Compound Interest

Percentage Accurate: 28.0% → 96.0%

Time: 15.9s

Alternatives: 16

Speedup: 8.1×

• 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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*r* N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   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

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. pow-to-exp N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. 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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied 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: 81.0% accurate, 0.2× speedup

Math

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

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)
       (* n (* 100.0 (/ (expm1 i) i)))
       (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 = n * (100.0 * (expm1(i) / i));
	} 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(n * Float64(100.0 * Float64(expm1(i) / i)));
	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[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

   4. Applied 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

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.1

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

   7. Applied rewrites 76.1%

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

   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

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. pow-to-exp N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. 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

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

      2. lift-+.f64 N/A

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

      3. rem-exp-log N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. rem-exp-log N/A

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

      9. lift-+.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. div-sub N/A

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

   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 80.4%

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

Alternative 3: 81.0% accurate, 0.3× speedup

Math

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

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)
       (* n (* 100.0 (/ (expm1 i) i)))
       (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 = n * (100.0 * (expm1(i) / i));
	} 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(n * Float64(100.0 * Float64(expm1(i) / i)));
	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[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

   4. Applied 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

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.1

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

   7. Applied rewrites 76.1%

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

   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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied 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 80.3%

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

Alternative 4: 81.0% accurate, 0.3× speedup

Math

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

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)
       (* n (* 100.0 (/ (expm1 i) i)))
       (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 = n * (100.0 * (expm1(i) / i));
	} 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(n * Float64(100.0 * Float64(expm1(i) / i)));
	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[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

   4. Applied 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

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.1

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

   7. Applied rewrites 76.1%

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

   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 3 regimes into one program.

4. Final simplification 80.3%

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

Alternative 5: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -2e-311)
     t_0
     (if (<= n 1.6e-142)
       (* 100.0 (/ (* n (- (log i) (log n))) (/ i n)))
       t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -2e-311) {
		tmp = t_0;
	} else if (n <= 1.6e-142) {
		tmp = 100.0 * ((n * (log(i) - log(n))) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -2e-311) {
		tmp = t_0;
	} else if (n <= 1.6e-142) {
		tmp = 100.0 * ((n * (Math.log(i) - Math.log(n))) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -2e-311:
		tmp = t_0
	elif n <= 1.6e-142:
		tmp = 100.0 * ((n * (math.log(i) - math.log(n))) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -2e-311)
		tmp = t_0;
	elseif (n <= 1.6e-142)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(log(i) - log(n))) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2e-311], t$95$0, If[LessEqual[n, 1.6e-142], N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.9999999999999e-311 or 1.5999999999999999e-142 < n

   1. Initial program 28.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if -1.9999999999999e-311 < n < 1.5999999999999999e-142

   1. Initial program 44.0%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-311}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-311}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(n \cdot \frac{\log i - \log n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -2e-311)
     t_0
     (if (<= n 1.6e-142)
       (* 100.0 (* n (* n (/ (- (log i) (log n)) i))))
       t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -2e-311) {
		tmp = t_0;
	} else if (n <= 1.6e-142) {
		tmp = 100.0 * (n * (n * ((log(i) - log(n)) / i)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -2e-311) {
		tmp = t_0;
	} else if (n <= 1.6e-142) {
		tmp = 100.0 * (n * (n * ((Math.log(i) - Math.log(n)) / i)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -2e-311:
		tmp = t_0
	elif n <= 1.6e-142:
		tmp = 100.0 * (n * (n * ((math.log(i) - math.log(n)) / i)))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -2e-311)
		tmp = t_0;
	elseif (n <= 1.6e-142)
		tmp = Float64(100.0 * Float64(n * Float64(n * Float64(Float64(log(i) - log(n)) / i))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2e-311], t$95$0, If[LessEqual[n, 1.6e-142], N[(100.0 * N[(n * N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.9999999999999e-311 or 1.5999999999999999e-142 < n

   1. Initial program 28.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if -1.9999999999999e-311 < n < 1.5999999999999999e-142

   1. Initial program 44.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. pow-to-exp N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-log1p.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. rem-exp-log N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. rem-exp-log N/A

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

      9. lift-+.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. div-sub N/A

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

   6. Applied rewrites 44.0%

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

   7. Taylor expanded in n around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. div-sub N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. lower-log.f64 90.6

