Compound Interest

Percentage Accurate: 28.6% → 93.2%

Time: 10.6s

Alternatives: 13

Speedup: 24.3×

• could not determine a ground truth

  1. i = 2.0164552468763506e-52
  2. n = 7.448967507267318e+37

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.6% 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: 93.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = 100.0 * (t_0 / (i / n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) * (100.0 / i)) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((100.0 * t_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;
	double t_1 = 100.0 * (t_0 / (i / n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) * (100.0 / i)) * n;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((100.0 * t_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
	t_1 = 100.0 * (t_0 / (i / n))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.expm1((math.log1p((i / n)) * n)) * (100.0 / i)) * n
	elif t_1 <= math.inf:
		tmp = ((100.0 * t_0) / i) * n
	else:
		tmp = 100.0 * n
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.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 26.1%

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

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

      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

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

   4. Applied rewrites 98.5%

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

   if -0.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 99.6%

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

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

      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

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

   4. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.0

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

   6. Applied rewrites 73.0%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if +inf.0 < (*.f64 #s(literal 100 binary64) (/.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 82.5

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

   5. Applied rewrites 82.5%

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

Alternative 2: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.05 \cdot 10^{-191}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\left(\left(\left(\log i - \log n\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -1.05e-191)
     (* 100.0 (* t_0 n))
     (if (<= n -2e-310)
       (* (/ (* 100.0 (- (pow (+ 1.0 (/ i n)) n) 1.0)) i) n)
       (if (<= n 1.8e-112)
         (* (* (* (- (log i) (log n)) n) (/ 100.0 i)) n)
         (* (* t_0 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -1.05e-191) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -2e-310) {
		tmp = ((100.0 * (pow((1.0 + (i / n)), n) - 1.0)) / i) * n;
	} else if (n <= 1.8e-112) {
		tmp = (((log(i) - log(n)) * n) * (100.0 / i)) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -1.05e-191) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -2e-310) {
		tmp = ((100.0 * (Math.pow((1.0 + (i / n)), n) - 1.0)) / i) * n;
	} else if (n <= 1.8e-112) {
		tmp = (((Math.log(i) - Math.log(n)) * n) * (100.0 / i)) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -1.05e-191:
		tmp = 100.0 * (t_0 * n)
	elif n <= -2e-310:
		tmp = ((100.0 * (math.pow((1.0 + (i / n)), n) - 1.0)) / i) * n
	elif n <= 1.8e-112:
		tmp = (((math.log(i) - math.log(n)) * n) * (100.0 / i)) * n
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -1.05e-191)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= -2e-310)
		tmp = Float64(Float64(Float64(100.0 * Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0)) / i) * n);
	elseif (n <= 1.8e-112)
		tmp = Float64(Float64(Float64(Float64(log(i) - log(n)) * n) * Float64(100.0 / i)) * n);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.05e-191], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2e-310], N[(N[(N[(100.0 * N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.8e-112], N[(N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.04999999999999993e-191

   1. Initial program 22.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -1.04999999999999993e-191 < n < -1.999999999999994e-310

   1. Initial program 78.1%

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

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

      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

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 78.1

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

   8. Applied rewrites 78.1%

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

   if -1.999999999999994e-310 < n < 1.8e-112

   1. Initial program 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in n around 0

      \[\leadsto \left(\color{blue}{\left(n \cdot \left(\log i + -1 \cdot \log n\right)\right)} \cdot \frac{100}{i}\right) \cdot n \]
   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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if 1.8e-112 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-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 87.8

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

   5. Applied rewrites 87.8%

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

Alternative 3: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.05 \cdot 10^{-191}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\left(\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -1.05e-191)
     (* 100.0 (* t_0 n))
     (if (<= n -2e-310)
       (* (/ (* 100.0 (- (pow (+ 1.0 (/ i n)) n) 1.0)) i) n)
       (if (<= n 1.8e-112)
         (* (* (* n (/ (- (log i) (log n)) i)) 100.0) n)
         (* (* t_0 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -1.05e-191) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -2e-310) {
		tmp = ((100.0 * (pow((1.0 + (i / n)), n) - 1.0)) / i) * n;
	} else if (n <= 1.8e-112) {
		tmp = ((n * ((log(i) - log(n)) / i)) * 100.0) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -1.05e-191) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -2e-310) {
		tmp = ((100.0 * (Math.pow((1.0 + (i / n)), n) - 1.0)) / i) * n;
	} else if (n <= 1.8e-112) {
		tmp = ((n * ((Math.log(i) - Math.log(n)) / i)) * 100.0) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -1.05e-191:
		tmp = 100.0 * (t_0 * n)
	elif n <= -2e-310:
		tmp = ((100.0 * (math.pow((1.0 + (i / n)), n) - 1.0)) / i) * n
	elif n <= 1.8e-112:
		tmp = ((n * ((math.log(i) - math.log(n)) / i)) * 100.0) * n
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -1.05e-191)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= -2e-310)
		tmp = Float64(Float64(Float64(100.0 * Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0)) / i) * n);
	elseif (n <= 1.8e-112)
		tmp = Float64(Float64(Float64(n * Float64(Float64(log(i) - log(n)) / i)) * 100.0) * n);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.05e-191], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2e-310], N[(N[(N[(100.0 * N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.8e-112], N[(N[(N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.04999999999999993e-191

