Compound Interest

Percentage Accurate: 28.0% → 96.0%

Time: 16.3s

Alternatives: 18

Speedup: 24.3×

• could not determine a ground truth

  1. i = 1.8957324430238944e-72
  2. n = 7.499628372452655e+39

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: 28.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \mathsf{fma}\left(t\_0, 100, -100\right)\\ t_2 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{n \cdot t\_1}{i}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-289}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

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 -2e-7)
     (/ (* n t_1) i)
     (if (<= t_2 2e-289)
       (* 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 <= -2e-7) {
		tmp = (n * t_1) / i;
	} else if (t_2 <= 2e-289) {
		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 <= -2e-7)
		tmp = Float64(Float64(n * t_1) / i);
	elseif (t_2 <= 2e-289)
		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, -2e-7], N[(N[(n * t$95$1), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$2, 2e-289], 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)) < -1.9999999999999999e-7

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. distribute-lft-in N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.9999999999999999e-7 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 2e-289

   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

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

      2. lift-pow.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 99.6

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

   4. Applied rewrites 99.6%

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

   if 2e-289 < (/.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 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-289}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(t\_0, 100, -100\right)}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-196)
     (* n (/ (fma t_0 100.0 -100.0) i))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY)
         (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
         (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-196) {
		tmp = n * (fma(t_0, 100.0, -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (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 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-196)
		tmp = Float64(n * Float64(fma(t_0, 100.0, -100.0) / i));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - 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[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-196], N[(n * N[(N[(t$95$0 * 100.0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] - N[(n / i), $MachinePrecision]), $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)) < -5.0000000000000005e-196

   1. Initial program 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 91.1

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

   4. Applied rewrites 91.1%

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

   if -5.0000000000000005e-196 < (/.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 18.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 75.7

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

   5. Applied rewrites 75.7%

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

   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 95.8%

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

      1. lift--.f64 N/A

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

      2. lift-pow.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 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lift-pow.f64 N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

   6. Applied rewrites 96.0%

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

   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. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-196}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{n}{i} - \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \mathsf{fma}\left(t\_0, 100, -100\right)\\ t_2 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;n \cdot \frac{t\_1}{i}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

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 -5e-196)
     (* n (/ t_1 i))
     (if (<= t_2 0.0)
       (* 100.0 (/ (expm1 i) (/ 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 <= -5e-196) {
		tmp = n * (t_1 / i);
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (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 <= -5e-196)
		tmp = Float64(n * Float64(t_1 / i));
	elseif (t_2 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / 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, -5e-196], N[(n * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(100.0 * N[(N[(Exp[i] - 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)) < -5.0000000000000005e-196

   1. Initial program 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 91.1

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

   4. Applied rewrites 91.1%

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

   if -5.0000000000000005e-196 < (/.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 18.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 75.7

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

   5. Applied rewrites 75.7%

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

   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 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   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. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-196}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ t_2 := n \cdot \frac{\mathsf{fma}\left(t\_0, 100, -100\right)}{i}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (/ (+ t_0 -1.0) (/ i n)))
        (t_2 (* n (/ (fma t_0 100.0 -100.0) i))))
   (if (<= t_1 -5e-196)
     t_2
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY) t_2 (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double t_2 = n * (fma(t_0, 100.0, -100.0) / i);
	double tmp;
	if (t_1 <= -5e-196) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	t_2 = Float64(n * Float64(fma(t_0, 100.0, -100.0) / i))
	tmp = 0.0
	if (t_1 <= -5e-196)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(N[(t$95$0 * 100.0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-196], t$95$2, If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(n * 100.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -5.0000000000000005e-196 or 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 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 93.5

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

   4. Applied rewrites 93.5%

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

   if -5.0000000000000005e-196 < (/.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 18.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 75.7

