Compound Interest

Percentage Accurate: 28.1% → 99.4%

Time: 21.3s

Alternatives: 18

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.1% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ t_2 := 100 \cdot \left(\frac{n}{i} \cdot t_0\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* 100.0 (* (/ n i) t_0))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 0.0)
       (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) (/ i n))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = 100.0 * ((n / i) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (100.0 * expm1((n * log1p((i / n))))) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(100.0 * Float64(Float64(n / i) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

      2. clear-num 99.9%

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

      3. sub-neg 99.9%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. neg-mul-1 99.9%

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

      3. distribute-rgt-out 99.9%

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

   5. Simplified 99.9%

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

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

   1. Initial program 25.3%

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

      1. associate-*r/ 25.3%

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

      2. *-commutative 25.3%

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

      3. pow-to-exp 25.3%

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

      4. expm1-def 35.9%

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

      5. add-log-exp 25.3%

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

      6. pow-to-exp 25.3%

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

      7. log-pow 35.9%

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

      8. log1p-udef 99.6%

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

   3. Applied egg-rr 99.6%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{n}{1 + \color{blue}{\left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]

      2. associate-+r+ 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      3. +-commutative 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. *-commutative 100.0%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. fma-def 100.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      6. *-commutative 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      7. unpow2 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      8. associate-*l* 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 100.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 2: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ t_2 := 100 \cdot \left(\frac{n}{i} \cdot t_0\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* 100.0 (* (/ n i) t_0))))
   (if (<= t_1 -5e-17)
     t_2
     (if (<= t_1 0.0)
       (* 100.0 (/ n (/ i (expm1 i))))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = 100.0 * ((n / i) * t_0);
	double tmp;
	if (t_1 <= -5e-17) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(100.0 * Float64(Float64(n / i) * t_0))
	tmp = 0.0
	if (t_1 <= -5e-17)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-17], t$95$2, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.9999999999999999e-17 or -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 98.5%

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

      1. div-sub 98.4%

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

      2. clear-num 98.4%

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

      3. sub-neg 98.4%

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

      4. div-inv 98.4%

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

      5. clear-num 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-commutative 98.4%

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

      2. neg-mul-1 98.4%

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

      3. distribute-rgt-out 98.5%

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

   5. Simplified 98.5%

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

   if -4.9999999999999999e-17 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 24.7%

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

   2. Taylor expanded in n around inf 41.6%

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

      1. *-commutative 41.6%

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

      2. associate-/l* 41.6%

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

      3. expm1-def 79.7%

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

   4. Simplified 79.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{n}{1 + \color{blue}{\left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]

      2. associate-+r+ 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      3. +-commutative 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. *-commutative 100.0%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. fma-def 100.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      6. *-commutative 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      7. unpow2 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      8. associate-*l* 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 100.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 3: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ t_2 := 100 \cdot \left(\frac{n}{i} \cdot t_0\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* 100.0 (* (/ n i) t_0))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 0.0)
       (* 100.0 (* (/ n i) (expm1 (* n (log1p (/ i n))))))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = 100.0 * ((n / i) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * ((n / i) * expm1((n * log1p((i / n)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(100.0 * Float64(Float64(n / i) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(Float64(n / i) * expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

      2. clear-num 99.9%

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

      3. sub-neg 99.9%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. neg-mul-1 99.9%

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

      3. distribute-rgt-out 99.9%

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

   5. Simplified 99.9%

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

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

   1. Initial program 25.3%

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

      1. clear-num 25.3%

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

      2. associate-/r/ 25.2%

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

      3. clear-num 25.2%

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

      4. pow-to-exp 25.2%

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

      5. expm1-def 35.7%

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

      6. add-log-exp 25.2%

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

      7. pow-to-exp 25.2%

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

      8. log-pow 35.7%

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

      9. log1p-udef 95.4%

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

   3. Applied egg-rr 95.4%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{n}{1 + \color{blue}{\left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]

      2. associate-+r+ 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      3. +-commutative 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. *-commutative 100.0%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. fma-def 100.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      6. *-commutative 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      7. unpow2 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      8. associate-*l* 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 100.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 4: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ t_2 := 100 \cdot \left(\frac{n}{i} \cdot t_0\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* 100.0 (* (/ n i) t_0))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 0.0)
       (* n (/ (expm1 (* n (log1p (/ i n)))) (/ i 100.0)))
       (if (<= t_1 INFINITY)
         t_2
         (*
          100.0
          (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = 100.0 * ((n / i) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = n * (expm1((n * log1p((i / n)))) / (i / 100.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(100.0 * Float64(Float64(n / i) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / 100.0)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(100.0 * N[(N[(n / i), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

      2. clear-num 99.9%

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

      3. sub-neg 99.9%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. neg-mul-1 99.9%

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

      3. distribute-rgt-out 99.9%

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

   5. Simplified 99.9%

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

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

   1. Initial program 25.3%

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

      1. associate-*r/ 25.3%

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

      2. *-commutative 25.3%

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

      3. pow-to-exp 25.3%

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

      4. expm1-def 35.9%

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

      5. add-log-exp 25.3%

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

      6. pow-to-exp 25.3%

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

      7. log-pow 35.9%

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

      8. log1p-udef 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. associate-/r/ 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-/l* 97.8%

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

   7. Simplified 97.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

      3. expm1-def 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{n}{1 + \color{blue}{\left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]

