Compound Interest

Percentage Accurate: 28.1% → 93.8%

Time: 12.0s

Alternatives: 19

Speedup: 8.1×

• could not determine a ground truth

  1. i = 3.304626849542179e-272
  2. n = 6.659343580009098e+39

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.1% 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: 93.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * ((Math.expm1((Math.log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.pow(((i / n) + 1.0), n) * 100.0) / (i / n)) + ((-100.0 / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) - 1.0) / (i / n)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 100.0 * ((math.expm1((math.log1p((i / n)) * n)) / i) * n)
	elif t_0 <= math.inf:
		tmp = ((math.pow(((i / n) + 1.0), n) * 100.0) / (i / n)) + ((-100.0 / i) * n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] + N[(N[(-100.0 / i), $MachinePrecision] * n), $MachinePrecision]), $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 24.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 24.1

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 97.4

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

   4. Applied rewrites 97.4%

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

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

   1. Initial program 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.5

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

   6. Applied rewrites 96.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

Alternative 2: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\log i - \log n\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-311) (not (<= n 2.6e-67)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* (* (* (- (log i) (log n)) n) (/ 100.0 i)) n)))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = (((log(i) - log(n)) * n) * (100.0 / i)) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = (((Math.log(i) - Math.log(n)) * n) * (100.0 / i)) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-311) or not (n <= 2.6e-67):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = (((math.log(i) - math.log(n)) * n) * (100.0 / i)) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-311) || !(n <= 2.6e-67))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(Float64(Float64(Float64(log(i) - log(n)) * n) * Float64(100.0 / i)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-311], N[Not[LessEqual[n, 2.6e-67]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.00000000000023e-311 or 2.5999999999999999e-67 < n

   1. Initial program 26.8%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -5.00000000000023e-311 < n < 2.5999999999999999e-67

   1. Initial program 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 75.2

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

   7. Applied rewrites 75.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\log i - \log n\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-311) (not (<= n 2.6e-67)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* (* (* n (/ (- (log i) (log n)) i)) 100.0) n)))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((n * ((log(i) - log(n)) / i)) * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((n * ((Math.log(i) - Math.log(n)) / i)) * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-311) or not (n <= 2.6e-67):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = ((n * ((math.log(i) - math.log(n)) / i)) * 100.0) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-311) || !(n <= 2.6e-67))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(Float64(Float64(n * Float64(Float64(log(i) - log(n)) / i)) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-311], N[Not[LessEqual[n, 2.6e-67]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.00000000000023e-311 or 2.5999999999999999e-67 < n

   1. Initial program 26.8%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -5.00000000000023e-311 < n < 2.5999999999999999e-67

   1. Initial program 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-add N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 N/A

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

      16. lower-log.f64 75.2

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

   7. Applied rewrites 75.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\left(\log i - \log n\right) \cdot n}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-311) (not (<= n 2.6e-67)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (* (/ (* (- (log i) (log n)) n) i) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((((log(i) - log(n)) * n) / i) * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((((Math.log(i) - Math.log(n)) * n) / i) * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-311) or not (n <= 2.6e-67):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((((math.log(i) - math.log(n)) * n) / i) * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-311) || !(n <= 2.6e-67))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(Float64(log(i) - log(n)) * n) / i) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-311], N[Not[LessEqual[n, 2.6e-67]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.00000000000023e-311 or 2.5999999999999999e-67 < n

   1. Initial program 26.8%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -5.00000000000023e-311 < n < 2.5999999999999999e-67

   1. Initial program 31.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.8

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 75.1

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

   7. Applied rewrites 75.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\left(\log i - \log n\right) \cdot n}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{\log i - \log n}{i} \cdot n\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-311) (not (<= n 2.6e-67)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (* (* (/ (- (log i) (log n)) i) n) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((((log(i) - log(n)) / i) * n) * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((((Math.log(i) - Math.log(n)) / i) * n) * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-311) or not (n <= 2.6e-67):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((((math.log(i) - math.log(n)) / i) * n) * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-311) || !(n <= 2.6e-67))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(Float64(log(i) - log(n)) / i) * n) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-311], N[Not[LessEqual[n, 2.6e-67]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.00000000000023e-311 or 2.5999999999999999e-67 < n

