Compound Interest

Percentage Accurate: 28.2% → 94.6%

Time: 13.0s

Alternatives: 19

Speedup: 24.3×

• could not determine a ground truth

  1. i = 6.705006426195702e-84
  2. n = 7.822323811878364e+37

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: 28.2% 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: 94.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 24.1%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-log1p.f64 99.1

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

   4. Applied rewrites 99.1%

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

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

   1. Initial program 96.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. distribute-neg-frac N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 97.1

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

   4. Applied rewrites 97.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 96.2%

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

Alternative 2: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 10^{-271}:\\ \;\;\;\;\frac{\left(1 - 1\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{1 \cdot i}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{\left(\log i - \log n\right) \cdot \left(\left(n \cdot n\right) \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -8.2e-126)
     t_0
     (if (<= n 1e-271)
       (/ (* (- 1.0 1.0) (* 100.0 n)) i)
       (if (<= n 1.08e-72)
         (* (/ (* 1.0 i) (/ i n)) 100.0)
         (if (<= n 4.7e-28)
           (/ (* (- (log i) (log n)) (* (* n n) 100.0)) i)
           t_0))))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -8.2e-126) {
		tmp = t_0;
	} else if (n <= 1e-271) {
		tmp = ((1.0 - 1.0) * (100.0 * n)) / i;
	} else if (n <= 1.08e-72) {
		tmp = ((1.0 * i) / (i / n)) * 100.0;
	} else if (n <= 4.7e-28) {
		tmp = ((log(i) - log(n)) * ((n * n) * 100.0)) / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -8.2e-126) {
		tmp = t_0;
	} else if (n <= 1e-271) {
		tmp = ((1.0 - 1.0) * (100.0 * n)) / i;
	} else if (n <= 1.08e-72) {
		tmp = ((1.0 * i) / (i / n)) * 100.0;
	} else if (n <= 4.7e-28) {
		tmp = ((Math.log(i) - Math.log(n)) * ((n * n) * 100.0)) / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -8.2e-126:
		tmp = t_0
	elif n <= 1e-271:
		tmp = ((1.0 - 1.0) * (100.0 * n)) / i
	elif n <= 1.08e-72:
		tmp = ((1.0 * i) / (i / n)) * 100.0
	elif n <= 4.7e-28:
		tmp = ((math.log(i) - math.log(n)) * ((n * n) * 100.0)) / i
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0)
	tmp = 0.0
	if (n <= -8.2e-126)
		tmp = t_0;
	elseif (n <= 1e-271)
		tmp = Float64(Float64(Float64(1.0 - 1.0) * Float64(100.0 * n)) / i);
	elseif (n <= 1.08e-72)
		tmp = Float64(Float64(Float64(1.0 * i) / Float64(i / n)) * 100.0);
	elseif (n <= 4.7e-28)
		tmp = Float64(Float64(Float64(log(i) - log(n)) * Float64(Float64(n * n) * 100.0)) / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -8.2e-126], t$95$0, If[LessEqual[n, 1e-271], N[(N[(N[(1.0 - 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.08e-72], N[(N[(N[(1.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 4.7e-28], N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * N[(N[(n * n), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.1999999999999995e-126 or 4.6999999999999996e-28 < n

   1. Initial program 21.6%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 91.9

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 91.9%

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

   if -8.1999999999999995e-126 < n < 9.99999999999999963e-272

   1. Initial program 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-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. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 65.6

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 71.8%

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

   if 9.99999999999999963e-272 < n < 1.07999999999999998e-72

   1. Initial program 13.0%

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

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i}}{\frac{i}{n}} \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i}}{\frac{i}{n}} \]

