Compound Interest

Percentage Accurate: 28.9% → 95.1%

Time: 11.0s

Alternatives: 16

Speedup: 24.3×

• could not determine a ground truth

  1. i = 8.592197882288067e-228
  2. n = 1.5336325297212603e+35

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} - 1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ t_2 := \frac{t\_0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-301}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ 1.0 (/ i n)) n) 1.0))
        (t_1 (/ t_0 (/ i n)))
        (t_2 (* (/ t_0 i) (* n 100.0))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e-301)
       (* (* (expm1 (* (log1p (/ i n)) n)) (/ 100.0 i)) n)
       (if (<= t_1 INFINITY) t_2 (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = (t_0 / i) * (n * 100.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e-301) {
		tmp = (expm1((log1p((i / n)) * n)) * (100.0 / i)) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = t_0 / (i / n);
	double t_2 = (t_0 / i) * (n * 100.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= 1e-301) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) * (100.0 / i)) * n;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) - 1.0
	t_1 = t_0 / (i / n)
	t_2 = (t_0 / i) * (n * 100.0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= 1e-301:
		tmp = (math.expm1((math.log1p((i / n)) * n)) * (100.0 / i)) * n
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	t_2 = Float64(Float64(t_0 / i) * Float64(n * 100.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e-301)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * Float64(100.0 / i)) * n);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e-301], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 * n), $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)) < -inf.0 or 1.00000000000000007e-301 < (/.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 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 43.5

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

   4. Applied rewrites 43.5%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log1p.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 1.00000000000000007e-301

   1. Initial program 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

Alternative 2: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e-311)
   (* (* (expm1 i) (/ 100.0 i)) n)
   (if (<= n 7.5e-111)
     (* (* n (/ (- (log i) (log n)) i)) (* n 100.0))
     (* (/ (expm1 i) i) (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e-311) {
		tmp = (expm1(i) * (100.0 / i)) * n;
	} else if (n <= 7.5e-111) {
		tmp = (n * ((log(i) - log(n)) / i)) * (n * 100.0);
	} else {
		tmp = (expm1(i) / i) * (100.0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5e-311) {
		tmp = (Math.expm1(i) * (100.0 / i)) * n;
	} else if (n <= 7.5e-111) {
		tmp = (n * ((Math.log(i) - Math.log(n)) / i)) * (n * 100.0);
	} else {
		tmp = (Math.expm1(i) / i) * (100.0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5e-311:
		tmp = (math.expm1(i) * (100.0 / i)) * n
	elif n <= 7.5e-111:
		tmp = (n * ((math.log(i) - math.log(n)) / i)) * (n * 100.0)
	else:
		tmp = (math.expm1(i) / i) * (100.0 * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e-311)
		tmp = Float64(Float64(expm1(i) * Float64(100.0 / i)) * n);
	elseif (n <= 7.5e-111)
		tmp = Float64(Float64(n * Float64(Float64(log(i) - log(n)) / i)) * Float64(n * 100.0));
	else
		tmp = Float64(Float64(expm1(i) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e-311], N[(N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 7.5e-111], N[(N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.00000000000023e-311

   1. Initial program 29.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-expm1.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if -5.00000000000023e-311 < n < 7.49999999999999965e-111

   1. Initial program 34.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 72.7

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 82.5

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

   7. Applied rewrites 82.5%

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

   if 7.49999999999999965e-111 < n

   1. Initial program 14.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 68.9

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

   5. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 88.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e-311)
   (* (* (expm1 i) (/ 100.0 i)) n)
   (if (<= n 7.5e-111)
     (* (* (* n (/ (- (log i) (log n)) i)) 100.0) n)
     (* (/ (expm1 i) i) (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e-311) {
		tmp = (expm1(i) * (100.0 / i)) * n;
	} else if (n <= 7.5e-111) {
		tmp = ((n * ((log(i) - log(n)) / i)) * 100.0) * n;
	} else {
		tmp = (expm1(i) / i) * (100.0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5e-311) {
		tmp = (Math.expm1(i) * (100.0 / i)) * n;
	} else if (n <= 7.5e-111) {
		tmp = ((n * ((Math.log(i) - Math.log(n)) / i)) * 100.0) * n;
	} else {
		tmp = (Math.expm1(i) / i) * (100.0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5e-311:
		tmp = (math.expm1(i) * (100.0 / i)) * n
	elif n <= 7.5e-111:
		tmp = ((n * ((math.log(i) - math.log(n)) / i)) * 100.0) * n
	else:
		tmp = (math.expm1(i) / i) * (100.0 * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e-311)
		tmp = Float64(Float64(expm1(i) * Float64(100.0 / i)) * n);
	elseif (n <= 7.5e-111)
		tmp = Float64(Float64(Float64(n * Float64(Float64(log(i) - log(n)) / i)) * 100.0) * n);
	else
		tmp = Float64(Float64(expm1(i) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e-311], N[(N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 7.5e-111], N[(N[(N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.00000000000023e-311