   1. Initial program 26.8%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -5.00000000000023e-311 < n < 2.5999999999999999e-67

   1. Initial program 31.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.8

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. div-add N/A

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

      3. associate-*r/ N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-log.f64 75.0

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

   7. Applied rewrites 75.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\frac{\log i - \log n}{i} \cdot n\right) \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot n\right) \cdot 100\right) \cdot \frac{\log i - \log n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-311) (not (<= n 2.6e-67)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* (* (* n n) 100.0) (/ (- (log i) (log n)) i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((n * n) * 100.0) * ((log(i) - log(n)) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-311) || !(n <= 2.6e-67)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((n * n) * 100.0) * ((Math.log(i) - Math.log(n)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-311) or not (n <= 2.6e-67):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = ((n * n) * 100.0) * ((math.log(i) - math.log(n)) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-311) || !(n <= 2.6e-67))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(Float64(Float64(n * n) * 100.0) * Float64(Float64(log(i) - log(n)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-311], N[Not[LessEqual[n, 2.6e-67]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(n * n), $MachinePrecision] * 100.0), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.00000000000023e-311 or 2.5999999999999999e-67 < n

   1. Initial program 26.8%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -5.00000000000023e-311 < n < 2.5999999999999999e-67

   1. Initial program 31.4%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-log.f64 69.5

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

   5. Applied rewrites 69.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-311} \lor \neg \left(n \leq 2.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot n\right) \cdot 100\right) \cdot \frac{\log i - \log n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{-233} \lor \neg \left(n \leq 1.95 \cdot 10^{-184}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.1e-233) (not (<= n 1.95e-184)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.1e-233) || !(n <= 1.95e-184)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.1e-233) || !(n <= 1.95e-184)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.1e-233) or not (n <= 1.95e-184):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.1e-233) || !(n <= 1.95e-184))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.1e-233], N[Not[LessEqual[n, 1.95e-184]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.0999999999999999e-233 or 1.94999999999999997e-184 < n

   1. Initial program 23.8%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if -2.0999999999999999e-233 < n < 1.94999999999999997e-184

   1. Initial program 57.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 76.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{-233} \lor \neg \left(n \leq 1.95 \cdot 10^{-184}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(100 \cdot \left(\left(0.5 - \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{i}{n}, 0.5 \cdot i\right) + 0.5}{n}\right) + 0.16666666666666666 \cdot i\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-193)
   (*
    (fma
     (*
      100.0
      (+
       (- 0.5 (/ (+ (fma -0.3333333333333333 (/ i n) (* 0.5 i)) 0.5) n))
       (* 0.16666666666666666 i)))
     i
     100.0)
    n)
   (if (<= n 1.95e-184)
     0.0
     (*
      (*
       (/
        (*
         (fma
          (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
          i
          1.0)
         i)
        i)
       100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-193) {
		tmp = fma((100.0 * ((0.5 - ((fma(-0.3333333333333333, (i / n), (0.5 * i)) + 0.5) / n)) + (0.16666666666666666 * i))), i, 100.0) * n;
	} else if (n <= 1.95e-184) {
		tmp = 0.0;
	} else {
		tmp = (((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e-193)
		tmp = Float64(fma(Float64(100.0 * Float64(Float64(0.5 - Float64(Float64(fma(-0.3333333333333333, Float64(i / n), Float64(0.5 * i)) + 0.5) / n)) + Float64(0.16666666666666666 * i))), i, 100.0) * n);
	elseif (n <= 1.95e-184)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e-193], N[(N[(N[(100.0 * N[(N[(0.5 - N[(N[(N[(-0.3333333333333333 * N[(i / n), $MachinePrecision] + N[(0.5 * i), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.95e-184], 0.0, N[(N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.09999999999999988e-193