   1. Initial program 22.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -1.04999999999999993e-191 < n < -1.999999999999994e-310

   1. Initial program 78.1%

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

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

      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

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 78.1

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

   8. Applied rewrites 78.1%

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

   if -1.999999999999994e-310 < n < 1.8e-112

   1. Initial program 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-log.f64 83.1

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

   7. Applied rewrites 83.1%

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

   if 1.8e-112 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-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 87.8

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

   5. Applied rewrites 87.8%

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

Alternative 4: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-190} \lor \neg \left(n \leq 8 \cdot 10^{-113}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.05e-190) (not (<= n 8e-113)))
   (* (* (/ (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 <= -1.05e-190) || !(n <= 8e-113)) {
		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 <= -1.05e-190) || !(n <= 8e-113)) {
		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 <= -1.05e-190) or not (n <= 8e-113):
		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 <= -1.05e-190) || !(n <= 8e-113))
		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, -1.05e-190], N[Not[LessEqual[n, 8e-113]], $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 < -1.04999999999999996e-190 or 7.99999999999999983e-113 < n

   1. Initial program 23.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if -1.04999999999999996e-190 < n < 7.99999999999999983e-113

   1. Initial program 52.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-190} \lor \neg \left(n \leq 8 \cdot 10^{-113}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.05 \cdot 10^{-190}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-113}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -1.05e-190)
     (* 100.0 (* t_0 n))
     (if (<= n 8e-113)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -1.05e-190) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= 8e-113) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -1.05e-190) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= 8e-113) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -1.05e-190:
		tmp = 100.0 * (t_0 * n)
	elif n <= 8e-113:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -1.05e-190)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= 8e-113)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.05e-190], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8e-113], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.04999999999999996e-190

   1. Initial program 22.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -1.04999999999999996e-190 < n < 7.99999999999999983e-113

   1. Initial program 52.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.5%

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

   if 7.99999999999999983e-113 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-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 87.8

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

   5. Applied rewrites 87.8%

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

Alternative 6: 64.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-190}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-113}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot i}{i}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.2e-190)
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (if (<= n 8e-113)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (* (* 100.0 (/ (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) i) i)) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.2e-190) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else if (n <= 8e-113) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * ((fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * i) / i)) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.2e-190)
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	elseif (n <= 8e-113)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(100.0 * Float64(Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * i) / i)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.2e-190], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8e-113], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2e-190

   1. Initial program 22.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 60.1%

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

   if -1.2e-190 < n < 7.99999999999999983e-113

   1. Initial program 52.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.5%

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

   if 7.99999999999999983e-113 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 52.4%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{i}}{\frac{i}{n}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 72.5

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

   10. Applied rewrites 72.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-190}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-113}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot i}{i}\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-190} \lor \neg \left(n \leq 8 \cdot 10^{-113}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.2e-190) (not (<= n 8e-113)))
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.2e-190) || !(n <= 8e-113)) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.2e-190) || !(n <= 8e-113))
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, 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, -1.2e-190], N[Not[LessEqual[n, 8e-113]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $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 < -1.2e-190 or 7.99999999999999983e-113 < n

   1. Initial program 23.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 65.4%

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

   if -1.2e-190 < n < 7.99999999999999983e-113

   1. Initial program 52.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.5%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-190} \lor \neg \left(n \leq 8 \cdot 10^{-113}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 4.5 \cdot 10^{+27}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 4.5e+27)
   (* 100.0 n)
   (* 100.0 (* (* (* i i) n) 0.16666666666666666))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 4.5e+27) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * (((i * i) * n) * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 4.5d+27) then
        tmp = 100.0d0 * n
    else
        tmp = 100.0d0 * (((i * i) * n) * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 4.5e+27) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * (((i * i) * n) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 4.5e+27:
		tmp = 100.0 * n
	else:
		tmp = 100.0 * (((i * i) * n) * 0.16666666666666666)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 4.5e+27)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(i * i) * n) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 4.5e+27)
		tmp = 100.0 * n;
	else
		tmp = 100.0 * (((i * i) * n) * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 4.5e+27], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(N[(N[(i * i), $MachinePrecision] * n), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 4.4999999999999999e27

   1. Initial program 22.3%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 4.4999999999999999e27 < i

   1. Initial program 51.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 40.2%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 40.2%

       \[\leadsto 100 \cdot \left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

3. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 56.3%

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

Alternative 10: 56.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

3. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 56.2%

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

Alternative 11: 55.9% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

3. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 56.2%

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

9. Taylor expanded in i around inf

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

11. Applied rewrites 55.9%

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

Alternative 12: 54.2% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

3. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 56.2%

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

9. Taylor expanded in i around 0

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

11. Applied rewrites 54.0%

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

Alternative 13: 49.0% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

3. Taylor expanded in i around 0

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

   1. lower-*.f64 48.7

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

5. Applied rewrites 48.7%

   \[\leadsto \color{blue}{100 \cdot n} \]
6. 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 2024343 
(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))))