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

   9. Applied rewrites 90.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-311}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(n \cdot \frac{\log i - \log n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 8.5 \cdot 10^{-181}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{+63}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{i}{n} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{n}}{\frac{\frac{i}{n}}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 8.5e-181)
   (* n (* 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 <= 8.5e-181) {
		tmp = n * (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;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 8.5e-181) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else if (i <= 4.5e+63) {
		tmp = 100.0 * ((n * Math.expm1(i)) / i);
	} else {
		tmp = 100.0 * ((((i / n) * (Math.pow((1.0 + (i / n)), n) / i)) + (-1.0 / n)) / ((i / n) / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 8.5e-181:
		tmp = n * (100.0 * (math.expm1(i) / i))
	elif i <= 4.5e+63:
		tmp = 100.0 * ((n * math.expm1(i)) / i)
	else:
		tmp = 100.0 * ((((i / n) * (math.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 <= 8.5e-181)
		tmp = Float64(n * 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, 8.5e-181], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $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 < 8.49999999999999953e-181

   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

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 83.1

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

   7. Applied rewrites 83.1%

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

   if 8.49999999999999953e-181 < 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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. frac-sub N/A

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

      15. lower-/.f64 N/A

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

   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 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 8.5 \cdot 10^{-181}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{+63}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{i}{n} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{n}}{\frac{\frac{i}{n}}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -8.8 \cdot 10^{-205}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -8.8e-205) t_0 (if (<= n 5.8e-130) (* 100.0 0.0) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -8.8e-205) {
		tmp = t_0;
	} else if (n <= 5.8e-130) {
		tmp = 100.0 * 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -8.8e-205) {
		tmp = t_0;
	} else if (n <= 5.8e-130) {
		tmp = 100.0 * 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -8.8e-205:
		tmp = t_0
	elif n <= 5.8e-130:
		tmp = 100.0 * 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -8.8e-205)
		tmp = t_0;
	elseif (n <= 5.8e-130)
		tmp = Float64(100.0 * 0.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -8.8e-205], t$95$0, If[LessEqual[n, 5.8e-130], N[(100.0 * 0.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.80000000000000036e-205 or 5.8e-130 < n

   1. Initial program 26.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 74.4

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

   5. Applied rewrites 74.4%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if -8.80000000000000036e-205 < n < 5.8e-130

   1. Initial program 50.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. sub-div N/A

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

      12. frac-sub N/A

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

      13. pow2 N/A

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

      14. 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 32.4%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.4

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

   7. Applied rewrites 52.4%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-sub N/A

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

      12. +-inverses 70.5

          \[\leadsto 100 \cdot \color{blue}{0} \]

   9. Applied rewrites 70.5%

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-205}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 0.0013:\\ \;\;\;\;100 \cdot \frac{\frac{1}{n}}{\frac{\frac{1}{n}}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -2.2e-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 <= -2.2e-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 <= -2.2e-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 <= -2.2e-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 <= -2.2e-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, -2.2e-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 < -2.1999999999999998e-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 -2.1999999999999998e-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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. sub-div N/A

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

      12. frac-sub N/A

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

      13. pow2 N/A

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

      14. 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 10: 65.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), n \cdot 0.5\right), n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e-132)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 5.8e-130)
     (* 100.0 0.0)
     (*
      100.0
      (fma
       i
       (fma i (* n (fma i 0.041666666666666664 0.16666666666666666)) (* n 0.5))
       n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e-132) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 5.8e-130) {
		tmp = 100.0 * 0.0;
	} else {
		tmp = 100.0 * fma(i, fma(i, (n * fma(i, 0.041666666666666664, 0.16666666666666666)), (n * 0.5)), n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e-132)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 5.8e-130)
		tmp = Float64(100.0 * 0.0);
	else
		tmp = Float64(100.0 * fma(i, fma(i, Float64(n * fma(i, 0.041666666666666664, 0.16666666666666666)), Float64(n * 0.5)), n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e-132], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-130], N[(100.0 * 0.0), $MachinePrecision], N[(100.0 * N[(i * N[(i * N[(n * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(n * 0.5), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.80000000000000002e-132

   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

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50 + \frac{50}{3} \cdot i, 100\right)} \]

      3. +-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\frac{50}{3} \cdot i + 50}, 100\right) \]

      4. *-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{50}{3}} + 50, 100\right) \]

      5. lower-fma.f64 54.3

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

   10. Applied rewrites 54.3%

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

   if -2.80000000000000002e-132 < n < 5.8e-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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. sub-div N/A

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

      12. frac-sub N/A

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

      13. pow2 N/A

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

      14. 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 34.4%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-sub N/A

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

      12. +-inverses 61.3

          \[\leadsto 100 \cdot \color{blue}{0} \]

   9. Applied rewrites 61.3%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if 5.8e-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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 72.1

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

   8. Applied rewrites 72.1%

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

Alternative 11: 65.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e-132)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 5.8e-130)
     (* 100.0 0.0)
     (*
      n
      (fma
       i
       (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
       100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e-132) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 5.8e-130) {
		tmp = 100.0 * 0.0;
	} else {
		tmp = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e-132)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 5.8e-130)
		tmp = Float64(100.0 * 0.0);
	else
		tmp = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e-132], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-130], N[(100.0 * 0.0), $MachinePrecision], N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.80000000000000002e-132

