666666668 \cdot \left({i}^{2} \cdot n\right) + \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)} \]
   7. Step-by-step derivation

      1. +-commutative 87.2%

         \[\leadsto \color{blue}{\left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right) + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right)} \]

      2. associate-*r* 87.2%

         \[\leadsto \left(\color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n\right) + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right) \]

      3. +-commutative 87.2%

         \[\leadsto \color{blue}{\left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)} + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right) \]

      4. distribute-rgt-in 87.2%

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right) \]

      5. *-commutative 87.2%

         \[\leadsto \color{blue}{\left(100 + 50 \cdot i\right) \cdot n} + 16.666666666666668 \cdot \left({i}^{2} \cdot n\right) \]

      6. associate-*r* 87.2%

         \[\leadsto \left(100 + 50 \cdot i\right) \cdot n + \color{blue}{\left(16.666666666666668 \cdot {i}^{2}\right) \cdot n} \]

      7. distribute-rgt-out 87.2%

         \[\leadsto \color{blue}{n \cdot \left(\left(100 + 50 \cdot i\right) + 16.666666666666668 \cdot {i}^{2}\right)} \]

      8. *-commutative 87.2%

         \[\leadsto n \cdot \left(\left(100 + 50 \cdot i\right) + \color{blue}{{i}^{2} \cdot 16.666666666666668}\right) \]

   8. Simplified 87.2%

      \[\leadsto \color{blue}{n \cdot \left(\left(100 + 50 \cdot i\right) + {i}^{2} \cdot 16.666666666666668\right)} \]

   if 2.49999999999999981e-25 < i

   1. Initial program 50.7%

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

      1. associate-*r/ 50.6%

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

      2. sub-neg 50.6%

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

      3. distribute-rgt-in 50.7%

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

      4. metadata-eval 50.7%

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

      5. metadata-eval 50.7%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in n around inf 50.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.000165:\\ \;\;\;\;\frac{100 \cdot e^{i} - 100}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;n \cdot \left({i}^{2} \cdot 16.666666666666668 + \left(100 + i \cdot 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{0.01}{n} + \frac{i}{n} \cdot -0.005}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-208}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.45e-242)
   (/ 1.0 (+ (/ 0.01 n) (* (/ i n) -0.005)))
   (if (<= n 5.5e-208)
     0.0
     (+ (* 100.0 n) (* 100.0 (* i (* n (- 0.5 (* 0.5 (/ 1.0 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.45e-242) {
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	} else if (n <= 5.5e-208) {
		tmp = 0.0;
	} else {
		tmp = (100.0 * n) + (100.0 * (i * (n * (0.5 - (0.5 * (1.0 / n))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.45d-242)) then
        tmp = 1.0d0 / ((0.01d0 / n) + ((i / n) * (-0.005d0)))
    else if (n <= 5.5d-208) then
        tmp = 0.0d0
    else
        tmp = (100.0d0 * n) + (100.0d0 * (i * (n * (0.5d0 - (0.5d0 * (1.0d0 / n))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.45e-242) {
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	} else if (n <= 5.5e-208) {
		tmp = 0.0;
	} else {
		tmp = (100.0 * n) + (100.0 * (i * (n * (0.5 - (0.5 * (1.0 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.45e-242:
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005))
	elif n <= 5.5e-208:
		tmp = 0.0
	else:
		tmp = (100.0 * n) + (100.0 * (i * (n * (0.5 - (0.5 * (1.0 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.45e-242)
		tmp = Float64(1.0 / Float64(Float64(0.01 / n) + Float64(Float64(i / n) * -0.005)));
	elseif (n <= 5.5e-208)
		tmp = 0.0;
	else
		tmp = Float64(Float64(100.0 * n) + Float64(100.0 * Float64(i * Float64(n * Float64(0.5 - Float64(0.5 * Float64(1.0 / n)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.45e-242)
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	elseif (n <= 5.5e-208)
		tmp = 0.0;
	else
		tmp = (100.0 * n) + (100.0 * (i * (n * (0.5 - (0.5 * (1.0 / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.45e-242], N[(1.0 / N[(N[(0.01 / n), $MachinePrecision] + N[(N[(i / n), $MachinePrecision] * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e-208], 0.0, N[(N[(100.0 * n), $MachinePrecision] + N[(100.0 * N[(i * N[(n * N[(0.5 - N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.45e-242

