Compound Interest

Percentage Accurate: 28.2% → 98.3%

Time: 12.1s

Alternatives: 19

Speedup: 8.1×

• could not determine a ground truth

  1. i = 2.1627698750747247e-280
  2. n = 9.70182971444766e+35

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ t_1 := 100 \cdot t\_0\\ \mathbf{if}\;t\_0 \leq -40000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot -100}{\frac{-1}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (- (pow (+ (/ i n) 1.0) n) 1.0) (/ i n))) (t_1 (* 100.0 t_0)))
   (if (<= t_0 -40000000000.0)
     t_1
     (if (<= t_0 5e-188)
       (/ (* (/ (expm1 (* (log1p (/ i n)) n)) i) -100.0) (/ -1.0 n))
       (if (<= t_0 INFINITY)
         t_1
         (* (/ 1.0 (fma (fma 0.0008333333333333334 i -0.005) i 0.01)) n))))))

C

double code(double i, double n) {
	double t_0 = (pow(((i / n) + 1.0), n) - 1.0) / (i / n);
	double t_1 = 100.0 * t_0;
	double tmp;
	if (t_0 <= -40000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-188) {
		tmp = ((expm1((log1p((i / n)) * n)) / i) * -100.0) / (-1.0 / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / Float64(i / n))
	t_1 = Float64(100.0 * t_0)
	tmp = 0.0
	if (t_0 <= -40000000000.0)
		tmp = t_1;
	elseif (t_0 <= 5e-188)
		tmp = Float64(Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * -100.0) / Float64(-1.0 / n));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -40000000000.0], t$95$1, If[LessEqual[t$95$0, 5e-188], N[(N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * -100.0), $MachinePrecision] / N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(N[(1.0 / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -4e10 or 5.0000000000000001e-188 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.9%

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

   if -4e10 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 5.0000000000000001e-188

   1. Initial program 24.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{-100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}}{\frac{-1}{n}}} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -40000000000:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot -100}{\frac{-1}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ t_1 := 100 \cdot t\_0\\ \mathbf{if}\;t\_0 \leq -40000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\left(\frac{100}{i} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (- (pow (+ (/ i n) 1.0) n) 1.0) (/ i n))) (t_1 (* 100.0 t_0)))
   (if (<= t_0 -40000000000.0)
     t_1
     (if (<= t_0 5e-188)
       (* (* (/ 100.0 i) (expm1 (* (log1p (/ i n)) n))) n)
       (if (<= t_0 INFINITY)
         t_1
         (* (/ 1.0 (fma (fma 0.0008333333333333334 i -0.005) i 0.01)) n))))))

C

double code(double i, double n) {
	double t_0 = (pow(((i / n) + 1.0), n) - 1.0) / (i / n);
	double t_1 = 100.0 * t_0;
	double tmp;
	if (t_0 <= -40000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-188) {
		tmp = ((100.0 / i) * expm1((log1p((i / n)) * n))) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / Float64(i / n))
	t_1 = Float64(100.0 * t_0)
	tmp = 0.0
	if (t_0 <= -40000000000.0)
		tmp = t_1;
	elseif (t_0 <= 5e-188)
		tmp = Float64(Float64(Float64(100.0 / i) * expm1(Float64(log1p(Float64(i / n)) * n))) * n);
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -40000000000.0], t$95$1, If[LessEqual[t$95$0, 5e-188], N[(N[(N[(100.0 / i), $MachinePrecision] * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(N[(1.0 / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -4e10 or 5.0000000000000001e-188 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.9%

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

   if -4e10 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 5.0000000000000001e-188

   1. Initial program 24.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 98.2

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

   4. Applied rewrites 98.2%

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

   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 n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -40000000000:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\left(\frac{100}{i} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{n}{i} \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (- (pow (+ (/ i n) 1.0) n) 1.0) (/ i n))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n)
     (if (<= t_0 INFINITY)
       (* (* (/ n i) 100.0) (expm1 i))
       (* (/ 1.0 (fma (fma 0.0008333333333333334 i -0.005) i 0.01)) n)))))

C

double code(double i, double n) {
	double t_0 = (pow(((i / n) + 1.0), n) - 1.0) / (i / n);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((n / i) * 100.0) * expm1(i);
	} else {
		tmp = (1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(n / i) * 100.0) * expm1(i));
	else
		tmp = Float64(Float64(1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(n / i), $MachinePrecision] * 100.0), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 17.5