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

   5. Applied rewrites 75.7%

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

   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. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-196}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.35 \cdot 10^{-180}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{+63}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{i}{n} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{n}}{\frac{\frac{i}{n}}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.35e-180)
   (* n (fma (/ (* i (exp i)) n) -50.0 (* 100.0 (/ (expm1 i) i))))
   (if (<= i 4.5e+63)
     (* 100.0 (/ (* n (expm1 i)) i))
     (*
      100.0
      (/
       (+ (* (/ i n) (/ (pow (+ 1.0 (/ i n)) n) i)) (/ -1.0 n))
       (/ (/ i n) n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.35e-180) {
		tmp = n * fma(((i * exp(i)) / n), -50.0, (100.0 * (expm1(i) / i)));
	} else if (i <= 4.5e+63) {
		tmp = 100.0 * ((n * expm1(i)) / i);
	} else {
		tmp = 100.0 * ((((i / n) * (pow((1.0 + (i / n)), n) / i)) + (-1.0 / n)) / ((i / n) / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.35e-180)
		tmp = Float64(n * fma(Float64(Float64(i * exp(i)) / n), -50.0, Float64(100.0 * Float64(expm1(i) / i))));
	elseif (i <= 4.5e+63)
		tmp = Float64(100.0 * Float64(Float64(n * expm1(i)) / i));
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(Float64(i / n) * Float64((Float64(1.0 + Float64(i / n)) ^ n) / i)) + Float64(-1.0 / n)) / Float64(Float64(i / n) / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.35e-180], N[(n * N[(N[(N[(i * N[Exp[i], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * -50.0 + N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.5e+63], N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(i / n), $MachinePrecision] * N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / N[(N[(i / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < 1.35000000000000007e-180

   1. Initial program 24.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto n \cdot \left(\color{blue}{\frac{i \cdot e^{i}}{n} \cdot -50} + 100 \cdot \frac{e^{i} - 1}{i}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, 100 \cdot \frac{e^{i} - 1}{i}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto n \cdot \mathsf{fma}\left(\color{blue}{\frac{i \cdot e^{i}}{n}}, -50, 100 \cdot \frac{e^{i} - 1}{i}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto n \cdot \mathsf{fma}\left(\frac{\color{blue}{i \cdot e^{i}}}{n}, -50, 100 \cdot \frac{e^{i} - 1}{i}\right) \]

      6. lower-exp.f64 N/A

         \[\leadsto n \cdot \mathsf{fma}\left(\frac{i \cdot \color{blue}{e^{i}}}{n}, -50, 100 \cdot \frac{e^{i} - 1}{i}\right) \]

      7. *-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, \color{blue}{\frac{e^{i} - 1}{i} \cdot 100}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, \color{blue}{\frac{e^{i} - 1}{i} \cdot 100}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, \color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \]

      10. lower-expm1.f64 82.9

          \[\leadsto n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \]

   5. Applied rewrites 82.9%

      \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, \frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right)} \]

   if 1.35000000000000007e-180 < i < 4.50000000000000017e63

   1. Initial program 21.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if 4.50000000000000017e63 < i

   1. Initial program 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-/r* N/A

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

      7. frac-sub N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*r/ N/A

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

   4. Applied rewrites 62.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.35 \cdot 10^{-180}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{+63}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{i}{n} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{n}}{\frac{\frac{i}{n}}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.6 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -6.6 \cdot 10^{-208}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.0013:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -2.6e+76)
     t_0
     (if (<= n -6.6e-208)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= n 0.0013) (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -2.6e+76) {
		tmp = t_0;
	} else if (n <= -6.6e-208) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / 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 <= -2.6e+76) {
		tmp = t_0;
	} else if (n <= -6.6e-208) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / 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 <= -2.6e+76:
		tmp = t_0
	elif n <= -6.6e-208:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif n <= 0.0013:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / 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 <= -2.6e+76)
		tmp = t_0;
	elseif (n <= -6.6e-208)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (n <= 0.0013)
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / 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, -2.6e+76], t$95$0, If[LessEqual[n, -6.6e-208], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.0013], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.5999999999999999e76 or 0.0012999999999999999 < n

   1. Initial program 27.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -2.5999999999999999e76 < n < -6.60000000000000013e-208

   1. Initial program 31.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if -6.60000000000000013e-208 < n < 0.0012999999999999999

