Compound Interest

Percentage Accurate: 28.9% → 96.0%

Time: 16.7s

Alternatives: 18

Speedup: 8.1×

• could not determine a ground truth

  1. i = 1.0540360963436131e-83
  2. n = -4.4542706690541265e+36

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: 28.9% 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: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \mathsf{fma}\left(t\_0, 100, -100\right)\\ t_2 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-13}:\\ \;\;\;\;n \cdot \frac{t\_1}{i}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (fma t_0 100.0 -100.0))
        (t_2 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_2 -1e-13)
     (* n (/ t_1 i))
     (if (<= t_2 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_2 INFINITY) (* t_1 (/ n i)) (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = fma(t_0, 100.0, -100.0);
	double t_2 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_2 <= -1e-13) {
		tmp = n * (t_1 / i);
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1 * (n / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = fma(t_0, 100.0, -100.0)
	t_2 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -1e-13)
		tmp = Float64(n * Float64(t_1 / i));
	elseif (t_2 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_2 <= Inf)
		tmp = Float64(t_1 * Float64(n / i));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0 + -100.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-13], N[(n * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$1 * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e-13

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   if -1e-13 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   4. Applied egg-rr 98.0%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 0.0

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 72.6

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

   7. Simplified 72.6%

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

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 73.8

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

   10. Simplified 73.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.6 \cdot 10^{+194}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+140}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)}, 100, \frac{n}{i} \cdot \left(-100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.6e+194)
   (* (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) 100.0)
   (if (<= i 3.8e+37)
     (* (* n 100.0) (/ (expm1 i) i))
     (if (<= i 7.6e+140)
       (* 100.0 (/ (- (* n (pow (/ i n) n)) n) i))
       (fma (pow (/ i n) (+ n -1.0)) 100.0 (* (/ n i) (- 100.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.6e+194) {
		tmp = ((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) * 100.0;
	} else if (i <= 3.8e+37) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (i <= 7.6e+140) {
		tmp = 100.0 * (((n * pow((i / n), n)) - n) / i);
	} else {
		tmp = fma(pow((i / n), (n + -1.0)), 100.0, ((n / i) * -100.0));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.6e+194)
		tmp = Float64(Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n)) * 100.0);
	elseif (i <= 3.8e+37)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (i <= 7.6e+140)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * (Float64(i / n) ^ n)) - n) / i));
	else
		tmp = fma((Float64(i / n) ^ Float64(n + -1.0)), 100.0, Float64(Float64(n / i) * Float64(-100.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.6e+194], N[(N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[i, 3.8e+37], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.6e+140], N[(100.0 * N[(N[(N[(n * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(i / n), $MachinePrecision], N[(n + -1.0), $MachinePrecision]], $MachinePrecision] * 100.0 + N[(N[(n / i), $MachinePrecision] * (-100.0)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -4.6000000000000001e194

   1. Initial program 99.2%

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

   if -4.6000000000000001e194 < i < 3.7999999999999999e37

   1. Initial program 15.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 81.4

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

   7. Simplified 81.4%

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

   if 3.7999999999999999e37 < i < 7.6000000000000002e140

   1. Initial program 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 41.0%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 90.2

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

   7. Simplified 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 90.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.3

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

   9. Applied egg-rr 90.3%

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

   if 7.6000000000000002e140 < i

   1. Initial program 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 48.3

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

   7. Simplified 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.6 \cdot 10^{+194}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+140}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)}, 100, \frac{n}{i} \cdot \left(-100\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.6 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+140}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)}, 100, \frac{n}{i} \cdot \left(-100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.6e+194)
   (* n (/ (fma (pow (+ 1.0 (/ i n)) n) 100.0 -100.0) i))
   (if (<= i 3.8e+37)
     (* (* n 100.0) (/ (expm1 i) i))
     (if (<= i 7.6e+140)
       (* 100.0 (/ (- (* n (pow (/ i n) n)) n) i))
       (fma (pow (/ i n) (+ n -1.0)) 100.0 (* (/ n i) (- 100.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.6e+194) {
		tmp = n * (fma(pow((1.0 + (i / n)), n), 100.0, -100.0) / i);
	} else if (i <= 3.8e+37) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (i <= 7.6e+140) {
		tmp = 100.0 * (((n * pow((i / n), n)) - n) / i);
	} else {
		tmp = fma(pow((i / n), (n + -1.0)), 100.0, ((n / i) * -100.0));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.6e+194)
		tmp = Float64(n * Float64(fma((Float64(1.0 + Float64(i / n)) ^ n), 100.0, -100.0) / i));
	elseif (i <= 3.8e+37)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (i <= 7.6e+140)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * (Float64(i / n) ^ n)) - n) / i));
	else
		tmp = fma((Float64(i / n) ^ Float64(n + -1.0)), 100.0, Float64(Float64(n / i) * Float64(-100.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.6e+194], N[(n * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] * 100.0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.8e+37], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.6e+140], N[(100.0 * N[(N[(N[(n * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(i / n), $MachinePrecision], N[(n + -1.0), $MachinePrecision]], $MachinePrecision] * 100.0 + N[(N[(n / i), $MachinePrecision] * (-100.0)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -4.6000000000000001e194