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

   5. Applied rewrites 17.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 84.3%

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

   if -inf.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 31.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 76.6

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 76.5%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]
   8. Step-by-step derivation

   9. Applied rewrites 76.5%

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

   11. Applied rewrites 74.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{n}{i} \cdot 100\right) \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -1.35e-192)
     t_0
     (if (<= n -2e-310)
       (* 100.0 (/ (- (pow (+ (/ i n) 1.0) n) 1.0) (/ i n)))
       (if (<= n 2.6e-98)
         (* (/ (* (- (log i) (log n)) n) (/ i n)) 100.0)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.35e-192) {
		tmp = t_0;
	} else if (n <= -2e-310) {
		tmp = 100.0 * ((pow(((i / n) + 1.0), n) - 1.0) / (i / n));
	} else if (n <= 2.6e-98) {
		tmp = (((log(i) - log(n)) * n) / (i / n)) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.35e-192) {
		tmp = t_0;
	} else if (n <= -2e-310) {
		tmp = 100.0 * ((Math.pow(((i / n) + 1.0), n) - 1.0) / (i / n));
	} else if (n <= 2.6e-98) {
		tmp = (((Math.log(i) - Math.log(n)) * n) / (i / n)) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -1.35e-192:
		tmp = t_0
	elif n <= -2e-310:
		tmp = 100.0 * ((math.pow(((i / n) + 1.0), n) - 1.0) / (i / n))
	elif n <= 2.6e-98:
		tmp = (((math.log(i) - math.log(n)) * n) / (i / n)) * 100.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -1.35e-192)
		tmp = t_0;
	elseif (n <= -2e-310)
		tmp = Float64(100.0 * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / Float64(i / n)));
	elseif (n <= 2.6e-98)
		tmp = Float64(Float64(Float64(Float64(log(i) - log(n)) * n) / Float64(i / n)) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.35e-192], t$95$0, If[LessEqual[n, -2e-310], N[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.6e-98], N[(N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.34999999999999996e-192 or 2.60000000000000013e-98 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -1.34999999999999996e-192 < n < -1.999999999999994e-310

   1. Initial program 83.5%

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

   if -1.999999999999994e-310 < n < 2.60000000000000013e-98

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(\frac{n}{i} \cdot n\right) \cdot \left(\log i - \log n\right)\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -1.35e-192)
     t_0
     (if (<= n -2e-310)
       (* 100.0 (/ (- (pow (+ (/ i n) 1.0) n) 1.0) (/ i n)))
       (if (<= n 2.6e-98)
         (* (* (* (/ n i) n) (- (log i) (log n))) 100.0)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.35e-192) {
		tmp = t_0;
	} else if (n <= -2e-310) {
		tmp = 100.0 * ((pow(((i / n) + 1.0), n) - 1.0) / (i / n));
	} else if (n <= 2.6e-98) {
		tmp = (((n / i) * n) * (log(i) - log(n))) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.35e-192) {
		tmp = t_0;
	} else if (n <= -2e-310) {
		tmp = 100.0 * ((Math.pow(((i / n) + 1.0), n) - 1.0) / (i / n));
	} else if (n <= 2.6e-98) {
		tmp = (((n / i) * n) * (Math.log(i) - Math.log(n))) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -1.35e-192:
		tmp = t_0
	elif n <= -2e-310:
		tmp = 100.0 * ((math.pow(((i / n) + 1.0), n) - 1.0) / (i / n))
	elif n <= 2.6e-98:
		tmp = (((n / i) * n) * (math.log(i) - math.log(n))) * 100.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -1.35e-192)
		tmp = t_0;
	elseif (n <= -2e-310)
		tmp = Float64(100.0 * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / Float64(i / n)));
	elseif (n <= 2.6e-98)
		tmp = Float64(Float64(Float64(Float64(n / i) * n) * Float64(log(i) - log(n))) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.35e-192], t$95$0, If[LessEqual[n, -2e-310], N[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.6e-98], N[(N[(N[(N[(n / i), $MachinePrecision] * n), $MachinePrecision] * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.34999999999999996e-192 or 2.60000000000000013e-98 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -1.34999999999999996e-192 < n < -1.999999999999994e-310

