Compound Interest

Percentage Accurate: 29.1% → 94.8%

Time: 15.5s

Alternatives: 11

Speedup: 8.1×

• could not determine a ground truth

  1. i = 5.280844226592954e-216
  2. n = -3.5403473076102204e+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: 29.1% 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: 94.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * (100.0 * (n / i))) - ((n * 100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = (t_0 * (100.0 * (n / i))) - ((n * 100.0) / i)
	else:
		tmp = n * 100.0
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 27.5%

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

      1. pow-to-exp N/A

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

      2. accelerator-lowering-expm1.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. accelerator-lowering-log1p.f64 N/A

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

      6. /-lowering-/.f64 99.7

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

   4. Applied egg-rr 99.7%

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

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

   1. Initial program 98.3%

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

      1. pow-to-exp N/A

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

      2. accelerator-lowering-expm1.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. accelerator-lowering-log1p.f64 N/A

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

      6. /-lowering-/.f64 60.0

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

   4. Applied egg-rr 60.0%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. pow-to-exp N/A

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

      9. un-div-inv N/A

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

      10. distribute-frac-neg2 N/A

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

      11. distribute-rgt-in N/A

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

   6. Applied egg-rr 98.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 68.7

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

   5. Simplified 68.7%

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

4. Final simplification 94.1%

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

Alternative 2: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-145}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (* 100.0 (expm1 i)) i))))
   (if (<= n -5e-310)
     t_0
     (if (<= n 1.2e-145)
       (* 100.0 (/ (* n (- (log i) (log n))) (/ i n)))
       t_0))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 * expm1(i)) / i);
	double tmp;
	if (n <= -5e-310) {
		tmp = t_0;
	} else if (n <= 1.2e-145) {
		tmp = 100.0 * ((n * (log(i) - log(n))) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * ((100.0 * Math.expm1(i)) / i);
	double tmp;
	if (n <= -5e-310) {
		tmp = t_0;
	} else if (n <= 1.2e-145) {
		tmp = 100.0 * ((n * (Math.log(i) - Math.log(n))) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * ((100.0 * math.expm1(i)) / i)
	tmp = 0
	if n <= -5e-310:
		tmp = t_0
	elif n <= 1.2e-145:
		tmp = 100.0 * ((n * (math.log(i) - math.log(n))) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 * expm1(i)) / i))
	tmp = 0.0
	if (n <= -5e-310)
		tmp = t_0;
	elseif (n <= 1.2e-145)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(log(i) - log(n))) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5e-310], t$95$0, If[LessEqual[n, 1.2e-145], N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.999999999999985e-310 or 1.20000000000000008e-145 < n

   1. Initial program 25.6%

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

   3. Taylor expanded in n around inf

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-expm1.f64 75.8

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

   5. Simplified 75.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-expm1.f64 83.8

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

   7. Applied egg-rr 83.8%

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

   if -4.999999999999985e-310 < n < 1.20000000000000008e-145

   1. Initial program 44.0%

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

   3. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. log-lowering-log.f64 88.8

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

   5. Simplified 88.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-145}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-143}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(n \cdot \frac{\log i - \log n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (* 100.0 (expm1 i)) i))))
   (if (<= n -5e-310)
     t_0
     (if (<= n 4.1e-143)
       (* 100.0 (* n (* n (/ (- (log i) (log n)) i))))
       t_0))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 * expm1(i)) / i);
	double tmp;
	if (n <= -5e-310) {
		tmp = t_0;
	} else if (n <= 4.1e-143) {
		tmp = 100.0 * (n * (n * ((log(i) - log(n)) / i)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * ((100.0 * Math.expm1(i)) / i);
	double tmp;
	if (n <= -5e-310) {
		tmp = t_0;
	} else if (n <= 4.1e-143) {
		tmp = 100.0 * (n * (n * ((Math.log(i) - Math.log(n)) / i)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * ((100.0 * math.expm1(i)) / i)
	tmp = 0
	if n <= -5e-310:
		tmp = t_0
	elif n <= 4.1e-143:
		tmp = 100.0 * (n * (n * ((math.log(i) - math.log(n)) / i)))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 * expm1(i)) / i))
	tmp = 0.0
	if (n <= -5e-310)
		tmp = t_0;
	elseif (n <= 4.1e-143)
		tmp = Float64(100.0 * Float64(n * Float64(n * Float64(Float64(log(i) - log(n)) / i))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5e-310], t$95$0, If[LessEqual[n, 4.1e-143], N[(100.0 * N[(n * N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.999999999999985e-310 or 4.1e-143 < n