   1. Initial program 29.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-expm1.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if -5.00000000000023e-311 < n < 7.49999999999999965e-111

   1. Initial program 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 72.7

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-log.f64 82.5

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

   7. Applied rewrites 82.5%

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

   if 7.49999999999999965e-111 < n

   1. Initial program 14.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 68.9

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

   5. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 88.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;\left(\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;\left(100 \cdot \left(n \cdot n\right)\right) \cdot \frac{\log i - \log n}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e-311)
   (* (* (expm1 i) (/ 100.0 i)) n)
   (if (<= n 7.5e-111)
     (* (* 100.0 (* n n)) (/ (- (log i) (log n)) i))
     (* (/ (expm1 i) i) (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e-311) {
		tmp = (expm1(i) * (100.0 / i)) * n;
	} else if (n <= 7.5e-111) {
		tmp = (100.0 * (n * n)) * ((log(i) - log(n)) / i);
	} else {
		tmp = (expm1(i) / i) * (100.0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -5e-311) {
		tmp = (Math.expm1(i) * (100.0 / i)) * n;
	} else if (n <= 7.5e-111) {
		tmp = (100.0 * (n * n)) * ((Math.log(i) - Math.log(n)) / i);
	} else {
		tmp = (Math.expm1(i) / i) * (100.0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -5e-311:
		tmp = (math.expm1(i) * (100.0 / i)) * n
	elif n <= 7.5e-111:
		tmp = (100.0 * (n * n)) * ((math.log(i) - math.log(n)) / i)
	else:
		tmp = (math.expm1(i) / i) * (100.0 * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e-311)
		tmp = Float64(Float64(expm1(i) * Float64(100.0 / i)) * n);
	elseif (n <= 7.5e-111)
		tmp = Float64(Float64(100.0 * Float64(n * n)) * Float64(Float64(log(i) - log(n)) / i));
	else
		tmp = Float64(Float64(expm1(i) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e-311], N[(N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 7.5e-111], N[(N[(100.0 * N[(n * n), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.00000000000023e-311

   1. Initial program 29.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-expm1.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if -5.00000000000023e-311 < n < 7.49999999999999965e-111

   1. Initial program 34.8%

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

   3. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if 7.49999999999999965e-111 < n

   1. Initial program 14.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 68.9

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

   5. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 88.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;\left(100 \cdot \left(n \cdot n\right)\right) \cdot \frac{\log i - \log n}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-226} \lor \neg \left(n \leq 4.5 \cdot 10^{-114}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -7e-226) (not (<= n 4.5e-114)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* (/ (- 1.0 1.0) i) (* 100.0 n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -7e-226) || !(n <= 4.5e-114)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -7e-226) || !(n <= 4.5e-114)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -7e-226) or not (n <= 4.5e-114):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = ((1.0 - 1.0) / i) * (100.0 * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -7e-226) || !(n <= 4.5e-114))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -7e-226], N[Not[LessEqual[n, 4.5e-114]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7e-226 or 4.49999999999999969e-114 < n

   1. Initial program 19.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -7e-226 < n < 4.49999999999999969e-114

   1. Initial program 44.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-226} \lor \neg \left(n \leq 4.5 \cdot 10^{-114}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-226}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.5e-226)
   (* (* (expm1 i) (/ 100.0 i)) n)
   (if (<= n 4.5e-114)
     (* (/ (- 1.0 1.0) i) (* 100.0 n))
     (* (/ (expm1 i) i) (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.5e-226) {
		tmp = (expm1(i) * (100.0 / i)) * n;
	} else if (n <= 4.5e-114) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else {
		tmp = (expm1(i) / i) * (100.0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.5e-226) {
		tmp = (Math.expm1(i) * (100.0 / i)) * n;
	} else if (n <= 4.5e-114) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else {
		tmp = (Math.expm1(i) / i) * (100.0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.5e-226:
		tmp = (math.expm1(i) * (100.0 / i)) * n
	elif n <= 4.5e-114:
		tmp = ((1.0 - 1.0) / i) * (100.0 * n)
	else:
		tmp = (math.expm1(i) / i) * (100.0 * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.5e-226)
		tmp = Float64(Float64(expm1(i) * Float64(100.0 / i)) * n);
	elseif (n <= 4.5e-114)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	else
		tmp = Float64(Float64(expm1(i) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.5e-226], N[(N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 4.5e-114], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.50000000000000044e-226