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

   4. Applied egg-rr 94.3%

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

   if -4.6000000000000001e194 < i < 3.7999999999999999e37

   1. Initial program 15.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 81.4

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

   7. Simplified 81.4%

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

   if 3.7999999999999999e37 < i < 7.6000000000000002e140

   1. Initial program 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 41.0%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 90.2

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

   7. Simplified 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 90.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.3

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

   9. Applied egg-rr 90.3%

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

   if 7.6000000000000002e140 < i

   1. Initial program 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 48.3

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

   7. Simplified 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.6 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+140}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)}, 100, \frac{n}{i} \cdot \left(-100\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.6 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.6e+194)
   (* n (/ (fma (pow (+ 1.0 (/ i n)) n) 100.0 -100.0) i))
   (if (<= i 3.8e+37)
     (* (* n 100.0) (/ (expm1 i) i))
     (if (<= i 7.5e+140)
       (* 100.0 (/ (- (* n (pow (/ i n) n)) n) i))
       (* 100.0 (- (pow (/ i n) (+ n -1.0)) (/ n i)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.6e+194) {
		tmp = n * (fma(pow((1.0 + (i / n)), n), 100.0, -100.0) / i);
	} else if (i <= 3.8e+37) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (i <= 7.5e+140) {
		tmp = 100.0 * (((n * pow((i / n), n)) - n) / i);
	} else {
		tmp = 100.0 * (pow((i / n), (n + -1.0)) - (n / i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.6e+194)
		tmp = Float64(n * Float64(fma((Float64(1.0 + Float64(i / n)) ^ n), 100.0, -100.0) / i));
	elseif (i <= 3.8e+37)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (i <= 7.5e+140)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * (Float64(i / n) ^ n)) - n) / i));
	else
		tmp = Float64(100.0 * Float64((Float64(i / n) ^ Float64(n + -1.0)) - Float64(n / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.6e+194], N[(n * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] * 100.0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.8e+37], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.5e+140], N[(100.0 * N[(N[(N[(n * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], N[(n + -1.0), $MachinePrecision]], $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -4.6000000000000001e194

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

   4. Applied egg-rr 94.3%

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

   if -4.6000000000000001e194 < i < 3.7999999999999999e37

   1. Initial program 15.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 81.4

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

   7. Simplified 81.4%

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

   if 3.7999999999999999e37 < i < 7.4999999999999997e140

   1. Initial program 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 41.0%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 90.2

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

   7. Simplified 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 90.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.3

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

   9. Applied egg-rr 90.3%

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

   if 7.4999999999999997e140 < i

   1. Initial program 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 48.3

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

   7. Simplified 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 48.3

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

   9. Applied egg-rr 83.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.6 \cdot 10^{+194}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{if}\;i \leq -3.5 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (* n (pow (/ i n) n)) n) i))))
   (if (<= i -3.5e+143)
     t_0
     (if (<= i 3.8e+37)
       (* (* n 100.0) (/ (expm1 i) i))
       (if (<= i 7.5e+140)
         t_0
         (* 100.0 (- (pow (/ i n) (+ n -1.0)) (/ n i))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (((n * pow((i / n), n)) - n) / i);
	double tmp;
	if (i <= -3.5e+143) {
		tmp = t_0;
	} else if (i <= 3.8e+37) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (i <= 7.5e+140) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (pow((i / n), (n + -1.0)) - (n / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (((n * Math.pow((i / n), n)) - n) / i);
	double tmp;
	if (i <= -3.5e+143) {
		tmp = t_0;
	} else if (i <= 3.8e+37) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else if (i <= 7.5e+140) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (Math.pow((i / n), (n + -1.0)) - (n / i));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (((n * math.pow((i / n), n)) - n) / i)
	tmp = 0
	if i <= -3.5e+143:
		tmp = t_0
	elif i <= 3.8e+37:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	elif i <= 7.5e+140:
		tmp = t_0
	else:
		tmp = 100.0 * (math.pow((i / n), (n + -1.0)) - (n / i))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(Float64(n * (Float64(i / n) ^ n)) - n) / i))
	tmp = 0.0
	if (i <= -3.5e+143)
		tmp = t_0;
	elseif (i <= 3.8e+37)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (i <= 7.5e+140)
		tmp = t_0;
	else
		tmp = Float64(100.0 * Float64((Float64(i / n) ^ Float64(n + -1.0)) - Float64(n / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(n * N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -3.5e+143], t$95$0, If[LessEqual[i, 3.8e+37], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.5e+140], t$95$0, N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], N[(n + -1.0), $MachinePrecision]], $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -3.50000000000000008e143 or 3.7999999999999999e37 < i < 7.4999999999999997e140