   1. Initial program 25.6%

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

   3. Taylor expanded in n around inf

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-expm1.f64 75.8

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

   5. Simplified 75.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-expm1.f64 83.8

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

   7. Applied egg-rr 83.8%

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

   if -4.999999999999985e-310 < n < 4.1e-143

   1. Initial program 44.0%

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

      1. pow-to-exp N/A

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

      2. accelerator-lowering-expm1.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. accelerator-lowering-log1p.f64 N/A

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

      6. /-lowering-/.f64 61.0

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

   4. Applied egg-rr 61.0%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. pow-to-exp N/A

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

      9. un-div-inv N/A

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

      10. distribute-frac-neg2 N/A

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

      11. distribute-rgt-in N/A

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

   6. Applied egg-rr 4.7%

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

   7. Taylor expanded in n around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. /-lowering-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. log-lowering-log.f64 N/A

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

      17. log-lowering-log.f64 88.8

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

   9. Simplified 88.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-143}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(n \cdot \frac{\log i - \log n}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -6.6 \cdot 10^{-256}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-146}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2.1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -1.35e-15)
     t_0
     (if (<= n -6.6e-256)
       t_1
       (if (<= n 1.65e-146) 0.0 (if (<= n 2.1) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.35e-15) {
		tmp = t_0;
	} else if (n <= -6.6e-256) {
		tmp = t_1;
	} else if (n <= 1.65e-146) {
		tmp = 0.0;
	} else if (n <= 2.1) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.35e-15) {
		tmp = t_0;
	} else if (n <= -6.6e-256) {
		tmp = t_1;
	} else if (n <= 1.65e-146) {
		tmp = 0.0;
	} else if (n <= 2.1) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -1.35e-15:
		tmp = t_0
	elif n <= -6.6e-256:
		tmp = t_1
	elif n <= 1.65e-146:
		tmp = 0.0
	elif n <= 2.1:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -1.35e-15)
		tmp = t_0;
	elseif (n <= -6.6e-256)
		tmp = t_1;
	elseif (n <= 1.65e-146)
		tmp = 0.0;
	elseif (n <= 2.1)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.35e-15], t$95$0, If[LessEqual[n, -6.6e-256], t$95$1, If[LessEqual[n, 1.65e-146], 0.0, If[LessEqual[n, 2.1], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.35000000000000005e-15 or 2.10000000000000009 < n

   1. Initial program 24.6%

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

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-expm1.f64 93.7

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

   5. Simplified 93.7%

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

   if -1.35000000000000005e-15 < n < -6.6e-256 or 1.65e-146 < n < 2.10000000000000009

   1. Initial program 25.6%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 67.8%

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

   if -6.6e-256 < n < 1.65e-146

   1. Initial program 45.5%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-lowering-neg.f64 7.4

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

   4. Applied egg-rr 7.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 79.6

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

   7. Simplified 79.6%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

      1. div0 79.6

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

   9. Applied egg-rr 79.6%

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

Alternative 5: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{-249}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-146}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (* 100.0 (expm1 i)) i))))
   (if (<= n -2.8e-249) t_0 (if (<= n 4.2e-146) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 * expm1(i)) / i);
	double tmp;
	if (n <= -2.8e-249) {
		tmp = t_0;
	} else if (n <= 4.2e-146) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * ((100.0 * Math.expm1(i)) / i);
	double tmp;
	if (n <= -2.8e-249) {
		tmp = t_0;
	} else if (n <= 4.2e-146) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * ((100.0 * math.expm1(i)) / i)
	tmp = 0
	if n <= -2.8e-249:
		tmp = t_0
	elif n <= 4.2e-146:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 * expm1(i)) / i))
	tmp = 0.0
	if (n <= -2.8e-249)
		tmp = t_0;
	elseif (n <= 4.2e-146)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.8e-249], t$95$0, If[LessEqual[n, 4.2e-146], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.7999999999999999e-249 or 4.1999999999999998e-146 < n

   1. Initial program 24.9%

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

   3. Taylor expanded in n around inf

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-expm1.f64 76.2

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

   5. Simplified 76.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-expm1.f64 84.7

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

   7. Applied egg-rr 84.7%

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

   if -2.7999999999999999e-249 < n < 4.1999999999999998e-146

   1. Initial program 45.5%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-lowering-neg.f64 7.4

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

   4. Applied egg-rr 7.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 79.6

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

   7. Simplified 79.6%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

      1. div0 79.6

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-249}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-146}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-166}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-144}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.4e-166)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 2.1e-144)
     0.0
     (*
      n
      (/
       (*
        i
        (fma
         i
         (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
         100.0))
       i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.4e-166) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 2.1e-144) {
		tmp = 0.0;
	} else {
		tmp = n * ((i * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0)) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.4e-166)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 2.1e-144)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(Float64(i * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.4e-166], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-144], 0.0, N[(n * N[(N[(i * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.4000000000000002e-166