   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

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50 + \frac{50}{3} \cdot i, 100\right)} \]

      3. +-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\frac{50}{3} \cdot i + 50}, 100\right) \]

      4. *-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{50}{3}} + 50, 100\right) \]

      5. lower-fma.f64 54.3

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

   10. Applied rewrites 54.3%

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

   if -2.80000000000000002e-132 < n < 5.8e-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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. sub-div N/A

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

      12. frac-sub N/A

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

      13. pow2 N/A

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

      14. 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 34.4%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-sub N/A

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

      12. +-inverses 61.3

          \[\leadsto 100 \cdot \color{blue}{0} \]

   9. Applied rewrites 61.3%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if 5.8e-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. Step-by-step derivation

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 83.7

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in i around 0

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + 100\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right), 100\right)} \]

      3. +-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + 50}, 100\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(i, \frac{50}{3} + \frac{25}{6} \cdot i, 50\right)}, 100\right) \]

      5. +-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \color{blue}{\frac{25}{6} \cdot i + \frac{50}{3}}, 50\right), 100\right) \]

      6. *-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{25}{6}} + \frac{50}{3}, 50\right), 100\right) \]

      7. lower-fma.f64 72.1

         \[\leadsto n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right)}, 50\right), 100\right) \]

   10. Applied rewrites 72.1%

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), i \cdot -50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e-132)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 5.8e-130) (* 100.0 0.0) (fma n (fma 50.0 i 100.0) (* i -50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e-132) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 5.8e-130) {
		tmp = 100.0 * 0.0;
	} else {
		tmp = fma(n, fma(50.0, i, 100.0), (i * -50.0));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e-132)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 5.8e-130)
		tmp = Float64(100.0 * 0.0);
	else
		tmp = fma(n, fma(50.0, i, 100.0), Float64(i * -50.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e-132], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-130], N[(100.0 * 0.0), $MachinePrecision], N[(n * N[(50.0 * i + 100.0), $MachinePrecision] + N[(i * -50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.80000000000000002e-132

   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

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50 + \frac{50}{3} \cdot i, 100\right)} \]

      3. +-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\frac{50}{3} \cdot i + 50}, 100\right) \]

      4. *-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{50}{3}} + 50, 100\right) \]

      5. lower-fma.f64 54.3

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

   10. Applied rewrites 54.3%

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

   if -2.80000000000000002e-132 < n < 5.8e-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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. sub-div N/A

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

      12. frac-sub N/A

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

      13. pow2 N/A

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

      14. 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 34.4%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-sub N/A

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

      12. +-inverses 61.3

          \[\leadsto 100 \cdot \color{blue}{0} \]

   9. Applied rewrites 61.3%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if 5.8e-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 i around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in n around 0

      \[\leadsto \color{blue}{-50 \cdot i + 100 \cdot \left(n \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + \frac{1}{2} \cdot i\right)\right) + -50 \cdot i} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left(1 + \frac{1}{2} \cdot i\right)} + -50 \cdot i \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(n \cdot 100\right)} \cdot \left(1 + \frac{1}{2} \cdot i\right) + -50 \cdot i \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)} + -50 \cdot i \]

      5. distribute-lft-in N/A

         \[\leadsto n \cdot \color{blue}{\left(100 \cdot 1 + 100 \cdot \left(\frac{1}{2} \cdot i\right)\right)} + -50 \cdot i \]

      6. metadata-eval N/A

         \[\leadsto n \cdot \left(\color{blue}{100} + 100 \cdot \left(\frac{1}{2} \cdot i\right)\right) + -50 \cdot i \]

      7. associate-*r* N/A

         \[\leadsto n \cdot \left(100 + \color{blue}{\left(100 \cdot \frac{1}{2}\right) \cdot i}\right) + -50 \cdot i \]

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 67.4

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

   8. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), i \cdot -50\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e-132)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 5.8e-130) (* 100.0 0.0) (* n (fma 50.0 i 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e-132) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 5.8e-130) {
		tmp = 100.0 * 0.0;
	} else {
		tmp = n * fma(50.0, i, 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e-132)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 5.8e-130)
		tmp = Float64(100.0 * 0.0);
	else
		tmp = Float64(n * fma(50.0, i, 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e-132], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-130], N[(100.0 * 0.0), $MachinePrecision], N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.80000000000000002e-132

   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

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50 + \frac{50}{3} \cdot i, 100\right)} \]

      3. +-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\frac{50}{3} \cdot i + 50}, 100\right) \]

      4. *-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{50}{3}} + 50, 100\right) \]

      5. lower-fma.f64 54.3

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

   10. Applied rewrites 54.3%

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

   if -2.80000000000000002e-132 < n < 5.8e-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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. sub-div N/A