      2. associate-+r+ 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      3. +-commutative 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. *-commutative 100.0%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. fma-def 100.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      6. *-commutative 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      7. unpow2 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      8. associate-*l* 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 100.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 5: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -8.6 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{-45}:\\ \;\;\;\;n \cdot \left(100 + \frac{i \cdot -50}{n}\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{+222}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{n} \cdot 33.333333333333336\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -8.6e-32)
     t_0
     (if (<= i 6.8e-45)
       (* n (+ 100.0 (/ (* i -50.0) n)))
       (if (<= i 3.1e+155)
         t_0
         (if (<= i 7.8e+222)
           (* 100.0 (/ n (+ 1.0 (* i -0.5))))
           (* (/ (* i i) n) 33.333333333333336)))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -8.6e-32) {
		tmp = t_0;
	} else if (i <= 6.8e-45) {
		tmp = n * (100.0 + ((i * -50.0) / n));
	} else if (i <= 3.1e+155) {
		tmp = t_0;
	} else if (i <= 7.8e+222) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = ((i * i) / n) * 33.333333333333336;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -8.6e-32) {
		tmp = t_0;
	} else if (i <= 6.8e-45) {
		tmp = n * (100.0 + ((i * -50.0) / n));
	} else if (i <= 3.1e+155) {
		tmp = t_0;
	} else if (i <= 7.8e+222) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = ((i * i) / n) * 33.333333333333336;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -8.6e-32:
		tmp = t_0
	elif i <= 6.8e-45:
		tmp = n * (100.0 + ((i * -50.0) / n))
	elif i <= 3.1e+155:
		tmp = t_0
	elif i <= 7.8e+222:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = ((i * i) / n) * 33.333333333333336
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -8.6e-32)
		tmp = t_0;
	elseif (i <= 6.8e-45)
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i * -50.0) / n)));
	elseif (i <= 3.1e+155)
		tmp = t_0;
	elseif (i <= 7.8e+222)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = Float64(Float64(Float64(i * i) / n) * 33.333333333333336);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -8.6e-32], t$95$0, If[LessEqual[i, 6.8e-45], N[(n * N[(100.0 + N[(N[(i * -50.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.1e+155], t$95$0, If[LessEqual[i, 7.8e+222], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * 33.333333333333336), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -8.5999999999999998e-32 or 6.80000000000000008e-45 < i < 3.09999999999999989e155

   1. Initial program 45.9%

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

   2. Taylor expanded in n around inf 65.1%

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

      1. expm1-def 68.3%

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

   4. Simplified 68.3%

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

   if -8.5999999999999998e-32 < i < 6.80000000000000008e-45

   1. Initial program 5.7%

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

      1. associate-/r/ 6.4%

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

      2. associate-*r* 6.4%

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

      3. *-commutative 6.4%

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

      4. associate-*r/ 6.4%

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

      5. sub-neg 6.4%

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

      6. distribute-lft-in 6.4%

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

      7. fma-def 6.4%

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

      8. metadata-eval 6.4%

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

      9. metadata-eval 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in i around 0 92.2%

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

      1. associate-*r* 92.2%

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

      2. *-commutative 92.2%

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

      3. associate-*r/ 92.2%

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

      4. metadata-eval 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in n around 0 92.2%

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

      1. associate-*r/ 92.2%

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

      2. *-commutative 92.2%

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

   9. Simplified 92.2%

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

   if 3.09999999999999989e155 < i < 7.7999999999999997e222

   1. Initial program 32.3%

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

   2. Taylor expanded in n around inf 16.9%

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

      1. *-commutative 16.9%

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

      2. associate-/l* 16.9%

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

      3. expm1-def 16.9%

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

   4. Simplified 16.9%

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

   5. Taylor expanded in i around 0 52.9%

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

      1. *-commutative 52.9%

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

   7. Simplified 52.9%

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

   if 7.7999999999999997e222 < i

   1. Initial program 82.0%

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

      1. associate-/r/ 82.0%

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

      2. associate-*r* 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r/ 82.0%

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

      5. sub-neg 82.0%

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

      6. distribute-lft-in 82.0%

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

      7. fma-def 82.0%

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

      8. metadata-eval 82.0%

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

      9. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in i around 0 54.8%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 54.8%

         \[\leadsto n \cdot \left(100 + \color{blue}{\left(100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      2. distribute-lft-out 54.8%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

   6. Simplified 54.8%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)} \]

   7. Taylor expanded in n around 0 55.0%

      \[\leadsto \color{blue}{33.333333333333336 \cdot \frac{{i}^{2}}{n}} \]
   8. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto \color{blue}{\frac{{i}^{2}}{n} \cdot 33.333333333333336} \]

      2. unpow2 55.0%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{n} \cdot 33.333333333333336 \]

   9. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{i \cdot i}{n} \cdot 33.333333333333336} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.6 \cdot 10^{-32}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{-45}:\\ \;\;\;\;n \cdot \left(100 + \frac{i \cdot -50}{n}\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+155}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 7.8 \cdot 10^{+222}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{n} \cdot 33.333333333333336\\ \end{array} \]

Alternative 6: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5.8 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \frac{n}{\frac{i}{100}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{-45}:\\ \;\;\;\;n \cdot \left(100 + \frac{i \cdot -50}{n}\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+155}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 4.8 \cdot 10^{+223}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{n} \cdot 33.333333333333336\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -5.8e-18)
   (* (expm1 i) (/ n (/ i 100.0)))
   (if (<= i 7e-45)
     (* n (+ 100.0 (/ (* i -50.0) n)))
     (if (<= i 3.1e+155)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= i 4.8e+223)
         (* 100.0 (/ n (+ 1.0 (* i -0.5))))
         (* (/ (* i i) n) 33.333333333333336))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -5.8e-18) {
		tmp = expm1(i) * (n / (i / 100.0));
	} else if (i <= 7e-45) {
		tmp = n * (100.0 + ((i * -50.0) / n));
	} else if (i <= 3.1e+155) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (i <= 4.8e+223) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = ((i * i) / n) * 33.333333333333336;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -5.8e-18) {
		tmp = Math.expm1(i) * (n / (i / 100.0));
	} else if (i <= 7e-45) {
		tmp = n * (100.0 + ((i * -50.0) / n));
	} else if (i <= 3.1e+155) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (i <= 4.8e+223) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = ((i * i) / n) * 33.333333333333336;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -5.8e-18:
		tmp = math.expm1(i) * (n / (i / 100.0))
	elif i <= 7e-45:
		tmp = n * (100.0 + ((i * -50.0) / n))
	elif i <= 3.1e+155:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif i <= 4.8e+223:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = ((i * i) / n) * 33.333333333333336
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -5.8e-18)
		tmp = Float64(expm1(i) * Float64(n / Float64(i / 100.0)));
	elseif (i <= 7e-45)
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i * -50.0) / n)));
	elseif (i <= 3.1e+155)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (i <= 4.8e+223)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = Float64(Float64(Float64(i * i) / n) * 33.333333333333336);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -5.8e-18], N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7e-45], N[(n * N[(100.0 + N[(N[(i * -50.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.1e+155], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.8e+223], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * 33.333333333333336), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -5.8e-18

   1. Initial program 61.7%

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

   2. Taylor expanded in n around inf 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-/l* 72.4%

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

      3. expm1-def 73.6%

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

   4. Simplified 73.6%

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

      1. associate-*l/ 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. *-un-lft-identity 70.0%