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 74.7

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 54.8%

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

   8. Taylor expanded in n around -inf

      \[\leadsto \mathsf{fma}\left(100 \cdot \left(\frac{1}{2} + \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{-1}{3} \cdot \frac{i}{n} + \frac{1}{2} \cdot i\right)}{n} + \frac{1}{6} \cdot i\right)\right), i, 100\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 58.2%

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

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in i around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-inverses N/A

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

      4. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]

   if 1.94999999999999997e-184 < n

   1. Initial program 19.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 70.3%

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

Alternative 9: 66.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\left(0.5 - \frac{\mathsf{fma}\left(1 + i, 0.5, -0.3333333333333333 \cdot \frac{i}{n}\right)}{n}\right) + 0.16666666666666666 \cdot i, i, 1\right) \cdot n\right)\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-193)
   (*
    100.0
    (*
     (fma
      (+
       (- 0.5 (/ (fma (+ 1.0 i) 0.5 (* -0.3333333333333333 (/ i n))) n))
       (* 0.16666666666666666 i))
      i
      1.0)
     n))
   (if (<= n 1.95e-184)
     0.0
     (*
      (*
       (/
        (*
         (fma
          (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
          i
          1.0)
         i)
        i)
       100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-193) {
		tmp = 100.0 * (fma(((0.5 - (fma((1.0 + i), 0.5, (-0.3333333333333333 * (i / n))) / n)) + (0.16666666666666666 * i)), i, 1.0) * n);
	} else if (n <= 1.95e-184) {
		tmp = 0.0;
	} else {
		tmp = (((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e-193)
		tmp = Float64(100.0 * Float64(fma(Float64(Float64(0.5 - Float64(fma(Float64(1.0 + i), 0.5, Float64(-0.3333333333333333 * Float64(i / n))) / n)) + Float64(0.16666666666666666 * i)), i, 1.0) * n));
	elseif (n <= 1.95e-184)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e-193], N[(100.0 * N[(N[(N[(N[(0.5 - N[(N[(N[(1.0 + i), $MachinePrecision] * 0.5 + N[(-0.3333333333333333 * N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * i), $MachinePrecision]), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e-184], 0.0, N[(N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.09999999999999988e-193

   1. Initial program 24.9%

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

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 24.8

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 75.4

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 54.8%

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

   8. Taylor expanded in n around -inf

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{-1}{3} \cdot \frac{i}{n} + \frac{1}{2} \cdot i\right)}{n} + \frac{1}{6} \cdot i\right), i, 1\right) \cdot n\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 58.2%

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

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in i around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-inverses N/A

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

      4. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]

   if 1.94999999999999997e-184 < n

   1. Initial program 19.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 70.3%

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

Alternative 10: 66.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-193)
   (fma (* n (fma 16.666666666666668 i 50.0)) i (* 100.0 n))
   (if (<= n 1.95e-184)
     0.0
     (*
      (*
       (/
        (*
         (fma
          (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
          i
          1.0)
         i)
        i)
       100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-193) {
		tmp = fma((n * fma(16.666666666666668, i, 50.0)), i, (100.0 * n));
	} else if (n <= 1.95e-184) {
		tmp = 0.0;
	} else {
		tmp = (((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e-193)
		tmp = fma(Float64(n * fma(16.666666666666668, i, 50.0)), i, Float64(100.0 * n));
	elseif (n <= 1.95e-184)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e-193], N[(N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e-184], 0.0, N[(N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.09999999999999988e-193

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 58.0%

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

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in i around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-inverses N/A

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

      4. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]

   if 1.94999999999999997e-184 < n

   1. Initial program 19.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 70.3%

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

Alternative 11: 65.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-193)
   (fma (* n (fma 16.666666666666668 i 50.0)) i (* 100.0 n))
   (if (<= n 1.95e-184)
     0.0
     (*
      100.0
      (*
       (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
       n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-193) {
		tmp = fma((n * fma(16.666666666666668, i, 50.0)), i, (100.0 * n));
	} else if (n <= 1.95e-184) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e-193)
		tmp = fma(Float64(n * fma(16.666666666666668, i, 50.0)), i, Float64(100.0 * n));
	elseif (n <= 1.95e-184)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e-193], N[(N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e-184], 0.0, N[(100.0 * N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.09999999999999988e-193