   1. Initial program 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 93.3

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

   7. Simplified 93.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 93.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.4

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

   9. Applied egg-rr 93.4%

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

   if -3.50000000000000008e143 < i < 3.7999999999999999e37

   1. Initial program 13.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 77.7

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

   4. Applied egg-rr 77.7%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 80.6

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

   7. Simplified 80.6%

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

   if 7.4999999999999997e140 < i

   1. Initial program 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 48.3

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

   7. Simplified 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 48.3

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

   9. Applied egg-rr 83.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{+143}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;100 \cdot \frac{n \cdot {\left(\frac{i}{n}\right)}^{n} - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 9.2 \cdot 10^{+134}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 9.2e+134)
   (* (* n 100.0) (/ (expm1 i) i))
   (* 100.0 (- (pow (/ i n) (+ n -1.0)) (/ n i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 9.2e+134) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else {
		tmp = 100.0 * (pow((i / n), (n + -1.0)) - (n / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 9.2e+134) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else {
		tmp = 100.0 * (Math.pow((i / n), (n + -1.0)) - (n / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 9.2e+134:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	else:
		tmp = 100.0 * (math.pow((i / n), (n + -1.0)) - (n / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 9.2e+134)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	else
		tmp = Float64(100.0 * Float64((Float64(i / n) ^ Float64(n + -1.0)) - Float64(n / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 9.2e+134], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], N[(n + -1.0), $MachinePrecision]], $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 9.1999999999999992e134

   1. Initial program 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 79.4

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

   4. Applied egg-rr 79.4%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 78.0

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

   7. Simplified 78.0%

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

   if 9.1999999999999992e134 < i

   1. Initial program 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 48.3

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

   7. Simplified 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 48.3

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

   9. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right) \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 9.2 \cdot 10^{+134}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} - \frac{n}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.3 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \left(100 \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2.3e-219)
     (* n (* 100.0 t_0))
     (if (<= n 1.05e-33)
       (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
       (* (* n 100.0) t_0)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2.3e-219) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2.3e-219) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2.3e-219:
		tmp = n * (100.0 * t_0)
	elif n <= 1.05e-33:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	else:
		tmp = (n * 100.0) * t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2.3e-219)
		tmp = Float64(n * Float64(100.0 * t_0));
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.3e-219], N[(n * N[(100.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.29999999999999988e-219

   1. Initial program 26.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 73.8

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

   4. Applied egg-rr 73.8%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 78.9

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

   7. Simplified 78.9%

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

      1. lift-*.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 78.9

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

   9. Applied egg-rr 78.9%

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

   if -2.29999999999999988e-219 < n < 1.05e-33

   1. Initial program 36.9%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 62.7%

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

   if 1.05e-33 < n

   1. Initial program 21.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 74.6

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

   4. Applied egg-rr 74.6%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 92.0

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

   7. Simplified 92.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.3 \cdot 10^{-219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n 100.0) (/ (expm1 i) i))))
   (if (<= n -2.3e-219)
     t_0
     (if (<= n 1.05e-33) (* 100.0 (/ (+ 1.0 -1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * (expm1(i) / i);
	double tmp;
	if (n <= -2.3e-219) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) * (Math.expm1(i) / i);
	double tmp;
	if (n <= -2.3e-219) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) * (math.expm1(i) / i)
	tmp = 0
	if n <= -2.3e-219:
		tmp = t_0
	elif n <= 1.05e-33:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * Float64(expm1(i) / i))
	tmp = 0.0
	if (n <= -2.3e-219)
		tmp = t_0;
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.3e-219], t$95$0, If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.29999999999999988e-219 or 1.05e-33 < n

   1. Initial program 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 74.1

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

   4. Applied egg-rr 74.1%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 83.9

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

   7. Simplified 83.9%

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

   if -2.29999999999999988e-219 < n < 1.05e-33

   1. Initial program 36.9%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 62.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{-219}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -4.2e-128)
     t_0
     (if (<= n 1.05e-33) (* 100.0 (/ (+ 1.0 -1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -4.2e-128) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -4.2e-128) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -4.2e-128:
		tmp = t_0
	elif n <= 1.05e-33:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -4.2e-128)
		tmp = t_0;
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.2e-128], t$95$0, If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.2000000000000002e-128 or 1.05e-33 < n