   1. Initial program 28.0%

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

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-expm1.f64 76.2

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

   5. Simplified 76.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-expm1.f64 84.8

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

   7. Applied egg-rr 84.8%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 61.0

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

   10. Simplified 61.0%

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

   if -4.4000000000000002e-166 < n < 2.1000000000000001e-144

   1. Initial program 48.2%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-lowering-neg.f64 13.8

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

   4. Applied egg-rr 13.8%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 74.1

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

   7. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

      1. div0 74.1

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

   9. Applied egg-rr 74.1%

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

   if 2.1000000000000001e-144 < n

   1. Initial program 17.0%

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

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-expm1.f64 81.9

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

   5. Simplified 81.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-expm1.f64 87.4

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

   7. Applied egg-rr 87.4%

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

   8. Taylor expanded in i around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{i \cdot \left(100 + i \cdot \left(50 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)}\right)\right)\right)\right)}{i} \cdot n \]

      3. metadata-eval N/A

         \[\leadsto \frac{i \cdot \left(100 + i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-50\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right)\right)\right)\right)}{i} \cdot n \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{i \cdot \left(100 + i \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-50 + -1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right)\right)\right)}\right)}{i} \cdot n \]

      5. +-commutative N/A

         \[\leadsto \frac{i \cdot \left(100 + i \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + -50\right)}\right)\right)\right)}{i} \cdot n \]

      6. metadata-eval N/A

         \[\leadsto \frac{i \cdot \left(100 + i \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + \color{blue}{\left(\mathsf{neg}\left(50\right)\right)}\right)\right)\right)\right)}{i} \cdot n \]

      7. sub-neg N/A

         \[\leadsto \frac{i \cdot \left(100 + i \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) - 50\right)}\right)\right)\right)}{i} \cdot n \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{i \cdot \left(100 + \color{blue}{\left(\mathsf{neg}\left(i \cdot \left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) - 50\right)\right)\right)}\right)}{i} \cdot n \]

      9. metadata-eval N/A

         \[\leadsto \frac{i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-100\right)\right)} + \left(\mathsf{neg}\left(i \cdot \left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) - 50\right)\right)\right)\right)}{i} \cdot n \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{i \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-100 + i \cdot \left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) - 50\right)\right)\right)\right)}}{i} \cdot n \]

      11. +-commutative N/A

          \[\leadsto \frac{i \cdot \left(\mathsf{neg}\left(\color{blue}{\left(i \cdot \left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) - 50\right) + -100\right)}\right)\right)}{i} \cdot n \]

      12. metadata-eval N/A

          \[\leadsto \frac{i \cdot \left(\mathsf{neg}\left(\left(i \cdot \left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) - 50\right) + \color{blue}{\left(\mathsf{neg}\left(100\right)\right)}\right)\right)\right)}{i} \cdot n \]

      13. sub-neg N/A

          \[\leadsto \frac{i \cdot \left(\mathsf{neg}\left(\color{blue}{\left(i \cdot \left(-1 \cdot \left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) - 50\right) - 100\right)}\right)\right)}{i} \cdot n \]

   10. Simplified 72.9%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-166}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-144}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-165}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-144}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i \cdot n, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.2e-165)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 2.6e-144)
     0.0
     (* 100.0 (fma (* i n) (fma i 0.16666666666666666 0.5) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.2e-165) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 2.6e-144) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * fma((i * n), fma(i, 0.16666666666666666, 0.5), n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.2e-165)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 2.6e-144)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * fma(Float64(i * n), fma(i, 0.16666666666666666, 0.5), n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.2e-165], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.6e-144], 0.0, N[(100.0 * N[(N[(i * n), $MachinePrecision] * N[(i * 0.16666666666666666 + 0.5), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2000000000000001e-165