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 58.0%

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

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in i around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-inverses N/A

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

      4. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]

   if 1.94999999999999997e-184 < n

   1. Initial program 19.6%

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

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 20.1

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \left(\color{blue}{\left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{n}^{2}} + i \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)} \cdot n\right) \]

   7. Applied rewrites 64.3%

      \[\leadsto 100 \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.4583333333333333}{n \cdot n} + 0.041666666666666664\right) - \left(\frac{0.25}{{n}^{3}} + \frac{0.25}{n}\right), i, \frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), i, 1\right)} \cdot n\right) \]

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 69.4%

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

Alternative 12: 65.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-193)
   (fma (* n (fma 16.666666666666668 i 50.0)) i (* 100.0 n))
   (if (<= n 1.95e-184)
     0.0
     (*
      (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 (n <= -1.1e-193) {
		tmp = fma((n * fma(16.666666666666668, i, 50.0)), i, (100.0 * n));
	} else if (n <= 1.95e-184) {
		tmp = 0.0;
	} 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 (n <= -1.1e-193)
		tmp = fma(Float64(n * fma(16.666666666666668, i, 50.0)), i, Float64(100.0 * n));
	elseif (n <= 1.95e-184)
		tmp = 0.0;
	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[n, -1.1e-193], N[(N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e-184], 0.0, 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 3 regimes

2. if n < -1.09999999999999988e-193

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 58.0%

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

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in i around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-inverses N/A

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

      4. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]

   if 1.94999999999999997e-184 < n

   1. Initial program 19.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 69.4%

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

Alternative 13: 64.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-193)
   (fma (* n (fma 16.666666666666668 i 50.0)) i (* 100.0 n))
   (if (<= n 1.95e-184)
     0.0
     (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-193) {
		tmp = fma((n * fma(16.666666666666668, i, 50.0)), i, (100.0 * n));
	} else if (n <= 1.95e-184) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e-193)
		tmp = fma(Float64(n * fma(16.666666666666668, i, 50.0)), i, Float64(100.0 * n));
	elseif (n <= 1.95e-184)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e-193], N[(N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e-184], 0.0, N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.09999999999999988e-193

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 58.0%

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

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in i around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-inverses N/A

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

      4. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]

   if 1.94999999999999997e-184 < n

   1. Initial program 19.6%

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

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 20.1

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 64.8%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 66.9%

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

Alternative 14: 64.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193} \lor \neg \left(n \leq 1.95 \cdot 10^{-184}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.1e-193) (not (<= n 1.95e-184)))
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.1e-193) || !(n <= 1.95e-184)) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.1e-193) || !(n <= 1.95e-184))
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.1e-193], N[Not[LessEqual[n, 1.95e-184]], $MachinePrecision]], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.09999999999999988e-193 or 1.94999999999999997e-184 < n

   1. Initial program 22.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 22.4%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 80.4

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

   7. Applied rewrites 80.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 62.1%

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

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in i around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-inverses N/A

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

      4. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193} \lor \neg \left(n \leq 1.95 \cdot 10^{-184}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-193)
   (fma (* n (fma 16.666666666666668 i 50.0)) i (* 100.0 n))
   (if (<= n 1.95e-184)
     0.0
     (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-193) {
		tmp = fma((n * fma(16.666666666666668, i, 50.0)), i, (100.0 * n));
	} else if (n <= 1.95e-184) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e-193)
		tmp = fma(Float64(n * fma(16.666666666666668, i, 50.0)), i, Float64(100.0 * n));
	elseif (n <= 1.95e-184)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e-193], N[(N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e-184], 0.0, N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.09999999999999988e-193

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in n around inf

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

      1. distribute-lft-out-- N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 58.0%

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

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in i around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-inverses N/A

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

      4. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]

   if 1.94999999999999997e-184 < n

   1. Initial program 19.6%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      5. sub-neg N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\frac{i}{n}} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\frac{i}{n}} \]