   1. Initial program 22.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 72.6

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

   4. Applied egg-rr 72.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 83.0

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

   7. Simplified 83.0%

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

   if -4.2000000000000002e-128 < n < 1.05e-33

   1. Initial program 40.0%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 61.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{-128}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;\frac{n}{i} \cdot \left(-100\right)\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+140}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i \cdot i, i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\frac{i}{i \cdot i}, n, -\frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.9)
   (* (/ n i) (- 100.0))
   (if (<= i 7.6e+140)
     (*
      (* n 100.0)
      (/
       (fma
        (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5)
        (* i i)
        i)
       i))
     (* 100.0 (fma (/ i (* i i)) n (- (/ n i)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.9) {
		tmp = (n / i) * -100.0;
	} else if (i <= 7.6e+140) {
		tmp = (n * 100.0) * (fma(fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), (i * i), i) / i);
	} else {
		tmp = 100.0 * fma((i / (i * i)), n, -(n / i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.9)
		tmp = Float64(Float64(n / i) * Float64(-100.0));
	elseif (i <= 7.6e+140)
		tmp = Float64(Float64(n * 100.0) * Float64(fma(fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), Float64(i * i), i) / i));
	else
		tmp = Float64(100.0 * fma(Float64(i / Float64(i * i)), n, Float64(-Float64(n / i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.9], N[(N[(n / i), $MachinePrecision] * (-100.0)), $MachinePrecision], If[LessEqual[i, 7.6e+140], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] * N[(i * i), $MachinePrecision] + i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(i / N[(i * i), $MachinePrecision]), $MachinePrecision] * n + (-N[(n / i), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.8999999999999999

   1. Initial program 62.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 72.1

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

   7. Simplified 72.1%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.7

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

   10. Simplified 67.7%

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

   if -1.8999999999999999 < i < 7.6000000000000002e140

   1. Initial program 15.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 76.6

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 80.6

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

   7. Simplified 80.6%

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

   8. Taylor expanded in i around 0

      \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}}{i} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{i \cdot \color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) + 1\right)}}{i} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) \cdot i + 1 \cdot i}}{i} \]

      3. *-lft-identity N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) \cdot i + \color{blue}{i}}{i} \]

      4. *-commutative N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\left(\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) \cdot i\right)} \cdot i + i}{i} \]

      5. associate-*l* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) \cdot \left(i \cdot i\right)} + i}{i} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right), i \cdot i, i\right)}}{i} \]

      7. +-commutative N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right) + \frac{1}{2}}, i \cdot i, i\right)}{i} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, \frac{1}{6} + \frac{1}{24} \cdot i, \frac{1}{2}\right)}, i \cdot i, i\right)}{i} \]

      9. +-commutative N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \color{blue}{\frac{1}{24} \cdot i + \frac{1}{6}}, \frac{1}{2}\right), i \cdot i, i\right)}{i} \]

      10. *-commutative N/A

          \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \color{blue}{i \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), i \cdot i, i\right)}{i} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(i, \frac{1}{24}, \frac{1}{6}\right)}, \frac{1}{2}\right), i \cdot i, i\right)}{i} \]

      12. lower-*.f64 77.4

          \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), \color{blue}{i \cdot i}, i\right)}{i} \]

   10. Simplified 77.4%

       \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i \cdot i, i\right)}}{i} \]

   if 7.6000000000000002e140 < i

   1. Initial program 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. frac-sub N/A

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

      16. lower-/.f64 N/A

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

   4. Applied egg-rr 7.8%

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

   5. Taylor expanded in i around 0

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

   7. Simplified 1.7%

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

   9. Applied egg-rr 60.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;\frac{n}{i} \cdot \left(-100\right)\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+140}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i \cdot i, i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\frac{i}{i \cdot i}, n, -\frac{n}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{i} \cdot \left(-100\right)\\ \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i \cdot i, i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ n i) (- 100.0))))
   (if (<= i -1.9)
     t_0
     (if (<= i 5e+154)
       (*
        (* n 100.0)
        (/
         (fma
          (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5)
          (* i i)
          i)
         i))
       t_0))))

C

double code(double i, double n) {
	double t_0 = (n / i) * -100.0;
	double tmp;
	if (i <= -1.9) {
		tmp = t_0;
	} else if (i <= 5e+154) {
		tmp = (n * 100.0) * (fma(fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), (i * i), i) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(n / i) * Float64(-100.0))
	tmp = 0.0
	if (i <= -1.9)
		tmp = t_0;
	elseif (i <= 5e+154)
		tmp = Float64(Float64(n * 100.0) * Float64(fma(fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), Float64(i * i), i) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n / i), $MachinePrecision] * (-100.0)), $MachinePrecision]}, If[LessEqual[i, -1.9], t$95$0, If[LessEqual[i, 5e+154], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] * N[(i * i), $MachinePrecision] + i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.8999999999999999 or 5.00000000000000004e154 < i

   1. Initial program 56.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 55.1%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 63.1

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

   7. Simplified 63.1%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 65.6

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

   10. Simplified 65.6%

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

   if -1.8999999999999999 < i < 5.00000000000000004e154

   1. Initial program 15.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 76.0

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 79.9

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

   7. Simplified 79.9%

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

   8. Taylor expanded in i around 0

      \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}}{i} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{i \cdot \color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) + 1\right)}}{i} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) \cdot i + 1 \cdot i}}{i} \]