   1. Initial program 28.3%

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

      1. associate-*r/ 28.3%

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

      2. sub-neg 28.3%

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

      3. distribute-rgt-in 28.3%

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

      4. metadata-eval 28.3%

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

      5. metadata-eval 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in i around 0 38.4%

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

      1. clear-num 38.4%

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

      2. inv-pow 38.4%

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

      3. *-un-lft-identity 38.4%

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

      4. distribute-lft-out 38.4%

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

      5. times-frac 38.3%

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

      6. metadata-eval 38.3%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 38.3%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 38.3%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 64.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 64.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 64.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   13. Taylor expanded in n around inf 66.4%

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

   if -1.45e-242 < n < 5.4999999999999997e-208

   1. Initial program 49.8%

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

      1. associate-*r/ 49.8%

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

      2. sub-neg 49.8%

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

      3. distribute-rgt-in 49.8%

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

      4. metadata-eval 49.8%

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

      5. metadata-eval 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in i around 0 81.6%

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

   6. Taylor expanded in i around 0 81.6%

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

   if 5.4999999999999997e-208 < n

   1. Initial program 14.4%

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

      1. associate-/r/ 14.8%

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

      2. *-commutative 14.8%

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

      3. sub-neg 14.8%

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

      4. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in i around 0 69.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{0.01}{n} + \frac{i}{n} \cdot -0.005}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-208}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + 100 \cdot \left(i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.7 \cdot 10^{+24}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{+137}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.15 \cdot 10^{+260}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.7e+24)
   0.0
   (if (<= i 2.5e-25)
     (* 100.0 n)
     (if (<= i 1.9e+137) 0.0 (if (<= i 2.15e+260) (* 50.0 (* i n)) 0.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.7e+24) {
		tmp = 0.0;
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 1.9e+137) {
		tmp = 0.0;
	} else if (i <= 2.15e+260) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-4.7d+24)) then
        tmp = 0.0d0
    else if (i <= 2.5d-25) then
        tmp = 100.0d0 * n
    else if (i <= 1.9d+137) then
        tmp = 0.0d0
    else if (i <= 2.15d+260) then
        tmp = 50.0d0 * (i * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -4.7e+24) {
		tmp = 0.0;
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else if (i <= 1.9e+137) {
		tmp = 0.0;
	} else if (i <= 2.15e+260) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -4.7e+24:
		tmp = 0.0
	elif i <= 2.5e-25:
		tmp = 100.0 * n
	elif i <= 1.9e+137:
		tmp = 0.0
	elif i <= 2.15e+260:
		tmp = 50.0 * (i * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.7e+24)
		tmp = 0.0;
	elseif (i <= 2.5e-25)
		tmp = Float64(100.0 * n);
	elseif (i <= 1.9e+137)
		tmp = 0.0;
	elseif (i <= 2.15e+260)
		tmp = Float64(50.0 * Float64(i * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -4.7e+24)
		tmp = 0.0;
	elseif (i <= 2.5e-25)
		tmp = 100.0 * n;
	elseif (i <= 1.9e+137)
		tmp = 0.0;
	elseif (i <= 2.15e+260)
		tmp = 50.0 * (i * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.7e+24], 0.0, If[LessEqual[i, 2.5e-25], N[(100.0 * n), $MachinePrecision], If[LessEqual[i, 1.9e+137], 0.0, If[LessEqual[i, 2.15e+260], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.7e24 or 2.49999999999999981e-25 < i < 1.89999999999999981e137 or 2.15000000000000012e260 < i

   1. Initial program 48.2%

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

      1. associate-*r/ 48.2%

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

      2. sub-neg 48.2%

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

      3. distribute-rgt-in 48.2%

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

      4. metadata-eval 48.2%

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

      5. metadata-eval 48.2%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in i around 0 33.8%