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 74.0%

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

   if 1.07999999999999998e-72 < n < 4.6999999999999996e-28

   1. Initial program 4.8%

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

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 4.8

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

   4. Applied rewrites 4.8%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 85.7

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

   7. Applied rewrites 85.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 10^{-271}:\\ \;\;\;\;\frac{\left(1 - 1\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{1 \cdot i}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{\left(\log i - \log n\right) \cdot \left(\left(n \cdot n\right) \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 10^{-271}:\\ \;\;\;\;\frac{\left(1 - 1\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{1 \cdot i}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{\log i - \log n}{i} \cdot \left(\left(n \cdot n\right) \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -8.2e-126)
     t_0
     (if (<= n 1e-271)
       (/ (* (- 1.0 1.0) (* 100.0 n)) i)
       (if (<= n 1.08e-72)
         (* (/ (* 1.0 i) (/ i n)) 100.0)
         (if (<= n 4.7e-28)
           (* (/ (- (log i) (log n)) i) (* (* n n) 100.0))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -8.2e-126) {
		tmp = t_0;
	} else if (n <= 1e-271) {
		tmp = ((1.0 - 1.0) * (100.0 * n)) / i;
	} else if (n <= 1.08e-72) {
		tmp = ((1.0 * i) / (i / n)) * 100.0;
	} else if (n <= 4.7e-28) {
		tmp = ((log(i) - log(n)) / i) * ((n * n) * 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -8.2e-126) {
		tmp = t_0;
	} else if (n <= 1e-271) {
		tmp = ((1.0 - 1.0) * (100.0 * n)) / i;
	} else if (n <= 1.08e-72) {
		tmp = ((1.0 * i) / (i / n)) * 100.0;
	} else if (n <= 4.7e-28) {
		tmp = ((Math.log(i) - Math.log(n)) / i) * ((n * n) * 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -8.2e-126:
		tmp = t_0
	elif n <= 1e-271:
		tmp = ((1.0 - 1.0) * (100.0 * n)) / i
	elif n <= 1.08e-72:
		tmp = ((1.0 * i) / (i / n)) * 100.0
	elif n <= 4.7e-28:
		tmp = ((math.log(i) - math.log(n)) / i) * ((n * n) * 100.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0)
	tmp = 0.0
	if (n <= -8.2e-126)
		tmp = t_0;
	elseif (n <= 1e-271)
		tmp = Float64(Float64(Float64(1.0 - 1.0) * Float64(100.0 * n)) / i);
	elseif (n <= 1.08e-72)
		tmp = Float64(Float64(Float64(1.0 * i) / Float64(i / n)) * 100.0);
	elseif (n <= 4.7e-28)
		tmp = Float64(Float64(Float64(log(i) - log(n)) / i) * Float64(Float64(n * n) * 100.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -8.2e-126], t$95$0, If[LessEqual[n, 1e-271], N[(N[(N[(1.0 - 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.08e-72], N[(N[(N[(1.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 4.7e-28], N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * N[(N[(n * n), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.1999999999999995e-126 or 4.6999999999999996e-28 < n

   1. Initial program 21.6%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 91.9

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 91.9%

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

   if -8.1999999999999995e-126 < n < 9.99999999999999963e-272

   1. Initial program 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-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. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 65.6

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 71.8%

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

   if 9.99999999999999963e-272 < n < 1.07999999999999998e-72

   1. Initial program 13.0%

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

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}}{\frac{i}{n}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i}}{\frac{i}{n}} \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i}}{\frac{i}{n}} \]

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 74.0%

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

   if 1.07999999999999998e-72 < n < 4.6999999999999996e-28

   1. Initial program 4.8%

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

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.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 85.5

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

   5. Applied rewrites 85.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 10^{-271}:\\ \;\;\;\;\frac{\left(1 - 1\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{1 \cdot i}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{\log i - \log n}{i} \cdot \left(\left(n \cdot n\right) \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(100 \cdot n\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\\ \mathbf{if}\;i \leq -1.05 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 1.85 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100, i, 100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* (* 100.0 n) (expm1 (* (log1p (/ i n)) n))) i)))
   (if (<= i -1.05e-52)
     t_0
     (if (<= i 1.85e-159)
       (fma (* (* (- 0.5 (/ 0.5 n)) n) 100.0) i (* 100.0 n))
       t_0))))

C

double code(double i, double n) {
	double t_0 = ((100.0 * n) * expm1((log1p((i / n)) * n))) / i;
	double tmp;
	if (i <= -1.05e-52) {
		tmp = t_0;
	} else if (i <= 1.85e-159) {
		tmp = fma((((0.5 - (0.5 / n)) * n) * 100.0), i, (100.0 * n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(100.0 * n) * expm1(Float64(log1p(Float64(i / n)) * n))) / i)
	tmp = 0.0
	if (i <= -1.05e-52)
		tmp = t_0;
	elseif (i <= 1.85e-159)
		tmp = fma(Float64(Float64(Float64(0.5 - Float64(0.5 / n)) * n) * 100.0), i, Float64(100.0 * n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(100.0 * n), $MachinePrecision] * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[i, -1.05e-52], t$95$0, If[LessEqual[i, 1.85e-159], N[(N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.0499999999999999e-52 or 1.8499999999999999e-159 < i

   1. Initial program 38.0%

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

      1. lift-*.f64 N/A

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

      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. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 38.1