      7. div-add N/A

         \[\leadsto \color{blue}{\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      13. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{-100}{\frac{i}{n}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{e^{i}}{i} - 100 \cdot \frac{1}{i}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto n \cdot \color{blue}{\left(100 \cdot \left(\frac{e^{i}}{i} - \frac{1}{i}\right)\right)} \]

      2. div-sub N/A

         \[\leadsto n \cdot \left(100 \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 84.6

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   7. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   8. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 66.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, i, 1\right) \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-193)
   (* (fma 50.0 i 100.0) n)
   (if (<= n 1.95e-184) 0.0 (* (* (fma 0.5 i 1.0) 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-193) {
		tmp = fma(50.0, i, 100.0) * n;
	} else if (n <= 1.95e-184) {
		tmp = 0.0;
	} else {
		tmp = (fma(0.5, i, 1.0) * 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e-193)
		tmp = Float64(fma(50.0, i, 100.0) * n);
	elseif (n <= 1.95e-184)
		tmp = 0.0;
	else
		tmp = Float64(Float64(fma(0.5, i, 1.0) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e-193], N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.95e-184], 0.0, N[(N[(N[(0.5 * i + 1.0), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.09999999999999988e-193

   1. Initial program 24.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      5. sub-neg N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\frac{i}{n}} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\frac{i}{n}} \]

      7. div-add N/A

         \[\leadsto \color{blue}{\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      13. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   4. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{-100}{\frac{i}{n}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{e^{i}}{i} - 100 \cdot \frac{1}{i}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto n \cdot \color{blue}{\left(100 \cdot \left(\frac{e^{i}}{i} - \frac{1}{i}\right)\right)} \]

      2. div-sub N/A

         \[\leadsto n \cdot \left(100 \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 76.8

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   7. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   8. Taylor expanded in i around 0

      \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 56.6%

       \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      5. sub-neg N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\frac{i}{n}} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\frac{i}{n}} \]

      7. div-add N/A

         \[\leadsto \color{blue}{\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      13. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{-100}{\frac{i}{n}}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{-100 \cdot n - \left(\mathsf{neg}\left(100\right)\right) \cdot n}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-100 \cdot n - \color{blue}{-100} \cdot n}{i} \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 70.0

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]

   if 1.94999999999999997e-184 < n

   1. Initial program 19.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      5. sub-neg N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\frac{i}{n}} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\frac{i}{n}} \]

      7. div-add N/A

         \[\leadsto \color{blue}{\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      13. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{-100}{\frac{i}{n}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{e^{i}}{i} - 100 \cdot \frac{1}{i}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto n \cdot \color{blue}{\left(100 \cdot \left(\frac{e^{i}}{i} - \frac{1}{i}\right)\right)} \]

      2. div-sub N/A

         \[\leadsto n \cdot \left(100 \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 84.6

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   7. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   8. Taylor expanded in i around 0

      \[\leadsto \left(\left(1 + \frac{1}{2} \cdot i\right) \cdot 100\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 65.5%

       \[\leadsto \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot 100\right) \cdot n \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193} \lor \neg \left(n \leq 1.95 \cdot 10^{-184}\right):\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.1e-193) (not (<= n 1.95e-184)))
   (* (fma 50.0 i 100.0) n)
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.1e-193) || !(n <= 1.95e-184)) {
		tmp = fma(50.0, i, 100.0) * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.1e-193) || !(n <= 1.95e-184))
		tmp = Float64(fma(50.0, i, 100.0) * n);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.1e-193], N[Not[LessEqual[n, 1.95e-184]], $MachinePrecision]], N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.09999999999999988e-193 or 1.94999999999999997e-184 < n

   1. Initial program 22.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      5. sub-neg N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\frac{i}{n}} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\frac{i}{n}} \]

      7. div-add N/A

         \[\leadsto \color{blue}{\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      13. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   4. Applied rewrites 22.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{-100}{\frac{i}{n}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(100 \cdot \frac{e^{i}}{i} - 100 \cdot \frac{1}{i}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto n \cdot \color{blue}{\left(100 \cdot \left(\frac{e^{i}}{i} - \frac{1}{i}\right)\right)} \]