      3. *-lft-identity N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\left(i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) \cdot i + \color{blue}{i}}{i} \]

      4. *-commutative N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\left(\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) \cdot i\right)} \cdot i + i}{i} \]

      5. associate-*l* N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) \cdot \left(i \cdot i\right)} + i}{i} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right), i \cdot i, i\right)}}{i} \]

      7. +-commutative N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right) + \frac{1}{2}}, i \cdot i, i\right)}{i} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(i, \frac{1}{6} + \frac{1}{24} \cdot i, \frac{1}{2}\right)}, i \cdot i, i\right)}{i} \]

      9. +-commutative N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \color{blue}{\frac{1}{24} \cdot i + \frac{1}{6}}, \frac{1}{2}\right), i \cdot i, i\right)}{i} \]

      10. *-commutative N/A

          \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \color{blue}{i \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), i \cdot i, i\right)}{i} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(i, \frac{1}{24}, \frac{1}{6}\right)}, \frac{1}{2}\right), i \cdot i, i\right)}{i} \]

      12. lower-*.f64 76.8

          \[\leadsto \left(100 \cdot n\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), \color{blue}{i \cdot i}, i\right)}{i} \]

   10. Simplified 76.8%

       \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i \cdot i, i\right)}}{i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;\frac{n}{i} \cdot \left(-100\right)\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i \cdot i, i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(-100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{i} \cdot \left(-100\right)\\ \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ n i) (- 100.0))))
   (if (<= i -1.9)
     t_0
     (if (<= i 5e+154)
       (fma
        100.0
        n
        (* (* i n) (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)))
       t_0))))

C

double code(double i, double n) {
	double t_0 = (n / i) * -100.0;
	double tmp;
	if (i <= -1.9) {
		tmp = t_0;
	} else if (i <= 5e+154) {
		tmp = fma(100.0, n, ((i * n) * fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(n / i) * Float64(-100.0))
	tmp = 0.0
	if (i <= -1.9)
		tmp = t_0;
	elseif (i <= 5e+154)
		tmp = fma(100.0, n, Float64(Float64(i * n) * fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n / i), $MachinePrecision] * (-100.0)), $MachinePrecision]}, If[LessEqual[i, -1.9], t$95$0, If[LessEqual[i, 5e+154], N[(100.0 * n + N[(N[(i * n), $MachinePrecision] * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.8999999999999999 or 5.00000000000000004e154 < i

   1. Initial program 56.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 55.1%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 63.1

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

   7. Simplified 63.1%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 65.6

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

   10. Simplified 65.6%

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

   if -1.8999999999999999 < i < 5.00000000000000004e154

   1. Initial program 15.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 76.0

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 79.9

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

   7. Simplified 79.9%

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

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(100, n, i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(100, n, \color{blue}{i \cdot \left(50 \cdot n\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(100, n, i \cdot \color{blue}{\left(n \cdot 50\right)} + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(100, n, \color{blue}{\left(i \cdot n\right) \cdot 50} + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + i \cdot \color{blue}{\left(\left(\frac{25}{6} \cdot \left(i \cdot n\right)\right) \cdot i + \left(\frac{50}{3} \cdot n\right) \cdot i\right)}\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \color{blue}{\left(\left(\left(\frac{25}{6} \cdot \left(i \cdot n\right)\right) \cdot i\right) \cdot i + \left(\left(\frac{50}{3} \cdot n\right) \cdot i\right) \cdot i\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right)\right)\right)} \cdot i + \left(\left(\frac{50}{3} \cdot n\right) \cdot i\right) \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\left(i \cdot \color{blue}{\left(\left(\frac{25}{6} \cdot i\right) \cdot n\right)}\right) \cdot i + \left(\left(\frac{50}{3} \cdot n\right) \cdot i\right) \cdot i\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\color{blue}{\left(\left(i \cdot \left(\frac{25}{6} \cdot i\right)\right) \cdot n\right)} \cdot i + \left(\left(\frac{50}{3} \cdot n\right) \cdot i\right) \cdot i\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot i\right)\right) \cdot \left(n \cdot i\right)} + \left(\left(\frac{50}{3} \cdot n\right) \cdot i\right) \cdot i\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\left(i \cdot \left(\frac{25}{6} \cdot i\right)\right) \cdot \color{blue}{\left(i \cdot n\right)} + \left(\left(\frac{50}{3} \cdot n\right) \cdot i\right) \cdot i\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\color{blue}{\left(i \cdot n\right) \cdot \left(i \cdot \left(\frac{25}{6} \cdot i\right)\right)} + \left(\left(\frac{50}{3} \cdot n\right) \cdot i\right) \cdot i\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\left(i \cdot n\right) \cdot \left(i \cdot \left(\frac{25}{6} \cdot i\right)\right) + \color{blue}{\left(\frac{50}{3} \cdot \left(n \cdot i\right)\right)} \cdot i\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\left(i \cdot n\right) \cdot \left(i \cdot \left(\frac{25}{6} \cdot i\right)\right) + \left(\frac{50}{3} \cdot \color{blue}{\left(i \cdot n\right)}\right) \cdot i\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\left(i \cdot n\right) \cdot \left(i \cdot \left(\frac{25}{6} \cdot i\right)\right) + \color{blue}{\left(\left(i \cdot n\right) \cdot \frac{50}{3}\right)} \cdot i\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot 50 + \left(\left(i \cdot n\right) \cdot \left(i \cdot \left(\frac{25}{6} \cdot i\right)\right) + \color{blue}{\left(i \cdot n\right) \cdot \left(\frac{50}{3} \cdot i\right)}\right)\right) \]