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

      2. associate-/r/ 70.0%

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

   8. Applied egg-rr 70.0%

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

      1. *-lft-identity 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/l* 73.6%

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

   10. Simplified 73.6%

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

   if -5.8e-18 < i < 7e-45

   1. Initial program 5.6%

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

      1. associate-/r/ 6.2%

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

      2. associate-*r* 6.2%

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

      3. *-commutative 6.2%

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

      4. associate-*r/ 6.2%

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

      5. sub-neg 6.2%

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

      6. distribute-lft-in 6.2%

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

      7. fma-def 6.2%

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

      8. metadata-eval 6.2%

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

      9. metadata-eval 6.2%

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

   3. Simplified 6.2%

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

   4. Taylor expanded in i around 0 91.7%

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

      1. associate-*r* 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*r/ 91.7%

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

      4. metadata-eval 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in n around 0 91.7%

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

      1. associate-*r/ 91.7%

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

      2. *-commutative 91.7%

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

   9. Simplified 91.7%

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

   if 7e-45 < i < 3.09999999999999989e155

   1. Initial program 32.4%

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

   2. Taylor expanded in n around inf 63.4%

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

      1. expm1-def 61.1%

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

   4. Simplified 61.1%

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

   if 3.09999999999999989e155 < i < 4.80000000000000022e223

   1. Initial program 32.3%

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

   2. Taylor expanded in n around inf 16.9%

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

      1. *-commutative 16.9%

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

      2. associate-/l* 16.9%

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

      3. expm1-def 16.9%

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

   4. Simplified 16.9%

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

   5. Taylor expanded in i around 0 52.9%

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

      1. *-commutative 52.9%

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

   7. Simplified 52.9%

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

   if 4.80000000000000022e223 < i

   1. Initial program 82.0%

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

      1. associate-/r/ 82.0%

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

      2. associate-*r* 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r/ 82.0%

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

      5. sub-neg 82.0%

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

      6. distribute-lft-in 82.0%

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

      7. fma-def 82.0%

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

      8. metadata-eval 82.0%

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

      9. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in i around 0 54.8%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 54.8%

         \[\leadsto n \cdot \left(100 + \color{blue}{\left(100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      2. distribute-lft-out 54.8%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

   6. Simplified 54.8%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)} \]

   7. Taylor expanded in n around 0 55.0%

      \[\leadsto \color{blue}{33.333333333333336 \cdot \frac{{i}^{2}}{n}} \]
   8. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto \color{blue}{\frac{{i}^{2}}{n} \cdot 33.333333333333336} \]

      2. unpow2 55.0%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{n} \cdot 33.333333333333336 \]

   9. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{i \cdot i}{n} \cdot 33.333333333333336} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.8 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \frac{n}{\frac{i}{100}}\\ \mathbf{elif}\;i \leq 7 \cdot 10^{-45}:\\ \;\;\;\;n \cdot \left(100 + \frac{i \cdot -50}{n}\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+155}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 4.8 \cdot 10^{+223}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{n} \cdot 33.333333333333336\\ \end{array} \]

Alternative 7: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{-268} \lor \neg \left(n \leq 5.1 \cdot 10^{-28}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3.2e-268) (not (<= n 5.1e-28)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/ (* n (* n -10000.0)) (- (* i (* n 50.0)) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3.2e-268) || !(n <= 5.1e-28)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -3.2e-268) || !(n <= 5.1e-28)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -3.2e-268) or not (n <= 5.1e-28):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3.2e-268) || !(n <= 5.1e-28))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(Float64(n * Float64(n * -10000.0)) / Float64(Float64(i * Float64(n * 50.0)) - Float64(n * 100.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -3.2e-268], N[Not[LessEqual[n, 5.1e-28]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(n * -10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision] - N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.1999999999999999e-268 or 5.10000000000000009e-28 < n

   1. Initial program 26.8%

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

   2. Taylor expanded in n around inf 36.2%

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

      1. *-commutative 36.2%

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

      2. associate-/l* 36.2%

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

      3. expm1-def 86.0%

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

   4. Simplified 86.0%

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

   if -3.1999999999999999e-268 < n < 5.10000000000000009e-28

   1. Initial program 31.0%

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

   2. Taylor expanded in n around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 23.2%

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

      3. expm1-def 30.6%

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

   4. Simplified 30.6%

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

   5. Taylor expanded in i around 0 29.6%

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

      1. flip-+ 49.2%

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

      2. associate-*r* 49.2%

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

      3. associate-*r* 49.2%

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

      4. *-commutative 49.2%

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

      5. *-commutative 49.2%

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

      6. associate-*r* 49.2%

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

      7. *-commutative 49.2%

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

   7. Applied egg-rr 49.2%

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

      1. associate-*l* 49.2%

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

      2. *-commutative 49.2%

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

      3. *-commutative 49.2%

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

      4. *-commutative 49.2%

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

      5. swap-sqr 48.9%

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

      6. metadata-eval 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot \color{blue}{10000}}{\left(50 \cdot n\right) \cdot i - n \cdot 100} \]

      7. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{\color{blue}{i \cdot \left(50 \cdot n\right)} - n \cdot 100} \]

      8. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \color{blue}{\left(n \cdot 50\right)} - n \cdot 100} \]

      9. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - \color{blue}{100 \cdot n}} \]

   9. Simplified 48.9%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n}} \]

   10. Taylor expanded in i around 0 73.7%

       \[\leadsto \frac{\color{blue}{-10000 \cdot {n}^{2}}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]
   11. Step-by-step derivation

       1. *-commutative 73.7%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot -10000}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       2. unpow2 73.7%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot -10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       3. associate-*r* 73.8%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   12. Simplified 73.8%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{-268} \lor \neg \left(n \leq 5.1 \cdot 10^{-28}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \end{array} \]

Alternative 8: 65.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(n \cdot 50\right)\\ t_1 := t_0 - n \cdot 100\\ \mathbf{if}\;n \leq -2.35 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{t_1}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(n \cdot 50\right) \cdot \left(i \cdot t_0\right) - \left(n \cdot n\right) \cdot 10000}{t_1}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* i (* n 50.0))) (t_1 (- t_0 (* n 100.0))))
   (if (<= n -2.35e-268)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 1.55e-15)
       (/ (* n (* n -10000.0)) t_1)
       (if (<= n 1.5e+151)
         (/ (- (* (* n 50.0) (* i t_0)) (* (* n n) 10000.0)) t_1)
         (*
          n
          (+
           100.0
           (*
            100.0
            (+ (* i (* i 0.16666666666666666)) (* i (- 0.5 (/ 0.5 n))))))))))))