   1. Initial program 33.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. sub-div N/A

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

      8. frac-sub N/A

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

      9. pow2 N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 63.5

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

   7. Applied rewrites 63.5%

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

Alternative 7: 78.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{if}\;n \leq -2.65 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{i \cdot 0.01}{t\_0}}\\ \mathbf{elif}\;n \leq 0.0013:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{t\_0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (expm1 i))))
   (if (<= n -2.65e-34)
     (/ 1.0 (/ (* i 0.01) t_0))
     (if (<= n 0.0013)
       (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n)))
       (* 100.0 (/ t_0 i))))))

C

double code(double i, double n) {
	double t_0 = n * expm1(i);
	double tmp;
	if (n <= -2.65e-34) {
		tmp = 1.0 / ((i * 0.01) / t_0);
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else {
		tmp = 100.0 * (t_0 / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * Math.expm1(i);
	double tmp;
	if (n <= -2.65e-34) {
		tmp = 1.0 / ((i * 0.01) / t_0);
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else {
		tmp = 100.0 * (t_0 / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * math.expm1(i)
	tmp = 0
	if n <= -2.65e-34:
		tmp = 1.0 / ((i * 0.01) / t_0)
	elif n <= 0.0013:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n))
	else:
		tmp = 100.0 * (t_0 / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * expm1(i))
	tmp = 0.0
	if (n <= -2.65e-34)
		tmp = Float64(1.0 / Float64(Float64(i * 0.01) / t_0));
	elseif (n <= 0.0013)
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / n)));
	else
		tmp = Float64(100.0 * Float64(t_0 / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.65e-34], N[(1.0 / N[(N[(i * 0.01), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.0013], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.6499999999999998e-34

   1. Initial program 35.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 80.0

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

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 80.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{i \cdot 0.01}{n \cdot \mathsf{expm1}\left(i\right)}}} \]

   if -2.6499999999999998e-34 < n < 0.0012999999999999999

   1. Initial program 30.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. sub-div N/A

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

      8. frac-sub N/A

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

      9. pow2 N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 23.0%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 59.5

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

   7. Applied rewrites 59.5%

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

   if 0.0012999999999999999 < n

   1. Initial program 22.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 94.2

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

   5. Applied rewrites 94.2%

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

Alternative 8: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 0.0013:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -3.2e-18)
     t_0
     (if (<= n 0.0013) (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -3.2e-18) {
		tmp = t_0;
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / 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 <= -3.2e-18) {
		tmp = t_0;
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / 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 <= -3.2e-18:
		tmp = t_0
	elif n <= 0.0013:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / 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 <= -3.2e-18)
		tmp = t_0;
	elseif (n <= 0.0013)
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / 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, -3.2e-18], t$95$0, If[LessEqual[n, 0.0013], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.1999999999999999e-18 or 0.0012999999999999999 < n

   1. Initial program 29.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -3.1999999999999999e-18 < n < 0.0012999999999999999

   1. Initial program 30.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. sub-div N/A

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

      8. frac-sub N/A

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

      9. pow2 N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 23.1%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 59.3

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

   7. Applied rewrites 59.3%

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

Alternative 9: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{if}\;n \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0 \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 0.0013:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot t\_0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (expm1 i))))
   (if (<= n -3.2e-18)
     (* t_0 (/ 100.0 i))
     (if (<= n 0.0013)
       (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n)))
       (/ (* 100.0 t_0) i)))))

C

double code(double i, double n) {
	double t_0 = n * expm1(i);
	double tmp;
	if (n <= -3.2e-18) {
		tmp = t_0 * (100.0 / i);
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else {
		tmp = (100.0 * t_0) / i;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * Math.expm1(i);
	double tmp;
	if (n <= -3.2e-18) {
		tmp = t_0 * (100.0 / i);
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else {
		tmp = (100.0 * t_0) / i;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * math.expm1(i)
	tmp = 0
	if n <= -3.2e-18:
		tmp = t_0 * (100.0 / i)
	elif n <= 0.0013:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n))
	else:
		tmp = (100.0 * t_0) / i
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * expm1(i))
	tmp = 0.0
	if (n <= -3.2e-18)
		tmp = Float64(t_0 * Float64(100.0 / i));
	elseif (n <= 0.0013)
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / n)));
	else
		tmp = Float64(Float64(100.0 * t_0) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.2e-18], N[(t$95$0 * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.0013], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * t$95$0), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.1999999999999999e-18