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

      12. frac-sub N/A

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

      13. pow2 N/A

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

      14. 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 34.4%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-sub N/A

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

      12. +-inverses 61.3

          \[\leadsto 100 \cdot \color{blue}{0} \]

   9. Applied rewrites 61.3%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if 5.8e-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

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 67.4

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

   8. Applied rewrites 67.4%

      \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(50, i, 100\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 61.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;100 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma 50.0 i 100.0))))
   (if (<= n -2.8e-132) t_0 (if (<= n 5.8e-130) (* 100.0 0.0) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(50.0, i, 100.0);
	double tmp;
	if (n <= -2.8e-132) {
		tmp = t_0;
	} else if (n <= 5.8e-130) {
		tmp = 100.0 * 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(50.0, i, 100.0))
	tmp = 0.0
	if (n <= -2.8e-132)
		tmp = t_0;
	elseif (n <= 5.8e-130)
		tmp = Float64(100.0 * 0.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.8e-132], t$95$0, If[LessEqual[n, 5.8e-130], N[(100.0 * 0.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.80000000000000002e-132 or 5.8e-130 < n

   1. Initial program 25.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in i around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 59.7

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

   8. Applied rewrites 59.7%

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

   if -2.80000000000000002e-132 < n < 5.8e-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

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. sub-div N/A

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

      12. frac-sub N/A

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

      13. pow2 N/A

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

      14. 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 34.4%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-sub N/A

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

      12. +-inverses 61.3

          \[\leadsto 100 \cdot \color{blue}{0} \]

   9. Applied rewrites 61.3%

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 57.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.4 \cdot 10^{+49}:\\ \;\;\;\;100 \cdot 0\\ \mathbf{elif}\;i \leq 6.6 \cdot 10^{-46}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -3.4e+49)
   (* 100.0 0.0)
   (if (<= i 6.6e-46) (* n 100.0) (* 100.0 0.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -3.4e+49) {
		tmp = 100.0 * 0.0;
	} else if (i <= 6.6e-46) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-3.4d+49)) then
        tmp = 100.0d0 * 0.0d0
    else if (i <= 6.6d-46) then
        tmp = n * 100.0d0
    else
        tmp = 100.0d0 * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -3.4e+49) {
		tmp = 100.0 * 0.0;
	} else if (i <= 6.6e-46) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -3.4e+49:
		tmp = 100.0 * 0.0
	elif i <= 6.6e-46:
		tmp = n * 100.0
	else:
		tmp = 100.0 * 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -3.4e+49)
		tmp = Float64(100.0 * 0.0);
	elseif (i <= 6.6e-46)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(100.0 * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -3.4e+49)
		tmp = 100.0 * 0.0;
	elseif (i <= 6.6e-46)
		tmp = n * 100.0;
	else
		tmp = 100.0 * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -3.4e+49], N[(100.0 * 0.0), $MachinePrecision], If[LessEqual[i, 6.6e-46], N[(n * 100.0), $MachinePrecision], N[(100.0 * 0.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < -3.4000000000000001e49 or 6.60000000000000027e-46 < i

   1. Initial program 59.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. sub-div N/A

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

      12. frac-sub N/A

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

      13. pow2 N/A

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

      14. 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 44.0%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 27.5

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

   7. Applied rewrites 27.5%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. div-sub N/A

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

      12. +-inverses 28.4

          \[\leadsto 100 \cdot \color{blue}{0} \]

   9. Applied rewrites 28.4%

      \[\leadsto 100 \cdot \color{blue}{0} \]

   if -3.4000000000000001e49 < i < 6.60000000000000027e-46

   1. Initial program 8.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

Alternative 16: 17.7% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 0.0))

C

double code(double i, double n) {
	return 100.0 * 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * 0.0d0
end function

Java

public static double code(double i, double n) {
	return 100.0 * 0.0;
}

Python

def code(i, n):
	return 100.0 * 0.0

Julia

function code(i, n)
	return Float64(100.0 * 0.0)
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * 0.0;
end

Wolfram

code[i_, n_] := N[(100.0 * 0.0), $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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift--.f64 N/A

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

   5. *-rgt-identity N/A

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

   6. div-inv N/A

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

   7. times-frac N/A

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

   8. div-inv N/A

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

   9. lift--.f64 N/A

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

   10. div-sub N/A

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

   11. sub-div N/A

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

   12. frac-sub N/A

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

   13. pow2 N/A

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

   14. 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 21.1%

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

5. Taylor expanded in i around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 13.3

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

7. Applied rewrites 13.3%

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

   1. metadata-eval N/A

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

   2. frac-times N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. div-sub N/A

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

   12. +-inverses 16.1

       \[\leadsto 100 \cdot \color{blue}{0} \]

9. Applied rewrites 16.1%

   \[\leadsto 100 \cdot \color{blue}{0} \]
10. 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))))