   10. Simplified 76.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(100, n, \left(n \cdot i\right) \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;\frac{n}{i} \cdot \left(-100\right)\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(100, n, \left(i \cdot n\right) \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(-100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{i} \cdot \left(-100\right)\\ \mathbf{if}\;i \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+154}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ n i) (- 100.0))))
   (if (<= i -1.6)
     t_0
     (if (<= i 5e+154)
       (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
       t_0))))

C

double code(double i, double n) {
	double t_0 = (n / i) * -100.0;
	double tmp;
	if (i <= -1.6) {
		tmp = t_0;
	} else if (i <= 5e+154) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(n / i) * Float64(-100.0))
	tmp = 0.0
	if (i <= -1.6)
		tmp = t_0;
	elseif (i <= 5e+154)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n / i), $MachinePrecision] * (-100.0)), $MachinePrecision]}, If[LessEqual[i, -1.6], t$95$0, If[LessEqual[i, 5e+154], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.6000000000000001 or 5.00000000000000004e154 < i

   1. Initial program 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 54.4%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 62.2

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

   7. Simplified 62.2%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.8

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

   10. Simplified 64.8%

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

   if -1.6000000000000001 < i < 5.00000000000000004e154

   1. Initial program 15.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 75.8

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

   4. Applied egg-rr 75.8%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 80.3

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

   7. Simplified 80.3%

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

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right) + 100 \cdot n} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right) \cdot i} + 100 \cdot n \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{50}{3} \cdot i\right) \cdot n} + 50 \cdot n\right) \cdot i + 100 \cdot n \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(n \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)} \cdot i + 100 \cdot n \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{n \cdot \left(\left(\frac{50}{3} \cdot i + 50\right) \cdot i\right)} + 100 \cdot n \]

      6. *-commutative N/A

         \[\leadsto n \cdot \left(\left(\frac{50}{3} \cdot i + 50\right) \cdot i\right) + \color{blue}{n \cdot 100} \]

      7. distribute-lft-out N/A

         \[\leadsto \color{blue}{n \cdot \left(\left(\frac{50}{3} \cdot i + 50\right) \cdot i + 100\right)} \]

      8. *-commutative N/A

         \[\leadsto n \cdot \left(\color{blue}{i \cdot \left(\frac{50}{3} \cdot i + 50\right)} + 100\right) \]

      9. +-commutative N/A

         \[\leadsto n \cdot \left(i \cdot \color{blue}{\left(50 + \frac{50}{3} \cdot i\right)} + 100\right) \]

      10. distribute-lft-out N/A

          \[\leadsto n \cdot \left(\color{blue}{\left(i \cdot 50 + i \cdot \left(\frac{50}{3} \cdot i\right)\right)} + 100\right) \]

      11. *-commutative N/A

          \[\leadsto n \cdot \left(\left(\color{blue}{50 \cdot i} + i \cdot \left(\frac{50}{3} \cdot i\right)\right) + 100\right) \]

      12. associate-+r+ N/A

          \[\leadsto n \cdot \color{blue}{\left(50 \cdot i + \left(i \cdot \left(\frac{50}{3} \cdot i\right) + 100\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{\left(100 + i \cdot \left(\frac{50}{3} \cdot i\right)\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + \left(100 + i \cdot \left(\frac{50}{3} \cdot i\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot i\right) + 100\right)}\right) \]

      16. associate-+r+ N/A

          \[\leadsto n \cdot \color{blue}{\left(\left(50 \cdot i + i \cdot \left(\frac{50}{3} \cdot i\right)\right) + 100\right)} \]

   10. Simplified 76.0%

       \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.6:\\ \;\;\;\;\frac{n}{i} \cdot \left(-100\right)\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+154}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(-100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n \cdot -100}{i}\\ \mathbf{if}\;i \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 5 \cdot 10^{+154}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* n -100.0) i)))
   (if (<= i -1.6)
     t_0
     (if (<= i 5e+154)
       (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
       t_0))))