C

double code(double i, double n) {
	double t_0 = i * (n * 50.0);
	double t_1 = t_0 - (n * 100.0);
	double tmp;
	if (n <= -2.35e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.55e-15) {
		tmp = (n * (n * -10000.0)) / t_1;
	} else if (n <= 1.5e+151) {
		tmp = (((n * 50.0) * (i * t_0)) - ((n * n) * 10000.0)) / t_1;
	} else {
		tmp = n * (100.0 + (100.0 * ((i * (i * 0.16666666666666666)) + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = i * (n * 50.0d0)
    t_1 = t_0 - (n * 100.0d0)
    if (n <= (-2.35d-268)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1.55d-15) then
        tmp = (n * (n * (-10000.0d0))) / t_1
    else if (n <= 1.5d+151) then
        tmp = (((n * 50.0d0) * (i * t_0)) - ((n * n) * 10000.0d0)) / t_1
    else
        tmp = n * (100.0d0 + (100.0d0 * ((i * (i * 0.16666666666666666d0)) + (i * (0.5d0 - (0.5d0 / n))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = i * (n * 50.0);
	double t_1 = t_0 - (n * 100.0);
	double tmp;
	if (n <= -2.35e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.55e-15) {
		tmp = (n * (n * -10000.0)) / t_1;
	} else if (n <= 1.5e+151) {
		tmp = (((n * 50.0) * (i * t_0)) - ((n * n) * 10000.0)) / t_1;
	} else {
		tmp = n * (100.0 + (100.0 * ((i * (i * 0.16666666666666666)) + (i * (0.5 - (0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = i * (n * 50.0)
	t_1 = t_0 - (n * 100.0)
	tmp = 0
	if n <= -2.35e-268:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1.55e-15:
		tmp = (n * (n * -10000.0)) / t_1
	elif n <= 1.5e+151:
		tmp = (((n * 50.0) * (i * t_0)) - ((n * n) * 10000.0)) / t_1
	else:
		tmp = n * (100.0 + (100.0 * ((i * (i * 0.16666666666666666)) + (i * (0.5 - (0.5 / n))))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(i * Float64(n * 50.0))
	t_1 = Float64(t_0 - Float64(n * 100.0))
	tmp = 0.0
	if (n <= -2.35e-268)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1.55e-15)
		tmp = Float64(Float64(n * Float64(n * -10000.0)) / t_1);
	elseif (n <= 1.5e+151)
		tmp = Float64(Float64(Float64(Float64(n * 50.0) * Float64(i * t_0)) - Float64(Float64(n * n) * 10000.0)) / t_1);
	else
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(Float64(i * Float64(i * 0.16666666666666666)) + Float64(i * Float64(0.5 - Float64(0.5 / n)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = i * (n * 50.0);
	t_1 = t_0 - (n * 100.0);
	tmp = 0.0;
	if (n <= -2.35e-268)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1.55e-15)
		tmp = (n * (n * -10000.0)) / t_1;
	elseif (n <= 1.5e+151)
		tmp = (((n * 50.0) * (i * t_0)) - ((n * n) * 10000.0)) / t_1;
	else
		tmp = n * (100.0 + (100.0 * ((i * (i * 0.16666666666666666)) + (i * (0.5 - (0.5 / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(n * 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.35e-268], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.55e-15], N[(N[(n * N[(n * -10000.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[n, 1.5e+151], N[(N[(N[(N[(n * 50.0), $MachinePrecision] * N[(i * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 10000.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(n * N[(100.0 + N[(100.0 * N[(N[(i * N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.34999999999999987e-268

   1. Initial program 30.1%

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

   2. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/l* 33.5%

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

      3. expm1-def 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in i around 0 63.0%

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

      1. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   if -2.34999999999999987e-268 < n < 1.5499999999999999e-15

   1. Initial program 30.5%

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

   2. Taylor expanded in n around inf 22.8%

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

      1. *-commutative 22.8%

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

      2. associate-/l* 22.8%

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

      3. expm1-def 31.9%

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

   4. Simplified 31.9%

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

   5. Taylor expanded in i around 0 31.0%

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

      1. flip-+ 50.2%

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

      2. associate-*r* 50.2%

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

      3. associate-*r* 50.2%

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

      4. *-commutative 50.2%

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

      5. *-commutative 50.2%

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

      6. associate-*r* 50.2%

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

      7. *-commutative 50.2%

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

   7. Applied egg-rr 50.2%

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

      1. associate-*l* 50.2%

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

      2. *-commutative 50.2%

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

      3. *-commutative 50.2%

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

      4. *-commutative 50.2%

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

      5. swap-sqr 49.9%

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

      6. metadata-eval 49.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot \color{blue}{10000}}{\left(50 \cdot n\right) \cdot i - n \cdot 100} \]

      7. *-commutative 49.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{\color{blue}{i \cdot \left(50 \cdot n\right)} - n \cdot 100} \]

      8. *-commutative 49.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \color{blue}{\left(n \cdot 50\right)} - n \cdot 100} \]

      9. *-commutative 49.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - \color{blue}{100 \cdot n}} \]

   9. Simplified 49.9%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n}} \]

   10. Taylor expanded in i around 0 74.2%

       \[\leadsto \frac{\color{blue}{-10000 \cdot {n}^{2}}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]
   11. Step-by-step derivation

       1. *-commutative 74.2%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot -10000}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       2. unpow2 74.2%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot -10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       3. associate-*r* 74.3%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   12. Simplified 74.3%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   if 1.5499999999999999e-15 < n < 1.5e151

   1. Initial program 27.6%

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

   2. Taylor expanded in n around inf 33.1%

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

      1. *-commutative 33.1%

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

      2. associate-/l* 33.1%

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

      3. expm1-def 91.5%

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

   4. Simplified 91.5%

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

   5. Taylor expanded in i around 0 62.0%

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

      1. flip-+ 85.9%

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

      2. associate-*r* 85.9%

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

      3. associate-*r* 85.9%

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

      4. *-commutative 85.9%

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

      5. *-commutative 85.9%

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

      6. associate-*r* 85.9%

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

      7. *-commutative 85.9%

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

   7. Applied egg-rr 85.9%

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

      1. associate-*l* 85.9%

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

      2. *-commutative 85.9%

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

      3. *-commutative 85.9%

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

      4. *-commutative 85.9%

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

      5. swap-sqr 85.5%

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

      6. metadata-eval 85.5%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot \color{blue}{10000}}{\left(50 \cdot n\right) \cdot i - n \cdot 100} \]