   1. Initial program 35.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 80.9%

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

   if -3.1999999999999999e-18 < n < 0.0012999999999999999

   1. Initial program 30.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. sub-div N/A

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

      8. frac-sub N/A

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

      9. pow2 N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 23.1%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 59.3

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

   7. Applied rewrites 59.3%

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

   if 0.0012999999999999999 < n

   1. Initial program 22.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 0.0013:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{if}\;n \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 0.0013:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n (expm1 i)) (/ 100.0 i))))
   (if (<= n -3.2e-18)
     t_0
     (if (<= n 0.0013) (* 100.0 (/ (/ 1.0 n) (/ (/ 1.0 n) n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (n * expm1(i)) * (100.0 / i);
	double tmp;
	if (n <= -3.2e-18) {
		tmp = t_0;
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * Math.expm1(i)) * (100.0 / i);
	double tmp;
	if (n <= -3.2e-18) {
		tmp = t_0;
	} else if (n <= 0.0013) {
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * math.expm1(i)) * (100.0 / i)
	tmp = 0
	if n <= -3.2e-18:
		tmp = t_0
	elif n <= 0.0013:
		tmp = 100.0 * ((1.0 / n) / ((1.0 / n) / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * expm1(i)) * Float64(100.0 / i))
	tmp = 0.0
	if (n <= -3.2e-18)
		tmp = t_0;
	elseif (n <= 0.0013)
		tmp = Float64(100.0 * Float64(Float64(1.0 / n) / Float64(Float64(1.0 / n) / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.2e-18], t$95$0, If[LessEqual[n, 0.0013], N[(100.0 * N[(N[(1.0 / n), $MachinePrecision] / N[(N[(1.0 / n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.1999999999999999e-18 or 0.0012999999999999999 < n

   1. Initial program 29.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 86.8

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

   5. Applied rewrites 86.8%

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

   7. Applied rewrites 86.8%

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

   if -3.1999999999999999e-18 < n < 0.0012999999999999999

   1. Initial program 30.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. sub-div N/A

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

      8. frac-sub N/A

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

      9. pow2 N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 23.1%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 59.3

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

   7. Applied rewrites 59.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{elif}\;n \leq 0.0013:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), n \cdot 100\right)\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot \frac{n \cdot 1 - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.6e-131)
   (fma i (* n (fma 16.666666666666668 i 50.0)) (* n 100.0))
   (if (<= n 7.2e-130)
     (* 100.0 (/ (- (* n 1.0) n) i))
     (*
      100.0
      (fma
       i
       (* n (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5))
       n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.6e-131) {
		tmp = fma(i, (n * fma(16.666666666666668, i, 50.0)), (n * 100.0));
	} else if (n <= 7.2e-130) {
		tmp = 100.0 * (((n * 1.0) - n) / i);
	} else {
		tmp = 100.0 * fma(i, (n * fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5)), n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.6e-131)
		tmp = fma(i, Float64(n * fma(16.666666666666668, i, 50.0)), Float64(n * 100.0));
	elseif (n <= 7.2e-130)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * 1.0) - n) / i));
	else
		tmp = Float64(100.0 * fma(i, Float64(n * fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5)), n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.6e-131], N[(i * N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.2e-130], N[(100.0 * N[(N[(N[(n * 1.0), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i * N[(n * N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.6e-131

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in i around 0

      \[\leadsto 100 \cdot n + \color{blue}{i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.3%

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right)}, n \cdot 100\right) \]

   if -1.6e-131 < n < 7.2000000000000003e-130

   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

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

      2. lift-pow.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 85.7

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

   4. Applied rewrites 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lift-pow.f64 N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. div-inv N/A

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

      16. associate-*r/ N/A

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

      17. lift-/.f64 N/A

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

      18. clear-num N/A

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 61.3%

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

   if 7.2000000000000003e-130 < n

   1. Initial program 19.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 72.1%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 72.1%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), n \cdot 100\right)\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot \frac{n \cdot 1 - n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (*
  100.0
  (fma
   i
   (* n (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5))
   n)))

C

double code(double i, double n) {
	return 100.0 * fma(i, (n * fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5)), n);
}

Julia

function code(i, n)
	return Float64(100.0 * fma(i, Float64(n * fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5)), n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(i * N[(n * N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.9%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-expm1.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in i around 0