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

   4. Applied rewrites 38.1%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-expm1.f64 88.9

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

   6. Applied rewrites 88.9%

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

   if -1.0499999999999999e-52 < i < 1.8499999999999999e-159

   1. Initial program 6.8%

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

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 7.4

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

   4. Applied rewrites 7.4%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot i} + 100 \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), i, 100 \cdot n\right)} \]

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 91.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.05 \cdot 10^{-52}:\\ \;\;\;\;\frac{\left(100 \cdot n\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\\ \mathbf{elif}\;i \leq 1.85 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n\right) \cdot 100, i, 100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(100 \cdot n\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -8.2e-126)
     t_0
     (if (<= n 2.3e-211) (* (/ (- 1.0 1.0) (/ i n)) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -8.2e-126) {
		tmp = t_0;
	} else if (n <= 2.3e-211) {
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -8.2e-126) {
		tmp = t_0;
	} else if (n <= 2.3e-211) {
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -8.2e-126:
		tmp = t_0
	elif n <= 2.3e-211:
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0)
	tmp = 0.0
	if (n <= -8.2e-126)
		tmp = t_0;
	elseif (n <= 2.3e-211)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / Float64(i / n)) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -8.2e-126], t$95$0, If[LessEqual[n, 2.3e-211], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.1999999999999995e-126 or 2.29999999999999988e-211 < n

   1. Initial program 19.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 85.8

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 85.9%

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

   if -8.1999999999999995e-126 < n < 2.29999999999999988e-211

   1. Initial program 58.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 71.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -8.2e-126)
     t_0
     (if (<= n 2.3e-211) (* (/ (- 1.0 1.0) (/ i n)) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -8.2e-126) {
		tmp = t_0;
	} else if (n <= 2.3e-211) {
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -8.2e-126) {
		tmp = t_0;
	} else if (n <= 2.3e-211) {
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -8.2e-126:
		tmp = t_0
	elif n <= 2.3e-211:
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -8.2e-126)
		tmp = t_0;
	elseif (n <= 2.3e-211)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / Float64(i / n)) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -8.2e-126], t$95$0, If[LessEqual[n, 2.3e-211], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.1999999999999995e-126 or 2.29999999999999988e-211 < n

   1. Initial program 19.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -8.1999999999999995e-126 < n < 2.29999999999999988e-211

   1. Initial program 58.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 71.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.8:\\ \;\;\;\;\mathsf{fma}\left(i, \frac{n}{i \cdot i}, \frac{-n}{i}\right) \cdot 100\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i\right) \cdot \left(100 \cdot n\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.8)
   (* (fma i (/ n (* i i)) (/ (- n) i)) 100.0)
   (if (<= i 2.2e-195)
     (fma
      n
      100.0
      (* (* (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) n) i))
     (/
      (*
       (*
        (fma
         (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
         i
         1.0)
        i)
       (* 100.0 n))
      i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.8) {
		tmp = fma(i, (n / (i * i)), (-n / i)) * 100.0;
	} else if (i <= 2.2e-195) {
		tmp = fma(n, 100.0, ((fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	} else {
		tmp = ((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) * (100.0 * n)) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.8)
		tmp = Float64(fma(i, Float64(n / Float64(i * i)), Float64(Float64(-n) / i)) * 100.0);
	elseif (i <= 2.2e-195)
		tmp = fma(n, 100.0, Float64(Float64(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) * Float64(100.0 * n)) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.8], N[(N[(i * N[(n / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[((-n) / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[i, 2.2e-195], N[(n * 100.0 + N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * n), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.7999999999999998

   1. Initial program 63.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-log1p.f64 97.3

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

   4. Applied rewrites 97.3%

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

   6. Applied rewrites 58.5%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

   if -2.7999999999999998 < i < 2.20000000000000005e-195

   1. Initial program 8.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-/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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 86.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50 \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 86.9%

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

   if 2.20000000000000005e-195 < i

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      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. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 25.2

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in n around inf

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

      1. lower-expm1.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 67.7%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \color{blue}{i}\right) \cdot \left(100 \cdot n\right)}{i} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.8:\\ \;\;\;\;\mathsf{fma}\left(i, \frac{n}{i \cdot i}, \frac{-n}{i}\right) \cdot 100\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i\right) \cdot \left(100 \cdot n\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.8:\\ \;\;\;\;\mathsf{fma}\left(i, \frac{n}{i \cdot i}, \frac{-n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.8)
   (* (fma i (/ n (* i i)) (/ (- n) i)) 100.0)
   (fma
    n
    100.0
    (* (* (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) n) i))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < -2.7999999999999998