      7. *-commutative 85.5%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{\color{blue}{i \cdot \left(50 \cdot n\right)} - n \cdot 100} \]

      8. *-commutative 85.5%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \color{blue}{\left(n \cdot 50\right)} - n \cdot 100} \]

      9. *-commutative 85.5%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - \color{blue}{100 \cdot n}} \]

   9. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n}} \]

   if 1.5e151 < n

   1. Initial program 17.2%

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

      1. associate-/r/ 17.8%

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

      2. associate-*r* 17.8%

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

      3. *-commutative 17.8%

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

      4. associate-*r/ 17.8%

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

      5. sub-neg 17.8%

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

      6. distribute-lft-in 17.8%

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

      7. fma-def 17.8%

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

      8. metadata-eval 17.8%

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

      9. metadata-eval 17.8%

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

   3. Simplified 17.8%

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

   4. Taylor expanded in i around 0 82.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 82.1%

         \[\leadsto n \cdot \left(100 + \color{blue}{\left(100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      2. distribute-lft-out 82.1%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

   6. Simplified 82.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)} \]

   7. Taylor expanded in n around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. unpow2 82.1%

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

      3. associate-*l* 82.1%

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

   9. Simplified 82.1%

      \[\leadsto n \cdot \left(100 + 100 \cdot \left(\color{blue}{i \cdot \left(i \cdot 0.16666666666666666\right)} + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.35 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 9: 66.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3.45 \cdot 10^{-28}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.1e-268)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 3.45e-28)
     (/ (* n (* n -10000.0)) (- (* i (* n 50.0)) (* n 100.0)))
     (*
      n
      (+
       100.0
       (*
        100.0
        (+ (* i (* i 0.16666666666666666)) (* i (- 0.5 (/ 0.5 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.1e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 3.45e-28) {
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	} else {
		tmp = n * (100.0 + (100.0 * ((i * (i * 0.16666666666666666)) + (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 <= (-4.1d-268)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 3.45d-28) then
        tmp = (n * (n * (-10000.0d0))) / ((i * (n * 50.0d0)) - (n * 100.0d0))
    else
        tmp = n * (100.0d0 + (100.0d0 * ((i * (i * 0.16666666666666666d0)) + (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 <= -4.1e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 3.45e-28) {
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	} else {
		tmp = n * (100.0 + (100.0 * ((i * (i * 0.16666666666666666)) + (i * (0.5 - (0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.1e-268:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 3.45e-28:
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0))
	else:
		tmp = n * (100.0 + (100.0 * ((i * (i * 0.16666666666666666)) + (i * (0.5 - (0.5 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.1e-268)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 3.45e-28)
		tmp = Float64(Float64(n * Float64(n * -10000.0)) / Float64(Float64(i * Float64(n * 50.0)) - Float64(n * 100.0)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(Float64(i * Float64(i * 0.16666666666666666)) + 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 <= -4.1e-268)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 3.45e-28)
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	else
		tmp = n * (100.0 + (100.0 * ((i * (i * 0.16666666666666666)) + (i * (0.5 - (0.5 / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.1e-268], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.45e-28], N[(N[(n * N[(n * -10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision] - N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(100.0 * N[(N[(i * N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.0999999999999999e-268

   1. Initial program 30.1%

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

   2. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/l* 33.5%

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

      3. expm1-def 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in i around 0 63.0%

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

      1. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   if -4.0999999999999999e-268 < n < 3.45000000000000001e-28

   1. Initial program 31.0%

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

   2. Taylor expanded in n around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 23.2%

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

      3. expm1-def 30.6%

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

   4. Simplified 30.6%

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

   5. Taylor expanded in i around 0 29.6%

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

      1. flip-+ 49.2%

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

      2. associate-*r* 49.2%

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

      3. associate-*r* 49.2%

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

      4. *-commutative 49.2%

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

      5. *-commutative 49.2%

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

      6. associate-*r* 49.2%

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

      7. *-commutative 49.2%

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

   7. Applied egg-rr 49.2%

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

      1. associate-*l* 49.2%

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

      2. *-commutative 49.2%

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

      3. *-commutative 49.2%

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

      4. *-commutative 49.2%

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

      5. swap-sqr 48.9%

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

      6. metadata-eval 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot \color{blue}{10000}}{\left(50 \cdot n\right) \cdot i - n \cdot 100} \]

      7. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{\color{blue}{i \cdot \left(50 \cdot n\right)} - n \cdot 100} \]

      8. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \color{blue}{\left(n \cdot 50\right)} - n \cdot 100} \]

      9. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - \color{blue}{100 \cdot n}} \]

   9. Simplified 48.9%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n}} \]

   10. Taylor expanded in i around 0 73.7%

       \[\leadsto \frac{\color{blue}{-10000 \cdot {n}^{2}}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]
   11. Step-by-step derivation

       1. *-commutative 73.7%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot -10000}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       2. unpow2 73.7%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot -10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       3. associate-*r* 73.8%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   12. Simplified 73.8%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   if 3.45000000000000001e-28 < n

   1. Initial program 21.6%

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

      1. associate-/r/ 22.0%

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

      2. associate-*r* 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-*r/ 22.0%

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

      5. sub-neg 22.0%

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

      6. distribute-lft-in 22.0%

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

      7. fma-def 22.0%

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

      8. metadata-eval 22.0%

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

      9. metadata-eval 22.0%

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

   3. Simplified 22.0%

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

   4. Taylor expanded in i around 0 78.2%

      \[\leadsto n \cdot \color{blue}{\left(100 + \left(100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 78.2%

         \[\leadsto n \cdot \left(100 + \color{blue}{\left(100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) + 100 \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      2. distribute-lft-out 78.2%

         \[\leadsto n \cdot \left(100 + \color{blue}{100 \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

   6. Simplified 78.2%

      \[\leadsto n \cdot \color{blue}{\left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)} \]