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

8. Applied rewrites 54.0%

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

9. Taylor expanded in i around 0

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

11. Applied rewrites 54.0%

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

Alternative 13: 55.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.04 \cdot 10^{-128}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.04e-128) (* n (fma 50.0 i 100.0)) (* (/ 100.0 i) (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.04e-128) {
		tmp = n * fma(50.0, i, 100.0);
	} else {
		tmp = (100.0 / i) * (i * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.04e-128)
		tmp = Float64(n * fma(50.0, i, 100.0));
	else
		tmp = Float64(Float64(100.0 / i) * Float64(i * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.04e-128], N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 / i), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.04e-128

   1. Initial program 22.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 60.1%

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

   if 1.04e-128 < i

   1. Initial program 44.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 53.4

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 40.1%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.04 \cdot 10^{-128}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \mathsf{fma}\left(i, n \cdot \left(0.041666666666666664 \cdot \left(i \cdot i\right)\right), n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (fma i (* n (* 0.041666666666666664 (* i i))) n)))

C

double code(double i, double n) {
	return 100.0 * fma(i, (n * (0.041666666666666664 * (i * i))), n);
}

Julia

function code(i, n)
	return Float64(100.0 * fma(i, Float64(n * Float64(0.041666666666666664 * Float64(i * i))), n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(i * N[(n * N[(0.041666666666666664 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.9%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-expm1.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in i around 0

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

8. Applied rewrites 54.0%

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

9. Taylor expanded in i around inf

   \[\leadsto 100 \cdot \mathsf{fma}\left(i, \frac{1}{24} \cdot \left({i}^{2} \cdot \color{blue}{n}\right), n\right) \]
10. Step-by-step derivation

11. Applied rewrites 53.9%

    \[\leadsto 100 \cdot \mathsf{fma}\left(i, n \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(i \cdot i\right)}\right), n\right) \]
12. Add Preprocessing

Alternative 15: 57.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), n \cdot 100\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (fma i (* n (fma 16.666666666666668 i 50.0)) (* n 100.0)))

C

double code(double i, double n) {
	return fma(i, (n * fma(16.666666666666668, i, 50.0)), (n * 100.0));
}

Julia

function code(i, n)
	return fma(i, Float64(n * fma(16.666666666666668, i, 50.0)), Float64(n * 100.0))
end

Wolfram

code[i_, n_] := N[(i * N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.9%

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

3. Taylor expanded in n around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-expm1.f64 68.4

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

5. Applied rewrites 68.4%

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

6. Taylor expanded in i around 0

   \[\leadsto 100 \cdot n + \color{blue}{i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
7. Step-by-step derivation

8. Applied rewrites 52.0%

   \[\leadsto \mathsf{fma}\left(i, \color{blue}{n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right)}, n \cdot 100\right) \]
9. Add Preprocessing

Alternative 16: 53.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 6.6 \cdot 10^{-46}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 6.6e-46) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 6.6e-46) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 6.6d-46) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 6.6e-46) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 6.6e-46:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 6.6e-46)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 6.6e-46)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 6.6e-46], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 6.60000000000000027e-46

   1. Initial program 21.9%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 6.60000000000000027e-46 < i

   1. Initial program 53.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 23.3%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 23.3%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6.6 \cdot 10^{-46}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ n \cdot \mathsf{fma}\left(50, i, 100\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n (fma 50.0 i 100.0)))

C

double code(double i, double n) {
	return n * fma(50.0, i, 100.0);
}

Julia

function code(i, n)
	return Float64(n * fma(50.0, i, 100.0))
end

Wolfram

code[i_, n_] := N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.9%

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

3. Taylor expanded in n around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-expm1.f64 68.4

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

5. Applied rewrites 68.4%

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

6. Taylor expanded in i around 0

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

8. Applied rewrites 51.6%

   \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(50, i, 100\right)} \]
9. Add Preprocessing

Alternative 18: 49.4% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

3. Taylor expanded in i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 47.3

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

5. Applied rewrites 47.3%

   \[\leadsto \color{blue}{n \cdot 100} \]
6. Add Preprocessing

Developer Target 1: 34.3% 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 2024219 
(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))))