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

   6. Taylor expanded in i around 0 33.8%

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

   if -4.7e24 < i < 2.49999999999999981e-25

   1. Initial program 7.3%

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

      1. associate-/r/ 7.9%

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

      2. *-commutative 7.9%

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

      3. sub-neg 7.9%

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

      4. metadata-eval 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in i around 0 84.2%

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

      1. *-commutative 84.2%

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

   7. Simplified 84.2%

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

   if 1.89999999999999981e137 < i < 2.15000000000000012e260

   1. Initial program 78.7%

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

      1. associate-/r/ 78.8%

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

      2. *-commutative 78.8%

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

      3. sub-neg 78.8%

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

      4. metadata-eval 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in i around 0 44.5%

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

   6. Taylor expanded in n around inf 44.6%

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

   7. Taylor expanded in i around inf 44.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.7 \cdot 10^{+24}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 1.9 \cdot 10^{+137}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.15 \cdot 10^{+260}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{0.01}{n} + \frac{i}{n} \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-208}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.35e-242)
   (/ 1.0 (+ (/ 0.01 n) (* (/ i n) -0.005)))
   (if (<= n 1.8e-208) 0.0 (* 100.0 (* n (+ 1.0 (* i (- 0.5 (/ 0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.35e-242) {
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	} else if (n <= 1.8e-208) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.35d-242)) then
        tmp = 1.0d0 / ((0.01d0 / n) + ((i / n) * (-0.005d0)))
    else if (n <= 1.8d-208) then
        tmp = 0.0d0
    else
        tmp = 100.0d0 * (n * (1.0d0 + (i * (0.5d0 - (0.5d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.35e-242) {
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	} else if (n <= 1.8e-208) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n * (1.0 + (i * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.35e-242:
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005))
	elif n <= 1.8e-208:
		tmp = 0.0
	else:
		tmp = 100.0 * (n * (1.0 + (i * (0.5 - (0.5 / n)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.35e-242)
		tmp = Float64(1.0 / Float64(Float64(0.01 / n) + Float64(Float64(i / n) * -0.005)));
	elseif (n <= 1.8e-208)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.35e-242)
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	elseif (n <= 1.8e-208)
		tmp = 0.0;
	else
		tmp = 100.0 * (n * (1.0 + (i * (0.5 - (0.5 / n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.35e-242], N[(1.0 / N[(N[(0.01 / n), $MachinePrecision] + N[(N[(i / n), $MachinePrecision] * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e-208], 0.0, N[(100.0 * N[(n * N[(1.0 + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.35e-242

   1. Initial program 28.3%

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

      1. associate-*r/ 28.3%

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

      2. sub-neg 28.3%

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

      3. distribute-rgt-in 28.3%

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

      4. metadata-eval 28.3%

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

      5. metadata-eval 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in i around 0 38.4%

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

      1. clear-num 38.4%

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

      2. inv-pow 38.4%

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

      3. *-un-lft-identity 38.4%

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

      4. distribute-lft-out 38.4%

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

      5. times-frac 38.3%

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

      6. metadata-eval 38.3%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 38.3%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 38.3%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 64.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 64.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 64.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   13. Taylor expanded in n around inf 66.4%

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

   if -1.35e-242 < n < 1.7999999999999999e-208

   1. Initial program 49.8%

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

      1. associate-*r/ 49.8%

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

      2. sub-neg 49.8%

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

      3. distribute-rgt-in 49.8%

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

      4. metadata-eval 49.8%

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

      5. metadata-eval 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in i around 0 81.6%

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

   6. Taylor expanded in i around 0 81.6%

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

   if 1.7999999999999999e-208 < n

   1. Initial program 14.4%

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

      1. associate-/r/ 14.8%

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

      2. *-commutative 14.8%

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

      3. sub-neg 14.8%

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

      4. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in i around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. metadata-eval 69.8%