   7. Taylor expanded in n around inf 78.2%

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

      1. *-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. associate-*l* 78.2%

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

   9. Simplified 78.2%

      \[\leadsto n \cdot \left(100 + 100 \cdot \left(\color{blue}{i \cdot \left(i \cdot 0.16666666666666666\right)} + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3.45 \cdot 10^{-28}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(i \cdot 0.16666666666666666\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 10: 66.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.1e-268)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 1.6e-23)
     (/ (* n (* n -10000.0)) (- (* i (* n 50.0)) (* n 100.0)))
     (* n (/ (* 100.0 (+ i (* (* i i) (- 0.5 (/ 0.5 n))))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.1e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.6e-23) {
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	} else {
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.1d-268)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1.6d-23) then
        tmp = (n * (n * (-10000.0d0))) / ((i * (n * 50.0d0)) - (n * 100.0d0))
    else
        tmp = n * ((100.0d0 * (i + ((i * i) * (0.5d0 - (0.5d0 / n))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.1e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.6e-23) {
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	} else {
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.1e-268:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1.6e-23:
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0))
	else:
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.1e-268)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1.6e-23)
		tmp = Float64(Float64(n * Float64(n * -10000.0)) / Float64(Float64(i * Float64(n * 50.0)) - Float64(n * 100.0)));
	else
		tmp = Float64(n * Float64(Float64(100.0 * Float64(i + Float64(Float64(i * i) * Float64(0.5 - Float64(0.5 / n))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.1e-268)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1.6e-23)
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	else
		tmp = n * ((100.0 * (i + ((i * i) * (0.5 - (0.5 / n))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.1e-268], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.6e-23], N[(N[(n * N[(n * -10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision] - N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(i + N[(N[(i * i), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.0999999999999999e-268

   1. Initial program 30.1%

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

   2. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/l* 33.5%

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

      3. expm1-def 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in i around 0 63.0%

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

      1. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   if -4.0999999999999999e-268 < n < 1.59999999999999988e-23

   1. Initial program 31.0%

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

   2. Taylor expanded in n around inf 23.2%

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

      1. *-commutative 23.2%

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

      2. associate-/l* 23.2%

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

      3. expm1-def 30.6%

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

   4. Simplified 30.6%

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

   5. Taylor expanded in i around 0 29.6%

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

      1. flip-+ 49.2%

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

      2. associate-*r* 49.2%

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

      3. associate-*r* 49.2%

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

      4. *-commutative 49.2%

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

      5. *-commutative 49.2%

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

      6. associate-*r* 49.2%

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

      7. *-commutative 49.2%

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

   7. Applied egg-rr 49.2%

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

      1. associate-*l* 49.2%

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

      2. *-commutative 49.2%

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

      3. *-commutative 49.2%

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

      4. *-commutative 49.2%

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

      5. swap-sqr 48.9%

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

      6. metadata-eval 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot \color{blue}{10000}}{\left(50 \cdot n\right) \cdot i - n \cdot 100} \]

      7. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{\color{blue}{i \cdot \left(50 \cdot n\right)} - n \cdot 100} \]

      8. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \color{blue}{\left(n \cdot 50\right)} - n \cdot 100} \]

      9. *-commutative 48.9%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - \color{blue}{100 \cdot n}} \]

   9. Simplified 48.9%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n}} \]

   10. Taylor expanded in i around 0 73.7%

       \[\leadsto \frac{\color{blue}{-10000 \cdot {n}^{2}}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]
   11. Step-by-step derivation

       1. *-commutative 73.7%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot -10000}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       2. unpow2 73.7%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot -10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       3. associate-*r* 73.8%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   12. Simplified 73.8%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   if 1.59999999999999988e-23 < n

   1. Initial program 21.6%

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

      1. associate-/r/ 22.0%

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

      2. associate-*r* 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-*r/ 22.0%

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

      5. sub-neg 22.0%

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

      6. distribute-lft-in 22.0%

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

      7. fma-def 22.0%

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

      8. metadata-eval 22.0%

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

      9. metadata-eval 22.0%

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

   3. Simplified 22.0%

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

   4. Taylor expanded in i around 0 75.5%

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

      1. distribute-lft-out 75.5%

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

      2. unpow2 75.5%

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

      3. associate-*r/ 75.5%

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

      4. metadata-eval 75.5%

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

   6. Simplified 75.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i + \left(i \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)}{i}\\ \end{array} \]

Alternative 11: 64.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4e-268)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 7.8e+45)
     (/ (* n (* n -10000.0)) (- (* i (* n 50.0)) (* n 100.0)))
     (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7.8e+45) {
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	} 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 <= (-4d-268)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 7.8d+45) then
        tmp = (n * (n * (-10000.0d0))) / ((i * (n * 50.0d0)) - (n * 100.0d0))
    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 <= -4e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7.8e+45) {
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4e-268:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 7.8e+45:
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4e-268)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 7.8e+45)
		tmp = Float64(Float64(n * Float64(n * -10000.0)) / Float64(Float64(i * Float64(n * 50.0)) - Float64(n * 100.0)));
	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 <= -4e-268)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 7.8e+45)
		tmp = (n * (n * -10000.0)) / ((i * (n * 50.0)) - (n * 100.0));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4e-268], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.8e+45], N[(N[(n * N[(n * -10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision] - N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.99999999999999983e-268

   1. Initial program 30.1%

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

   2. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/l* 33.5%

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

      3. expm1-def 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in i around 0 63.0%

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

      1. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   if -3.99999999999999983e-268 < n < 7.7999999999999999e45

   1. Initial program 33.0%

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

   2. Taylor expanded in n around inf 24.0%

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

      1. *-commutative 24.0%

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

      2. associate-/l* 24.0%

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

      3. expm1-def 42.7%

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

   4. Simplified 42.7%

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

   5. Taylor expanded in i around 0 36.4%

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

      1. flip-+ 55.5%

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

      2. associate-*r* 55.5%

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

      3. associate-*r* 55.5%

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

      4. *-commutative 55.5%

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

      5. *-commutative 55.5%

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

      6. associate-*r* 55.5%

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

      7. *-commutative 55.5%

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

   7. Applied egg-rr 55.5%

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

      1. associate-*l* 55.5%

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

      2. *-commutative 55.5%

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

      3. *-commutative 55.5%

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

      4. *-commutative 55.5%

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

      5. swap-sqr 55.2%

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

      6. metadata-eval 55.2%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot \color{blue}{10000}}{\left(50 \cdot n\right) \cdot i - n \cdot 100} \]

      7. *-commutative 55.2%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{\color{blue}{i \cdot \left(50 \cdot n\right)} - n \cdot 100} \]

      8. *-commutative 55.2%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \color{blue}{\left(n \cdot 50\right)} - n \cdot 100} \]