   1. Initial program 63.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-log1p.f64 97.3

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

   4. Applied rewrites 97.3%

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

   6. Applied rewrites 58.5%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

   if -2.7999999999999998 < i

   1. Initial program 16.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 75.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50 \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 75.7%

       \[\leadsto \mathsf{fma}\left(n, 100, \left(n \cdot \mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right)\right) \cdot i\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.8:\\ \;\;\;\;\mathsf{fma}\left(i, \frac{n}{i \cdot i}, \frac{-n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \mathbf{if}\;n \leq -2.95 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (fma
          n
          100.0
          (*
           (* (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) n)
           i))))
   (if (<= n -2.95e-120)
     t_0
     (if (<= n 2.3e-211) (* (/ (- 1.0 1.0) (/ i n)) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(n, 100.0, ((fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	double tmp;
	if (n <= -2.95e-120) {
		tmp = t_0;
	} else if (n <= 2.3e-211) {
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = fma(n, 100.0, Float64(Float64(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i))
	tmp = 0.0
	if (n <= -2.95e-120)
		tmp = t_0;
	elseif (n <= 2.3e-211)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / Float64(i / n)) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * 100.0 + N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * n), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.95e-120], t$95$0, If[LessEqual[n, 2.3e-211], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.94999999999999989e-120 or 2.29999999999999988e-211 < n

   1. Initial program 19.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 69.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50 \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.4%

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

   if -2.94999999999999989e-120 < n < 2.29999999999999988e-211

   1. Initial program 58.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 71.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \mathbf{if}\;n \leq -2.95 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{\left(1 - 1\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (fma
          n
          100.0
          (*
           (* (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) n)
           i))))
   (if (<= n -2.95e-120)
     t_0
     (if (<= n 2.3e-211) (/ (* (- 1.0 1.0) (* 100.0 n)) i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(n, 100.0, ((fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	double tmp;
	if (n <= -2.95e-120) {
		tmp = t_0;
	} else if (n <= 2.3e-211) {
		tmp = ((1.0 - 1.0) * (100.0 * n)) / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = fma(n, 100.0, Float64(Float64(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i))
	tmp = 0.0
	if (n <= -2.95e-120)
		tmp = t_0;
	elseif (n <= 2.3e-211)
		tmp = Float64(Float64(Float64(1.0 - 1.0) * Float64(100.0 * n)) / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * 100.0 + N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * n), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.95e-120], t$95$0, If[LessEqual[n, 2.3e-211], N[(N[(N[(1.0 - 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.94999999999999989e-120 or 2.29999999999999988e-211 < n

   1. Initial program 19.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 69.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50 \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.4%

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

   if -2.94999999999999989e-120 < n < 2.29999999999999988e-211

   1. Initial program 58.2%

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

      1. lift-*.f64 N/A

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

      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. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.4

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 71.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.95 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{\left(1 - 1\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.95 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{\left(1 - 1\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
           i
           100.0)
          n)))
   (if (<= n -2.95e-120)
     t_0
     (if (<= n 2.3e-211) (/ (* (- 1.0 1.0) (* 100.0 n)) i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -2.95e-120) {
		tmp = t_0;
	} else if (n <= 2.3e-211) {
		tmp = ((1.0 - 1.0) * (100.0 * n)) / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -2.95e-120)
		tmp = t_0;
	elseif (n <= 2.3e-211)
		tmp = Float64(Float64(Float64(1.0 - 1.0) * Float64(100.0 * n)) / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.95e-120], t$95$0, If[LessEqual[n, 2.3e-211], N[(N[(N[(1.0 - 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.94999999999999989e-120 or 2.29999999999999988e-211 < n

   1. Initial program 19.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 69.4%

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

   if -2.94999999999999989e-120 < n < 2.29999999999999988e-211

   1. Initial program 58.2%

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

      1. lift-*.f64 N/A

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

      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. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.4

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 71.3%

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

Alternative 12: 63.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < -2.7999999999999998

   1. Initial program 63.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 30.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 30.7

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

   6. Applied rewrites 30.7%

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

   7. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

         \[\leadsto n \cdot \frac{100 \cdot \left(\color{blue}{0} \cdot i\right)}{{i}^{2}} \]

      9. mul0-lft N/A

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{0}}{{i}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto n \cdot \frac{\color{blue}{0}}{{i}^{2}} \]

      11. lower-/.f64 N/A

          \[\leadsto n \cdot \color{blue}{\frac{0}{{i}^{2}}} \]

      12. unpow2 N/A

          \[\leadsto n \cdot \frac{0}{\color{blue}{i \cdot i}} \]