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

   7. Simplified 69.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{0.01}{n} + \frac{i}{n} \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-208}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-242}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-211}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.2e-242)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 1.7e-211) 0.0 (+ (* 100.0 n) (* 50.0 (* i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.2e-242) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.7e-211) {
		tmp = 0.0;
	} else {
		tmp = (100.0 * n) + (50.0 * (i * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.2d-242)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 1.7d-211) then
        tmp = 0.0d0
    else
        tmp = (100.0d0 * n) + (50.0d0 * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.2e-242) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.7e-211) {
		tmp = 0.0;
	} else {
		tmp = (100.0 * n) + (50.0 * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.2e-242:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 1.7e-211:
		tmp = 0.0
	else:
		tmp = (100.0 * n) + (50.0 * (i * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.2e-242)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 1.7e-211)
		tmp = 0.0;
	else
		tmp = Float64(Float64(100.0 * n) + Float64(50.0 * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.2e-242)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 1.7e-211)
		tmp = 0.0;
	else
		tmp = (100.0 * n) + (50.0 * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.2e-242], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.7e-211], 0.0, N[(N[(100.0 * n), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2e-242

   1. Initial program 28.3%

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

      1. associate-*r/ 28.3%

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

      2. sub-neg 28.3%

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

      3. distribute-rgt-in 28.3%

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

      4. metadata-eval 28.3%

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

      5. metadata-eval 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in i around 0 38.4%

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

      1. clear-num 38.4%

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

      2. inv-pow 38.4%

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

      3. *-un-lft-identity 38.4%

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

      4. distribute-lft-out 38.4%

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

      5. times-frac 38.3%

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

      6. metadata-eval 38.3%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 38.3%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 38.3%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 64.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 64.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 64.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   13. Taylor expanded in n around inf 66.3%

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

       1. *-commutative 66.3%

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

   15. Simplified 66.3%

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

   if -1.2e-242 < n < 1.7e-211

   1. Initial program 49.8%

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

      1. associate-*r/ 49.8%

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

      2. sub-neg 49.8%

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

      3. distribute-rgt-in 49.8%

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

      4. metadata-eval 49.8%

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

      5. metadata-eval 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in i around 0 81.6%

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

   6. Taylor expanded in i around 0 81.6%

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

   if 1.7e-211 < n

   1. Initial program 14.4%

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

      1. associate-/r/ 14.8%

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

      2. *-commutative 14.8%

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

      3. sub-neg 14.8%

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

      4. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in i around 0 69.8%

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

   6. Taylor expanded in n around inf 69.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-242}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-211}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{0.01}{n} + \frac{i}{n} \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.4e-242)
   (/ 1.0 (+ (/ 0.01 n) (* (/ i n) -0.005)))
   (if (<= n 3.9e-209) 0.0 (+ (* 100.0 n) (* 50.0 (* i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.4e-242) {
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	} else if (n <= 3.9e-209) {
		tmp = 0.0;
	} else {
		tmp = (100.0 * n) + (50.0 * (i * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.4d-242)) then
        tmp = 1.0d0 / ((0.01d0 / n) + ((i / n) * (-0.005d0)))
    else if (n <= 3.9d-209) then
        tmp = 0.0d0
    else
        tmp = (100.0d0 * n) + (50.0d0 * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.4e-242) {
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	} else if (n <= 3.9e-209) {
		tmp = 0.0;
	} else {
		tmp = (100.0 * n) + (50.0 * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.4e-242:
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005))
	elif n <= 3.9e-209:
		tmp = 0.0
	else:
		tmp = (100.0 * n) + (50.0 * (i * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.4e-242)
		tmp = Float64(1.0 / Float64(Float64(0.01 / n) + Float64(Float64(i / n) * -0.005)));
	elseif (n <= 3.9e-209)
		tmp = 0.0;
	else
		tmp = Float64(Float64(100.0 * n) + Float64(50.0 * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.4e-242)
		tmp = 1.0 / ((0.01 / n) + ((i / n) * -0.005));
	elseif (n <= 3.9e-209)
		tmp = 0.0;
	else
		tmp = (100.0 * n) + (50.0 * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.4e-242], N[(1.0 / N[(N[(0.01 / n), $MachinePrecision] + N[(N[(i / n), $MachinePrecision] * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.9e-209], 0.0, N[(N[(100.0 * n), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.39999999999999992e-242

   1. Initial program 28.3%

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

      1. associate-*r/ 28.3%

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

      2. sub-neg 28.3%

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

      3. distribute-rgt-in 28.3%

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

      4. metadata-eval 28.3%

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

      5. metadata-eval 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in i around 0 38.4%