   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

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in n around inf

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

      1. lower-expm1.f64 77.8

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

   7. Applied rewrites 77.8%

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

   if -7.50000000000000044e-226 < n < 4.49999999999999969e-114

   1. Initial program 44.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if 4.49999999999999969e-114 < n

   1. Initial program 14.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 68.9

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

   5. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 88.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-226}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-226}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.5e-226)
   (* (* (expm1 i) (/ 100.0 i)) n)
   (if (<= n 4.5e-114)
     (* (/ (- 1.0 1.0) i) (* 100.0 n))
     (* (* (/ (expm1 i) i) 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.5e-226) {
		tmp = (expm1(i) * (100.0 / i)) * n;
	} else if (n <= 4.5e-114) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else {
		tmp = ((expm1(i) / i) * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.5e-226) {
		tmp = (Math.expm1(i) * (100.0 / i)) * n;
	} else if (n <= 4.5e-114) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.5e-226:
		tmp = (math.expm1(i) * (100.0 / i)) * n
	elif n <= 4.5e-114:
		tmp = ((1.0 - 1.0) / i) * (100.0 * n)
	else:
		tmp = ((math.expm1(i) / i) * 100.0) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.5e-226)
		tmp = Float64(Float64(expm1(i) * Float64(100.0 / i)) * n);
	elseif (n <= 4.5e-114)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	else
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.5e-226], N[(N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 4.5e-114], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.50000000000000044e-226

   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

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in n around inf

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

      1. lower-expm1.f64 77.8

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

   7. Applied rewrites 77.8%

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

   if -7.50000000000000044e-226 < n < 4.49999999999999969e-114

   1. Initial program 44.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if 4.49999999999999969e-114 < n

   1. Initial program 14.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-226}:\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-183} \lor \neg \left(n \leq 4.5 \cdot 10^{-114}\right):\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.05e-183) (not (<= n 4.5e-114)))
   (*
    100.0
    (*
     (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
     n))
   (* (/ (- 1.0 1.0) i) (* 100.0 n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.05e-183) || !(n <= 4.5e-114)) {
		tmp = 100.0 * (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n);
	} else {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.05e-183) || !(n <= 4.5e-114))
		tmp = Float64(100.0 * Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n));
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.05e-183], N[Not[LessEqual[n, 4.5e-114]], $MachinePrecision]], N[(100.0 * N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.0500000000000001e-183 or 4.49999999999999969e-114 < n

   1. Initial program 18.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 83.9

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 66.6%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot n\right) \]

   if -1.0500000000000001e-183 < n < 4.49999999999999969e-114

   1. Initial program 44.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 65.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-183} \lor \neg \left(n \leq 4.5 \cdot 10^{-114}\right):\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-183} \lor \neg \left(n \leq 4.5 \cdot 10^{-114}\right):\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.05e-183) (not (<= n 4.5e-114)))
   (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n))
   (* (/ (- 1.0 1.0) i) (* 100.0 n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.05e-183) || !(n <= 4.5e-114)) {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	} else {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.05e-183) || !(n <= 4.5e-114))
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.05e-183], N[Not[LessEqual[n, 4.5e-114]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.0500000000000001e-183 or 4.49999999999999969e-114 < n

   1. Initial program 18.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 83.9

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 64.3%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

   if -1.0500000000000001e-183 < n < 4.49999999999999969e-114

   1. Initial program 44.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 65.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-183} \lor \neg \left(n \leq 4.5 \cdot 10^{-114}\right):\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-183}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, 0.16666666666666666, 0.5 \cdot n\right), i, n\right)\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.05e-183)
   (* 100.0 (fma (fma (* n i) 0.16666666666666666 (* 0.5 n)) i n))
   (if (<= n 4.5e-114)
     (* (/ (- 1.0 1.0) i) (* 100.0 n))
     (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.05e-183) {
		tmp = 100.0 * fma(fma((n * i), 0.16666666666666666, (0.5 * n)), i, n);
	} else if (n <= 4.5e-114) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.05e-183)
		tmp = Float64(100.0 * fma(fma(Float64(n * i), 0.16666666666666666, Float64(0.5 * n)), i, n));
	elseif (n <= 4.5e-114)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	else
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.05e-183], N[(100.0 * N[(N[(N[(n * i), $MachinePrecision] * 0.16666666666666666 + N[(0.5 * n), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.5e-114], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.0500000000000001e-183