      9. *-commutative 55.2%

         \[\leadsto \frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - \color{blue}{100 \cdot n}} \]

   9. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 50\right) \cdot \left(i \cdot \left(i \cdot \left(n \cdot 50\right)\right)\right) - \left(n \cdot n\right) \cdot 10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n}} \]

   10. Taylor expanded in i around 0 70.0%

       \[\leadsto \frac{\color{blue}{-10000 \cdot {n}^{2}}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]
   11. Step-by-step derivation

       1. *-commutative 70.0%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot -10000}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       2. unpow2 70.0%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot -10000}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

       3. associate-*r* 70.1%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   12. Simplified 70.1%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot -10000\right)}}{i \cdot \left(n \cdot 50\right) - 100 \cdot n} \]

   if 7.7999999999999999e45 < n

   1. Initial program 17.1%

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

      1. associate-/r/ 17.6%

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

      2. associate-*r* 17.6%

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

      3. *-commutative 17.6%

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

      4. associate-*r/ 17.6%

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

      5. sub-neg 17.6%

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

      6. distribute-lft-in 17.6%

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

      7. fma-def 17.6%

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

      8. metadata-eval 17.6%

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

      9. metadata-eval 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in i around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*r/ 77.2%

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

      4. metadata-eval 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in n around inf 77.2%

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

      1. *-commutative 77.2%

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

   9. Simplified 77.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot -10000\right)}{i \cdot \left(n \cdot 50\right) - n \cdot 100}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 12: 60.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{i} \cdot -200\\ \mathbf{if}\;i \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{+49}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+155} \lor \neg \left(i \leq 1.45 \cdot 10^{+238}\right):\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ n i) -200.0)))
   (if (<= i -2.0)
     t_0
     (if (<= i 2.2e+49)
       (* n 100.0)
       (if (or (<= i 3.1e+155) (not (<= i 1.45e+238)))
         (* 50.0 (* i n))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = (n / i) * -200.0;
	double tmp;
	if (i <= -2.0) {
		tmp = t_0;
	} else if (i <= 2.2e+49) {
		tmp = n * 100.0;
	} else if ((i <= 3.1e+155) || !(i <= 1.45e+238)) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (n / i) * (-200.0d0)
    if (i <= (-2.0d0)) then
        tmp = t_0
    else if (i <= 2.2d+49) then
        tmp = n * 100.0d0
    else if ((i <= 3.1d+155) .or. (.not. (i <= 1.45d+238))) then
        tmp = 50.0d0 * (i * n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (n / i) * -200.0;
	double tmp;
	if (i <= -2.0) {
		tmp = t_0;
	} else if (i <= 2.2e+49) {
		tmp = n * 100.0;
	} else if ((i <= 3.1e+155) || !(i <= 1.45e+238)) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n / i) * -200.0
	tmp = 0
	if i <= -2.0:
		tmp = t_0
	elif i <= 2.2e+49:
		tmp = n * 100.0
	elif (i <= 3.1e+155) or not (i <= 1.45e+238):
		tmp = 50.0 * (i * n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n / i) * -200.0)
	tmp = 0.0
	if (i <= -2.0)
		tmp = t_0;
	elseif (i <= 2.2e+49)
		tmp = Float64(n * 100.0);
	elseif ((i <= 3.1e+155) || !(i <= 1.45e+238))
		tmp = Float64(50.0 * Float64(i * n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n / i) * -200.0;
	tmp = 0.0;
	if (i <= -2.0)
		tmp = t_0;
	elseif (i <= 2.2e+49)
		tmp = n * 100.0;
	elseif ((i <= 3.1e+155) || ~((i <= 1.45e+238)))
		tmp = 50.0 * (i * n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]}, If[LessEqual[i, -2.0], t$95$0, If[LessEqual[i, 2.2e+49], N[(n * 100.0), $MachinePrecision], If[Or[LessEqual[i, 3.1e+155], N[Not[LessEqual[i, 1.45e+238]], $MachinePrecision]], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2 or 3.09999999999999989e155 < i < 1.4500000000000001e238

   1. Initial program 56.6%

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

   2. Taylor expanded in n around inf 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-/l* 59.4%

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

      3. expm1-def 59.4%

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

   4. Simplified 59.4%

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

   5. Taylor expanded in i around 0 36.3%

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

      1. *-commutative 36.3%

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

   7. Simplified 36.3%

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

   8. Taylor expanded in i around inf 36.3%

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

   if -2 < i < 2.2000000000000001e49

   1. Initial program 11.6%

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

   2. Taylor expanded in i around 0 79.2%

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

      1. *-commutative 79.2%

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

   4. Simplified 79.2%

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

   if 2.2000000000000001e49 < i < 3.09999999999999989e155 or 1.4500000000000001e238 < i

   1. Initial program 39.6%

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

   2. Taylor expanded in n around inf 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-/l* 68.2%

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

      3. expm1-def 68.2%

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

   4. Simplified 68.2%

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

   5. Taylor expanded in i around 0 41.5%

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

   6. Taylor expanded in i around inf 41.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{+49}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+155} \lor \neg \left(i \leq 1.45 \cdot 10^{+238}\right):\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \]

Alternative 13: 63.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -1.55 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -4.1 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-136}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -1.55e+55)
     t_0
     (if (<= n -4.1e-268)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 9.5e-136) (* 100.0 (/ 0.0 (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.55e+55) {
		tmp = t_0;
	} else if (n <= -4.1e-268) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 9.5e-136) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-1.55d+55)) then
        tmp = t_0
    else if (n <= (-4.1d-268)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (n <= 9.5d-136) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.55e+55) {
		tmp = t_0;
	} else if (n <= -4.1e-268) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 9.5e-136) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -1.55e+55:
		tmp = t_0
	elif n <= -4.1e-268:
		tmp = 100.0 * (i / (i / n))
	elif n <= 9.5e-136:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -1.55e+55)
		tmp = t_0;
	elseif (n <= -4.1e-268)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 9.5e-136)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -1.55e+55)
		tmp = t_0;
	elseif (n <= -4.1e-268)
		tmp = 100.0 * (i / (i / n));
	elseif (n <= 9.5e-136)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.55e+55], t$95$0, If[LessEqual[n, -4.1e-268], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e-136], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.54999999999999997e55 or 9.5000000000000007e-136 < n