   1. Initial program 28.0%

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

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-expm1.f64 76.2

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

   5. Simplified 76.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-expm1.f64 84.8

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

   7. Applied egg-rr 84.8%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 61.0

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

   10. Simplified 61.0%

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

   if -1.2000000000000001e-165 < n < 2.6000000000000001e-144

   1. Initial program 48.2%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-lowering-neg.f64 13.8

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

   4. Applied egg-rr 13.8%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 74.1

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

   7. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

      1. div0 74.1

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

   9. Applied egg-rr 74.1%

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

   if 2.6000000000000001e-144 < n

   1. Initial program 17.0%

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

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

   5. Simplified 72.2%

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

   6. Taylor expanded in n around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 72.3

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

   8. Simplified 72.3%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-165}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-144}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i \cdot n, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{if}\;n \leq -6 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-145}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i (fma i 16.666666666666668 50.0) 100.0))))
   (if (<= n -6e-164) t_0 (if (<= n 1.4e-145) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	double tmp;
	if (n <= -6e-164) {
		tmp = t_0;
	} else if (n <= 1.4e-145) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0))
	tmp = 0.0
	if (n <= -6e-164)
		tmp = t_0;
	elseif (n <= 1.4e-145)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6e-164], t$95$0, If[LessEqual[n, 1.4e-145], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.0000000000000002e-164 or 1.4000000000000001e-145 < n

   1. Initial program 22.9%

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

   3. Taylor expanded in n around inf

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-expm1.f64 78.9

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

   5. Simplified 78.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. accelerator-lowering-expm1.f64 86.0

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

   7. Applied egg-rr 86.0%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 66.3

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

   10. Simplified 66.3%

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

   if -6.0000000000000002e-164 < n < 1.4000000000000001e-145

   1. Initial program 48.2%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-lowering-neg.f64 13.8

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

   4. Applied egg-rr 13.8%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 74.1

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

   7. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

      1. div0 74.1

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

   9. Applied egg-rr 74.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-164}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-145}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{if}\;n \leq -4.5 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-142}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i 50.0 100.0))))
   (if (<= n -4.5e-166) t_0 (if (<= n 3e-142) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, 50.0, 100.0);
	double tmp;
	if (n <= -4.5e-166) {
		tmp = t_0;
	} else if (n <= 3e-142) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, 50.0, 100.0))
	tmp = 0.0
	if (n <= -4.5e-166)
		tmp = t_0;
	elseif (n <= 3e-142)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.5e-166], t$95$0, If[LessEqual[n, 3e-142], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.4999999999999998e-166 or 3.0000000000000001e-142 < n

   1. Initial program 22.9%

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

   3. Taylor expanded in n around inf

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-expm1.f64 78.9

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

   5. Simplified 78.9%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 60.5

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

   8. Simplified 60.5%

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

   if -4.4999999999999998e-166 < n < 3.0000000000000001e-142

   1. Initial program 48.2%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-lowering-neg.f64 13.8

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

   4. Applied egg-rr 13.8%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 74.1

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

   7. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

      1. div0 74.1

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

   9. Applied egg-rr 74.1%

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

Alternative 10: 59.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.8 \cdot 10^{+14}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.9:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.8e+14) 0.0 (if (<= i 0.9) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.8e+14) {
		tmp = 0.0;
	} else if (i <= 0.9) {
		tmp = n * 100.0;
	} 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 <= (-2.8d+14)) then
        tmp = 0.0d0
    else if (i <= 0.9d0) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.8e+14) {
		tmp = 0.0;
	} else if (i <= 0.9) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.8e+14:
		tmp = 0.0
	elif i <= 0.9:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.8e+14)
		tmp = 0.0;
	elseif (i <= 0.9)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.8e+14)
		tmp = 0.0;
	elseif (i <= 0.9)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.8e+14], 0.0, If[LessEqual[i, 0.9], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -2.8e14 or 0.900000000000000022 < i

   1. Initial program 50.5%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-lowering-neg.f64 39.9

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

   4. Applied egg-rr 39.9%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 33.9

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

   7. Simplified 33.9%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

      1. div0 33.9

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

   9. Applied egg-rr 33.9%

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

   if -2.8e14 < i < 0.900000000000000022

   1. Initial program 10.3%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

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

Alternative 11: 17.8% 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 28.2%

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

   1. div-sub N/A

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

   2. clear-num N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-*r* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. pow-lowering-pow.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. distribute-neg-frac2 N/A

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

   15. /-lowering-/.f64 N/A

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

   16. neg-lowering-neg.f64 19.8

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

4. Applied egg-rr 19.8%

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

5. Taylor expanded in i around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 21.1

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

7. Simplified 21.1%

   \[\leadsto \color{blue}{\frac{0}{i}} \]
8. Step-by-step derivation

   1. div0 21.1

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

9. Applied egg-rr 21.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 34.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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))))