      13. lower-*.f64 33.1

          \[\leadsto n \cdot \frac{0}{\color{blue}{i \cdot i}} \]

   9. Applied rewrites 33.1%

      \[\leadsto \color{blue}{n \cdot \frac{0}{i \cdot i}} \]

   if -2.7999999999999998 < i

   1. Initial program 16.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 75.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.8:\\ \;\;\;\;\frac{0}{i \cdot i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+84}:\\ \;\;\;\;\frac{0}{i \cdot i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2e+84)
   (* (/ 0.0 (* i i)) n)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2e+84) {
		tmp = (0.0 / (i * i)) * n;
	} else {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2e+84)
		tmp = Float64(Float64(0.0 / Float64(i * i)) * n);
	else
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2e+84], N[(N[(0.0 / N[(i * i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2.00000000000000012e84

   1. Initial program 75.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 31.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 31.6

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

   6. Applied rewrites 31.6%

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

   7. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

         \[\leadsto n \cdot \frac{100 \cdot \left(\color{blue}{0} \cdot i\right)}{{i}^{2}} \]

      9. mul0-lft N/A

         \[\leadsto n \cdot \frac{100 \cdot \color{blue}{0}}{{i}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto n \cdot \frac{\color{blue}{0}}{{i}^{2}} \]

      11. lower-/.f64 N/A

          \[\leadsto n \cdot \color{blue}{\frac{0}{{i}^{2}}} \]

      12. unpow2 N/A

          \[\leadsto n \cdot \frac{0}{\color{blue}{i \cdot i}} \]

      13. lower-*.f64 42.7

          \[\leadsto n \cdot \frac{0}{\color{blue}{i \cdot i}} \]

   9. Applied rewrites 42.7%

      \[\leadsto \color{blue}{n \cdot \frac{0}{i \cdot i}} \]

   if -2.00000000000000012e84 < i

   1. Initial program 16.8%

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

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 17.2

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

   4. Applied rewrites 17.2%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot i} + 100 \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), i, 100 \cdot n\right)} \]

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 70.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot \color{blue}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+84}:\\ \;\;\;\;\frac{0}{i \cdot i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.3% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.5%

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

   1. lift-*.f64 N/A

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

   2. *-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. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 25.9

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

4. Applied rewrites 25.9%

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

5. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot n} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot i} + 100 \cdot n \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), i, 100 \cdot n\right)} \]

7. Applied rewrites 56.3%

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

8. Taylor expanded in n around inf

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

10. Applied rewrites 60.3%

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

Alternative 15: 54.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{+45}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(50 \cdot i\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 5e+45) (* 100.0 n) (* (* 50.0 i) n)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 5e+45) {
		tmp = 100.0 * n;
	} else {
		tmp = (50.0 * i) * n;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 5d+45) then
        tmp = 100.0d0 * n
    else
        tmp = (50.0d0 * i) * n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 5e+45) {
		tmp = 100.0 * n;
	} else {
		tmp = (50.0 * i) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 5e+45:
		tmp = 100.0 * n
	else:
		tmp = (50.0 * i) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 5e+45)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(Float64(50.0 * i) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 5e+45)
		tmp = 100.0 * n;
	else
		tmp = (50.0 * i) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 5e+45], N[(100.0 * n), $MachinePrecision], N[(N[(50.0 * i), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5e45

   1. Initial program 22.1%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

   if 5e45 < i

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 34.5%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 34.5%

       \[\leadsto \left(50 \cdot i\right) \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 54.8% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.5%

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

3. Taylor expanded in n around inf

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

   1. associate-/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. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 79.0

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

5. Applied rewrites 79.0%

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

7. Applied rewrites 79.0%

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

8. Taylor expanded in i around 0

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

10. Applied rewrites 58.3%

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

Alternative 17: 54.7% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.5%

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

3. Taylor expanded in n around inf

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

   1. associate-/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. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in i around 0

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

8. Applied rewrites 61.5%

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

9. Taylor expanded in i around 0

   \[\leadsto \mathsf{fma}\left(50 \cdot n, i, n \cdot 100\right) \]
10. Step-by-step derivation

11. Applied rewrites 58.3%

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

12. Final simplification 58.3%

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

Alternative 18: 54.8% 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 25.5%

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

3. Taylor expanded in n around inf

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

   1. associate-/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. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in i around 0

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

8. Applied rewrites 58.3%

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

Alternative 19: 48.7% 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 25.5%

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

3. Taylor expanded in i around 0

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

   1. lower-*.f64 53.2

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

5. Applied rewrites 53.2%

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

Developer Target 1: 34.4% 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 2024235 
(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))))