   1. Initial program 23.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 78.8

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 60.2%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 60.2%

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

   if -1.0500000000000001e-183 < n < 4.49999999999999969e-114

   1. Initial program 44.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 65.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if 4.49999999999999969e-114 < n

   1. Initial program 14.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 67.9%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-226}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right)\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-235}:\\ \;\;\;\;\frac{100}{i} \cdot \left(n \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.65e-226)
   (* 100.0 (* (fma (* 0.16666666666666666 i) i 1.0) n))
   (if (<= n 2e-235) (* (/ 100.0 i) (* n i)) (* 100.0 (fma (* n i) 0.5 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.65e-226) {
		tmp = 100.0 * (fma((0.16666666666666666 * i), i, 1.0) * n);
	} else if (n <= 2e-235) {
		tmp = (100.0 / i) * (n * i);
	} else {
		tmp = 100.0 * fma((n * i), 0.5, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.65e-226)
		tmp = Float64(100.0 * Float64(fma(Float64(0.16666666666666666 * i), i, 1.0) * n));
	elseif (n <= 2e-235)
		tmp = Float64(Float64(100.0 / i) * Float64(n * i));
	else
		tmp = Float64(100.0 * fma(Float64(n * i), 0.5, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.65e-226], N[(100.0 * N[(N[(N[(0.16666666666666666 * i), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2e-235], N[(N[(100.0 / i), $MachinePrecision] * N[(n * i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * i), $MachinePrecision] * 0.5 + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.6500000000000002e-226

   1. Initial program 24.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 59.0%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

   9. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 58.3%

       \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \]

   if -2.6500000000000002e-226 < n < 1.9999999999999999e-235

   1. Initial program 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 3.3

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

   7. Applied rewrites 3.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 18.9

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

   9. Applied rewrites 18.9%

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

   10. Taylor expanded in i around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 43.5

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

   12. Applied rewrites 43.5%

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

   if 1.9999999999999999e-235 < n

   1. Initial program 16.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 60.7%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 60.2%

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

Alternative 12: 56.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n)))

C

double code(double i, double n) {
	return 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
}

Julia

function code(i, n)
	return Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in n around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-expm1.f64 73.4

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

5. Applied rewrites 73.4%

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

6. Taylor expanded in i around 0

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

8. Applied rewrites 55.5%

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]
9. Add Preprocessing

Alternative 13: 55.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (* (fma (* 0.16666666666666666 i) i 1.0) n)))

C

double code(double i, double n) {
	return 100.0 * (fma((0.16666666666666666 * i), i, 1.0) * n);
}

Julia

function code(i, n)
	return Float64(100.0 * Float64(fma(Float64(0.16666666666666666 * i), i, 1.0) * n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[(0.16666666666666666 * i), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in n around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-expm1.f64 73.4

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

5. Applied rewrites 73.4%

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

6. Taylor expanded in i around 0

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

8. Applied rewrites 55.5%

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

9. Taylor expanded in i around inf

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]
10. Step-by-step derivation

11. Applied rewrites 54.9%

    \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \]
12. Add Preprocessing

Alternative 14: 54.0% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in n around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-expm1.f64 73.4

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

5. Applied rewrites 73.4%

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

6. Taylor expanded in i around 0

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

8. Applied rewrites 55.5%

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

9. Taylor expanded in i around 0

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

11. Applied rewrites 54.7%

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

Alternative 15: 54.0% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow-to-exp N/A

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

   13. lower-expm1.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-/.f64 74.6

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

4. Applied rewrites 74.6%

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

5. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 54.4

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

7. Applied rewrites 54.4%

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

8. Taylor expanded in n around inf

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

10. Applied rewrites 54.7%

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

Alternative 16: 48.5% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in i around 0

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

   1. lower-*.f64 46.6

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

5. Applied rewrites 46.6%

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

Developer Target 1: 34.7% 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 2024329 
(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))))