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

      1. clear-num 38.4%

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

      2. inv-pow 38.4%

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

      3. *-un-lft-identity 38.4%

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

      4. distribute-lft-out 38.4%

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

      5. times-frac 38.3%

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

      6. metadata-eval 38.3%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 38.3%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 38.3%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 64.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 64.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 64.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   13. Taylor expanded in n around inf 66.4%

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

   if -1.39999999999999992e-242 < n < 3.9e-209

   1. Initial program 49.8%

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

      1. associate-*r/ 49.8%

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

      2. sub-neg 49.8%

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

      3. distribute-rgt-in 49.8%

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

      4. metadata-eval 49.8%

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

      5. metadata-eval 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in i around 0 81.6%

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

   6. Taylor expanded in i around 0 81.6%

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

   if 3.9e-209 < n

   1. Initial program 14.4%

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

      1. associate-/r/ 14.8%

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

      2. *-commutative 14.8%

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

      3. sub-neg 14.8%

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

      4. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in i around 0 69.8%

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

   6. Taylor expanded in n around inf 69.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{0.01}{n} + \frac{i}{n} \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-209}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-125} \lor \neg \left(n \leq 5 \cdot 10^{-208}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.65e-125) (not (<= n 5e-208)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.65e-125) || !(n <= 5e-208)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.65d-125)) .or. (.not. (n <= 5d-208))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.65e-125) || !(n <= 5e-208)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.65e-125) or not (n <= 5e-208):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.65e-125) || !(n <= 5e-208))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.65e-125) || ~((n <= 5e-208)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.65e-125], N[Not[LessEqual[n, 5e-208]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.65e-125 or 4.99999999999999963e-208 < n

   1. Initial program 20.2%

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

      1. associate-/r/ 20.6%

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

      2. *-commutative 20.6%

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

      3. sub-neg 20.6%

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

      4. metadata-eval 20.6%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in i around 0 63.5%

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

   6. Taylor expanded in n around inf 63.5%

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

      1. *-commutative 63.5%

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

   8. Simplified 63.5%

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

   if -1.65e-125 < n < 4.99999999999999963e-208

   1. Initial program 48.4%

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

      1. associate-*r/ 48.4%

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

      2. sub-neg 48.4%

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

      3. distribute-rgt-in 48.4%

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

      4. metadata-eval 48.4%

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

      5. metadata-eval 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in i around 0 67.9%

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

   6. Taylor expanded in i around 0 67.9%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-125} \lor \neg \left(n \leq 5 \cdot 10^{-208}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-242}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-211}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.45e-242)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 1.15e-211) 0.0 (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.45e-242) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.15e-211) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.45d-242)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 1.15d-211) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.45e-242) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.15e-211) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.45e-242:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 1.15e-211:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.45e-242)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 1.15e-211)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.45e-242)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 1.15e-211)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.45e-242], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-211], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.45e-242

   1. Initial program 28.3%

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

      1. associate-*r/ 28.3%

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

      2. sub-neg 28.3%

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

      3. distribute-rgt-in 28.3%

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

      4. metadata-eval 28.3%

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

      5. metadata-eval 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in i around 0 38.4%

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

      1. clear-num 38.4%

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

      2. inv-pow 38.4%

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

      3. *-un-lft-identity 38.4%

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

      4. distribute-lft-out 38.4%

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

      5. times-frac 38.3%

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

      6. metadata-eval 38.3%

         \[\leadsto {\left(\color{blue}{0.01} \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)}\right)}^{-1} \]

      7. *-commutative 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{n}\right) \cdot {i}^{2}}}\right)}^{-1} \]

      8. div-inv 38.3%

         \[\leadsto {\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \color{blue}{\frac{0.5}{n}}\right) \cdot {i}^{2}}\right)}^{-1} \]

   7. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{{\left(0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 38.3%

         \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \left(0.5 - \frac{0.5}{n}\right) \cdot {i}^{2}}}} \]

      2. *-commutative 38.3%

         \[\leadsto \frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + \color{blue}{{i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   9. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{i + {i}^{2} \cdot \left(0.5 - \frac{0.5}{n}\right)}}} \]