      2. div-sub N/A

         \[\leadsto n \cdot \left(100 \cdot \color{blue}{\frac{e^{i} - 1}{i}}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 80.4

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   7. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   8. Taylor expanded in i around 0

      \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 60.7%

       \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]

   if -1.09999999999999988e-193 < n < 1.94999999999999997e-184

   1. Initial program 56.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      5. sub-neg N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\frac{i}{n}} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\frac{i}{n}} \]

      7. div-add N/A

         \[\leadsto \color{blue}{\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      13. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{-100}{\frac{i}{n}}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{-100 \cdot n - \left(\mathsf{neg}\left(100\right)\right) \cdot n}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-100 \cdot n - \color{blue}{-100} \cdot n}{i} \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 70.0

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.0%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-193} \lor \neg \left(n \leq 1.95 \cdot 10^{-184}\right):\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -8.8 \cdot 10^{+21}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -8.8e+21) 0.0 (if (<= i 1.25e-15) (* 100.0 n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -8.8e+21) {
		tmp = 0.0;
	} else if (i <= 1.25e-15) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-8.8d+21)) then
        tmp = 0.0d0
    else if (i <= 1.25d-15) then
        tmp = 100.0d0 * n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -8.8e+21) {
		tmp = 0.0;
	} else if (i <= 1.25e-15) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -8.8e+21:
		tmp = 0.0
	elif i <= 1.25e-15:
		tmp = 100.0 * n
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -8.8e+21)
		tmp = 0.0;
	elseif (i <= 1.25e-15)
		tmp = Float64(100.0 * n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -8.8e+21)
		tmp = 0.0;
	elseif (i <= 1.25e-15)
		tmp = 100.0 * n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -8.8e+21], 0.0, If[LessEqual[i, 1.25e-15], N[(100.0 * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -8.8e21 or 1.25e-15 < i

   1. Initial program 53.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      5. sub-neg N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\frac{i}{n}} \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\frac{i}{n}} \]

      7. div-add N/A

         \[\leadsto \color{blue}{\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      13. +-commutative N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   4. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{-100}{\frac{i}{n}}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\color{blue}{-100 \cdot n - \left(\mathsf{neg}\left(100\right)\right) \cdot n}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-100 \cdot n - \color{blue}{-100} \cdot n}{i} \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 26.4

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 26.4%

       \[\leadsto 0 \]

   if -8.8e21 < i < 1.25e-15

   1. Initial program 8.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. lower-*.f64 80.3

         \[\leadsto \color{blue}{100 \cdot n} \]

   5. Applied rewrites 80.3%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 18.2% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 27.4%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

   2. lift-/.f64 N/A

      \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

   5. sub-neg N/A

      \[\leadsto \frac{100 \cdot \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\frac{i}{n}} \]

   6. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n} + 100 \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\frac{i}{n}} \]

   7. div-add N/A

      \[\leadsto \color{blue}{\frac{100 \cdot {\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   9. lower-+.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   13. +-commutative N/A

       \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} \cdot 100}{\frac{i}{n}} + \frac{100 \cdot \left(\mathsf{neg}\left(1\right)\right)}{\frac{i}{n}} \]

4. Applied rewrites 27.3%

   \[\leadsto \color{blue}{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot 100}{\frac{i}{n}} + \frac{-100}{\frac{i}{n}}} \]

5. Taylor expanded in i around 0

   \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\color{blue}{-100 \cdot n - \left(\mathsf{neg}\left(100\right)\right) \cdot n}}{i} \]

   2. metadata-eval N/A

      \[\leadsto \frac{-100 \cdot n - \color{blue}{-100} \cdot n}{i} \]

   3. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0}}{i} \]

   4. lower-/.f64 15.9

      \[\leadsto \color{blue}{\frac{0}{i}} \]

7. Applied rewrites 15.9%

   \[\leadsto \color{blue}{\frac{0}{i}} \]

8. Taylor expanded in i around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 15.9%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 34.1% 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 2024305 
(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))))