C

double code(double i, double n) {
	double t_0 = (n * -100.0) / i;
	double tmp;
	if (i <= -1.6) {
		tmp = t_0;
	} else if (i <= 5e+154) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(n * -100.0) / i)
	tmp = 0.0
	if (i <= -1.6)
		tmp = t_0;
	elseif (i <= 5e+154)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * -100.0), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[i, -1.6], t$95$0, If[LessEqual[i, 5e+154], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.6000000000000001 or 5.00000000000000004e154 < i

   1. Initial program 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 54.4%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 62.2

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

   7. Simplified 62.2%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.7

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

   10. Simplified 64.7%

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

   if -1.6000000000000001 < i < 5.00000000000000004e154

   1. Initial program 15.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 75.8

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

   4. Applied egg-rr 75.8%

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

   5. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 80.3

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

   7. Simplified 80.3%

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

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right) + 100 \cdot n} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right) \cdot i} + 100 \cdot n \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{50}{3} \cdot i\right) \cdot n} + 50 \cdot n\right) \cdot i + 100 \cdot n \]

      4. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(n \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)} \cdot i + 100 \cdot n \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{n \cdot \left(\left(\frac{50}{3} \cdot i + 50\right) \cdot i\right)} + 100 \cdot n \]

      6. *-commutative N/A

         \[\leadsto n \cdot \left(\left(\frac{50}{3} \cdot i + 50\right) \cdot i\right) + \color{blue}{n \cdot 100} \]

      7. distribute-lft-out N/A

         \[\leadsto \color{blue}{n \cdot \left(\left(\frac{50}{3} \cdot i + 50\right) \cdot i + 100\right)} \]

      8. *-commutative N/A

         \[\leadsto n \cdot \left(\color{blue}{i \cdot \left(\frac{50}{3} \cdot i + 50\right)} + 100\right) \]

      9. +-commutative N/A

         \[\leadsto n \cdot \left(i \cdot \color{blue}{\left(50 + \frac{50}{3} \cdot i\right)} + 100\right) \]

      10. distribute-lft-out N/A

          \[\leadsto n \cdot \left(\color{blue}{\left(i \cdot 50 + i \cdot \left(\frac{50}{3} \cdot i\right)\right)} + 100\right) \]

      11. *-commutative N/A

          \[\leadsto n \cdot \left(\left(\color{blue}{50 \cdot i} + i \cdot \left(\frac{50}{3} \cdot i\right)\right) + 100\right) \]

      12. associate-+r+ N/A

          \[\leadsto n \cdot \color{blue}{\left(50 \cdot i + \left(i \cdot \left(\frac{50}{3} \cdot i\right) + 100\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{\left(100 + i \cdot \left(\frac{50}{3} \cdot i\right)\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + \left(100 + i \cdot \left(\frac{50}{3} \cdot i\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot i\right) + 100\right)}\right) \]

      16. associate-+r+ N/A

          \[\leadsto n \cdot \color{blue}{\left(\left(50 \cdot i + i \cdot \left(\frac{50}{3} \cdot i\right)\right) + 100\right)} \]

   10. Simplified 76.0%

       \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 70.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n \cdot -100}{i}\\ \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 1.65 \cdot 10^{+78}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* n -100.0) i)))
   (if (<= i -1.9) t_0 (if (<= i 1.65e+78) (* n (fma i 50.0 100.0)) t_0))))

C

double code(double i, double n) {
	double t_0 = (n * -100.0) / i;
	double tmp;
	if (i <= -1.9) {
		tmp = t_0;
	} else if (i <= 1.65e+78) {
		tmp = n * fma(i, 50.0, 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(n * -100.0) / i)
	tmp = 0.0
	if (i <= -1.9)
		tmp = t_0;
	elseif (i <= 1.65e+78)
		tmp = Float64(n * fma(i, 50.0, 100.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * -100.0), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[i, -1.9], t$95$0, If[LessEqual[i, 1.65e+78], N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.8999999999999999 or 1.65e78 < i

   1. Initial program 55.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. lift-pow.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-sub N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lower--.f64 N/A

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

   4. Applied egg-rr 54.0%

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

   5. Taylor expanded in i around inf

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

      1. lower-/.f64 64.4

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

   7. Simplified 64.4%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 60.0

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

   10. Simplified 60.0%

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

   if -1.8999999999999999 < i < 1.65e78

   1. Initial program 13.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      7. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      8. lower-log1p.f64 76.8

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied egg-rr 76.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{e^{i} - 1}{i}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{e^{i} - 1}{i}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right)} \cdot \frac{e^{i} - 1}{i} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      6. lower-expm1.f64 81.5

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   7. Simplified 81.5%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{100 \cdot n + 50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* N/A

         \[\leadsto 100 \cdot n + \color{blue}{\left(50 \cdot i\right) \cdot n} \]