   10. Taylor expanded in i around 0 64.2%

       \[\leadsto \frac{1}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) + 0.01 \cdot \frac{1}{n}}} \]
   11. Step-by-step derivation

       1. *-commutative 64.2%

          \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01} + 0.01 \cdot \frac{1}{n}} \]

       2. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       3. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       4. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       5. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{\color{blue}{0.5}}{n}\right)\right) \cdot 0.01 + 0.01 \cdot \frac{1}{n}} \]

       6. associate-*r/ 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \color{blue}{\frac{0.01 \cdot 1}{n}}} \]

       7. metadata-eval 64.2%

          \[\leadsto \frac{1}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{\color{blue}{0.01}}{n}} \]

   12. Simplified 64.2%

       \[\leadsto \frac{1}{\color{blue}{\left(i \cdot \left(\frac{0.5}{{n}^{2}} - \frac{0.5}{n}\right)\right) \cdot 0.01 + \frac{0.01}{n}}} \]

   13. Taylor expanded in n around inf 66.3%

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

       1. *-commutative 66.3%

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

   15. Simplified 66.3%

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

   if -1.45e-242 < n < 1.14999999999999994e-211

   1. Initial program 49.8%

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

      1. associate-*r/ 49.8%

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

      2. sub-neg 49.8%

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

      3. distribute-rgt-in 49.8%

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

      4. metadata-eval 49.8%

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

      5. metadata-eval 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in i around 0 81.6%

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

   6. Taylor expanded in i around 0 81.6%

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

   if 1.14999999999999994e-211 < n

   1. Initial program 14.4%

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

      1. associate-/r/ 14.8%

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

      2. *-commutative 14.8%

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

      3. sub-neg 14.8%

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

      4. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in i around 0 69.8%

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

   6. Taylor expanded in n around inf 69.7%

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

      1. *-commutative 69.7%

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

   8. Simplified 69.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-242}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-211}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.9 \cdot 10^{+23}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.9e+23) 0.0 (if (<= i 2.5e-25) (* 100.0 n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.9e+23) {
		tmp = 0.0;
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2.9d+23)) then
        tmp = 0.0d0
    else if (i <= 2.5d-25) then
        tmp = 100.0d0 * n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.9e+23) {
		tmp = 0.0;
	} else if (i <= 2.5e-25) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.9e+23:
		tmp = 0.0
	elif i <= 2.5e-25:
		tmp = 100.0 * n
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.9e+23)
		tmp = 0.0;
	elseif (i <= 2.5e-25)
		tmp = Float64(100.0 * n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.9e+23)
		tmp = 0.0;
	elseif (i <= 2.5e-25)
		tmp = 100.0 * n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.9e+23], 0.0, If[LessEqual[i, 2.5e-25], N[(100.0 * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -2.90000000000000013e23 or 2.49999999999999981e-25 < i

   1. Initial program 52.7%

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

      1. associate-*r/ 52.7%

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

      2. sub-neg 52.7%

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

      3. distribute-rgt-in 52.7%

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

      4. metadata-eval 52.7%

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

      5. metadata-eval 52.7%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in i around 0 30.2%

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

   6. Taylor expanded in i around 0 30.2%

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

   if -2.90000000000000013e23 < i < 2.49999999999999981e-25

   1. Initial program 7.3%

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

      1. associate-/r/ 7.9%

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

      2. *-commutative 7.9%

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

      3. sub-neg 7.9%

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

      4. metadata-eval 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in i around 0 84.2%

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

      1. *-commutative 84.2%

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

   7. Simplified 84.2%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.9 \cdot 10^{+23}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 17.9% 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 24.2%

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

   1. associate-*r/ 24.2%

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

   2. sub-neg 24.2%

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

   3. distribute-rgt-in 24.2%

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

   4. metadata-eval 24.2%

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

   5. metadata-eval 24.2%

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

3. Simplified 24.2%

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

5. Taylor expanded in i around 0 15.7%

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

6. Taylor expanded in i around 0 16.1%

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

7. Final simplification 16.1%

   \[\leadsto 0 \]
8. Add Preprocessing

Developer target: 35.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024048 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))