   1. Initial program 21.9%

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

      1. associate-/r/ 22.3%

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

      2. associate-*r* 22.3%

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

      3. *-commutative 22.3%

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

      4. associate-*r/ 22.3%

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

      5. sub-neg 22.3%

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

      6. distribute-lft-in 22.3%

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

      7. fma-def 22.3%

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

      8. metadata-eval 22.3%

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

      9. metadata-eval 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in i around 0 65.0%

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

      1. associate-*r* 65.0%

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

      2. *-commutative 65.0%

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

      3. associate-*r/ 65.0%

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

      4. metadata-eval 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in n around inf 65.0%

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

      1. *-commutative 65.0%

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

   9. Simplified 65.0%

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

   if -1.54999999999999997e55 < n < -4.0999999999999999e-268

   1. Initial program 36.4%

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

   2. Taylor expanded in i around 0 66.8%

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

   if -4.0999999999999999e-268 < n < 9.5000000000000007e-136

   1. Initial program 41.1%

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

   2. Taylor expanded in i around 0 73.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+55}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -4.1 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-136}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 14: 63.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+55} \lor \neg \left(n \leq 6 \cdot 10^{+45}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.4e+55) (not (<= n 6e+45)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e+55) || !(n <= 6e+45)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.4d+55)) .or. (.not. (n <= 6d+45))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e+55) || !(n <= 6e+45)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.4e+55) or not (n <= 6e+45):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.4e+55) || !(n <= 6e+45))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.4e+55) || ~((n <= 6e+45)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.4e+55], N[Not[LessEqual[n, 6e+45]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.4e55 or 6.00000000000000021e45 < n

   1. Initial program 21.6%

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

      1. associate-/r/ 22.2%

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

      2. associate-*r* 22.2%

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

      3. *-commutative 22.2%

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

      4. associate-*r/ 22.2%

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

      5. sub-neg 22.2%

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

      6. distribute-lft-in 22.2%

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

      7. fma-def 22.2%

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

      8. metadata-eval 22.2%

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

      9. metadata-eval 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in i around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. *-commutative 66.5%

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

      3. associate-*r/ 66.5%

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

      4. metadata-eval 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in n around inf 66.5%

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

      1. *-commutative 66.5%

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

   9. Simplified 66.5%

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

   if -1.4e55 < n < 6.00000000000000021e45

   1. Initial program 34.5%

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

   2. Taylor expanded in i around 0 61.2%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+55} \lor \neg \left(n \leq 6 \cdot 10^{+45}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 15: 63.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{-136}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.7e-268)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 7.6e-136) (* 100.0 (/ 0.0 (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.7e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7.6e-136) {
		tmp = 100.0 * (0.0 / (i / n));
	} 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 <= (-3.7d-268)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 7.6d-136) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    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 <= -3.7e-268) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 7.6e-136) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.7e-268:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 7.6e-136:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.7e-268)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 7.6e-136)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	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 <= -3.7e-268)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 7.6e-136)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.7e-268], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.6e-136], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.70000000000000018e-268

   1. Initial program 30.1%

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

   2. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/l* 33.5%

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

      3. expm1-def 81.3%

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

   4. Simplified 81.3%

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

   5. Taylor expanded in i around 0 63.0%

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

      1. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   if -3.70000000000000018e-268 < n < 7.6000000000000005e-136

   1. Initial program 41.1%

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

   2. Taylor expanded in i around 0 73.5%

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

   if 7.6000000000000005e-136 < n

   1. Initial program 19.0%

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

      1. associate-/r/ 19.3%

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

      2. associate-*r* 19.3%

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

      3. *-commutative 19.3%

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

      4. associate-*r/ 19.3%

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

      5. sub-neg 19.3%

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

      6. distribute-lft-in 19.3%

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

      7. fma-def 19.3%

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

      8. metadata-eval 19.3%

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

      9. metadata-eval 19.3%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in i around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*r/ 71.0%

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

      4. metadata-eval 71.0%

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

   6. Simplified 71.0%

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

   7. Taylor expanded in n around inf 71.1%

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

      1. *-commutative 71.1%

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

   9. Simplified 71.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-268}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{-136}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 16: 58.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 1.3 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2.0) (not (<= i 1.3e+58))) (* (/ n i) -200.0) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 1.3e+58)) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-2.0d0)) .or. (.not. (i <= 1.3d+58))) then
        tmp = (n / i) * (-200.0d0)
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 1.3e+58)) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2.0) or not (i <= 1.3e+58):
		tmp = (n / i) * -200.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2.0) || !(i <= 1.3e+58))
		tmp = Float64(Float64(n / i) * -200.0);
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -2.0) || ~((i <= 1.3e+58)))
		tmp = (n / i) * -200.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2.0], N[Not[LessEqual[i, 1.3e+58]], $MachinePrecision]], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2 or 1.29999999999999994e58 < i

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf 61.3%

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

      1. *-commutative 61.3%

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

      2. associate-/l* 61.3%

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

      3. expm1-def 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in i around 0 30.5%

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

      1. *-commutative 30.5%

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

   7. Simplified 30.5%

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

   8. Taylor expanded in i around inf 30.5%

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

   if -2 < i < 1.29999999999999994e58

   1. Initial program 11.4%

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

   2. Taylor expanded in i around 0 78.3%

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

      1. *-commutative 78.3%

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

   4. Simplified 78.3%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 1.3 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 17: 2.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.6%

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

   1. associate-/r/ 28.0%

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

   2. associate-*r* 28.0%

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

   3. *-commutative 28.0%

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

   4. associate-*r/ 28.0%

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

   5. sub-neg 28.0%

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

   6. distribute-lft-in 28.0%

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

   7. fma-def 28.0%

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

   8. metadata-eval 28.0%

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

   9. metadata-eval 28.0%

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

3. Simplified 28.0%

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

4. Taylor expanded in i around 0 54.9%

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

   1. associate-*r* 54.9%

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

   2. *-commutative 54.9%

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

   3. associate-*r/ 54.9%

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

   4. metadata-eval 54.9%

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

6. Simplified 54.9%

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

7. Taylor expanded in n around 0 2.7%

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

   1. *-commutative 2.7%

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

9. Simplified 2.7%

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

10. Final simplification 2.7%

    \[\leadsto i \cdot -50 \]

Alternative 18: 50.0% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.6%

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

2. Taylor expanded in i around 0 49.4%

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

   1. *-commutative 49.4%

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

4. Simplified 49.4%

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

5. Final simplification 49.4%

   \[\leadsto n \cdot 100 \]

Developer target: 33.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

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

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023194 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))