      3. distribute-rgt-out N/A

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

      5. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(50 \cdot i + 100\right)} \]

      6. *-commutative N/A

         \[\leadsto n \cdot \left(\color{blue}{i \cdot 50} + 100\right) \]

      7. lower-fma.f64 75.7

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50, 100\right)} \]

   10. Simplified 75.7%

       \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(i, 50, 100\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 59.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{if}\;n \leq -1.02 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i 50.0 100.0))))
   (if (<= n -1.02e-156) t_0 (if (<= n 1.05e-33) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, 50.0, 100.0);
	double tmp;
	if (n <= -1.02e-156) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, 50.0, 100.0))
	tmp = 0.0
	if (n <= -1.02e-156)
		tmp = t_0;
	elseif (n <= 1.05e-33)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.02e-156], t$95$0, If[LessEqual[n, 1.05e-33], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.02e-156 or 1.05e-33 < n

   1. Initial program 22.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

      3. pow-to-exp N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      7. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      8. lower-log1p.f64 73.1

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied egg-rr 73.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{e^{i} - 1}{i}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{e^{i} - 1}{i}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right)} \cdot \frac{e^{i} - 1}{i} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      6. lower-expm1.f64 83.9

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   7. Simplified 83.9%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{100 \cdot n + 50 \cdot \left(i \cdot n\right)} \]

      2. associate-*r* N/A

         \[\leadsto 100 \cdot n + \color{blue}{\left(50 \cdot i\right) \cdot n} \]

      3. distribute-rgt-out N/A

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right)} \]

      5. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(50 \cdot i + 100\right)} \]

      6. *-commutative N/A

         \[\leadsto n \cdot \left(\color{blue}{i \cdot 50} + 100\right) \]

      7. lower-fma.f64 65.4

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50, 100\right)} \]

   10. Simplified 65.4%

       \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(i, 50, 100\right)} \]

   if -1.02e-156 < n < 1.05e-33

   1. Initial program 40.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

      3. pow-to-exp N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      7. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      8. lower-log1p.f64 77.1

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied egg-rr 77.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \color{blue}{0} \]
   6. Step-by-step derivation

   7. Simplified 62.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.0065:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{-29}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.0065) 0.0 (if (<= i 9.5e-29) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.0065) {
		tmp = 0.0;
	} else if (i <= 9.5e-29) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-0.0065d0)) then
        tmp = 0.0d0
    else if (i <= 9.5d-29) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.0065) {
		tmp = 0.0;
	} else if (i <= 9.5e-29) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.0065:
		tmp = 0.0
	elif i <= 9.5e-29:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.0065)
		tmp = 0.0;
	elseif (i <= 9.5e-29)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.0065)
		tmp = 0.0;
	elseif (i <= 9.5e-29)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.0065], 0.0, If[LessEqual[i, 9.5e-29], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -0.0064999999999999997 or 9.50000000000000023e-29 < i

   1. Initial program 50.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

      3. pow-to-exp N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      7. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      8. lower-log1p.f64 75.1

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied egg-rr 75.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \color{blue}{0} \]
   6. Step-by-step derivation

   7. Simplified 34.2%

      \[\leadsto \color{blue}{0} \]

   if -0.0064999999999999997 < i < 9.50000000000000023e-29

   1. Initial program 9.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

      3. pow-to-exp N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      7. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

      8. lower-log1p.f64 73.4

         \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   4. Applied egg-rr 73.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{e^{i} - 1}{i}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{e^{i} - 1}{i}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right)} \cdot \frac{e^{i} - 1}{i} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(100 \cdot n\right) \cdot \color{blue}{\frac{e^{i} - 1}{i}} \]

      6. lower-expm1.f64 85.4

         \[\leadsto \left(100 \cdot n\right) \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   7. Simplified 85.4%

      \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   9. Step-by-step derivation

      1. lower-*.f64 84.8

         \[\leadsto \color{blue}{100 \cdot n} \]

   10. Simplified 84.8%

       \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.0065:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{-29}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 17.8% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 27.2%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. lift-+.f64 N/A

      \[\leadsto 100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

   3. pow-to-exp N/A

      \[\leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \]

   4. lower-expm1.f64 N/A

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

   5. *-commutative N/A

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   6. lower-*.f64 N/A

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(\color{blue}{n \cdot \log \left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   7. lift-+.f64 N/A

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \log \color{blue}{\left(1 + \frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

   8. lower-log1p.f64 74.1

      \[\leadsto 100 \cdot \frac{\mathsf{expm1}\left(n \cdot \color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)}\right)}{\frac{i}{n}} \]

4. Applied egg-rr 74.1%

   \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}{\frac{i}{n}} \]

5. Taylor expanded in n around 0

   \[\leadsto \color{blue}{0} \]
6. Step-by-step derivation

7. Simplified 20.6%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 35.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024214 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))