   1. Initial program 83.5%

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

   if -1.999999999999994e-310 < n < 2.60000000000000013e-98

   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 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   7. Applied rewrites 77.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(\frac{n}{i} \cdot n\right) \cdot \left(\log i - \log n\right)\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot n}{i} \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.35e-192)
   (* (* (/ (expm1 i) i) 100.0) n)
   (if (<= n 1.44e-251)
     0.0
     (if (<= n 5e-48)
       (*
        (/
         1.0
         (fma
          (fma
           (fma (* i i) -1.388888888888889e-5 0.0008333333333333334)
           i
           -0.005)
          i
          0.01))
        n)
       (* (/ (* (expm1 i) n) i) 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.35e-192) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (n <= 1.44e-251) {
		tmp = 0.0;
	} else if (n <= 5e-48) {
		tmp = (1.0 / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n;
	} else {
		tmp = ((expm1(i) * n) / i) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.35e-192)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (n <= 1.44e-251)
		tmp = 0.0;
	elseif (n <= 5e-48)
		tmp = Float64(Float64(1.0 / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n);
	else
		tmp = Float64(Float64(Float64(expm1(i) * n) / i) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.35e-192], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.44e-251], 0.0, If[LessEqual[n, 5e-48], N[(N[(1.0 / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.34999999999999996e-192

   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-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -1.34999999999999996e-192 < n < 1.44000000000000009e-251

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 30.8%

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

   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. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto 0 \]

   if 1.44000000000000009e-251 < n < 4.9999999999999999e-48

   1. Initial program 15.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 35.7

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

   5. Applied rewrites 35.7%

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

   7. Applied rewrites 35.7%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 68.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n \]

   if 4.9999999999999999e-48 < n

   1. Initial program 17.6%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot n}{i} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -1.35e-192)
     t_0
     (if (<= n 1.44e-251)
       0.0
       (if (<= n 5e-48)
         (*
          (/
           1.0
           (fma
            (fma
             (fma (* i i) -1.388888888888889e-5 0.0008333333333333334)
             i
             -0.005)
            i
            0.01))
          n)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.35e-192) {
		tmp = t_0;
	} else if (n <= 1.44e-251) {
		tmp = 0.0;
	} else if (n <= 5e-48) {
		tmp = (1.0 / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -1.35e-192)
		tmp = t_0;
	elseif (n <= 1.44e-251)
		tmp = 0.0;
	elseif (n <= 5e-48)
		tmp = Float64(Float64(1.0 / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.35e-192], t$95$0, If[LessEqual[n, 1.44e-251], 0.0, If[LessEqual[n, 5e-48], N[(N[(1.0 / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.34999999999999996e-192 or 4.9999999999999999e-48 < n

   1. Initial program 22.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -1.34999999999999996e-192 < n < 1.44000000000000009e-251

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 30.8%

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

   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. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto 0 \]

   if 1.44000000000000009e-251 < n < 4.9999999999999999e-48

   1. Initial program 15.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 35.7

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

   5. Applied rewrites 35.7%

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

   7. Applied rewrites 35.7%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 68.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+124}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq -1.26 \cdot 10^{-194}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (fma (fma 0.0008333333333333334 i -0.005) i 0.01)) n)))
   (if (<= n -2.7e+124)
     (*
      (*
       (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
       100.0)
      n)
     (if (<= n -1.26e-194)
       t_0
       (if (<= n 1.44e-251)
         0.0
         (if (<= n 5.1e-14)
           t_0
           (*
            (fma
             (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
             i
             100.0)
            n)))))))

C

double code(double i, double n) {
	double t_0 = (1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n;
	double tmp;
	if (n <= -2.7e+124) {
		tmp = (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n;
	} else if (n <= -1.26e-194) {
		tmp = t_0;
	} else if (n <= 1.44e-251) {
		tmp = 0.0;
	} else if (n <= 5.1e-14) {
		tmp = t_0;
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n)
	tmp = 0.0
	if (n <= -2.7e+124)
		tmp = Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n);
	elseif (n <= -1.26e-194)
		tmp = t_0;
	elseif (n <= 1.44e-251)
		tmp = 0.0;
	elseif (n <= 5.1e-14)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(1.0 / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.7e+124], N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -1.26e-194], t$95$0, If[LessEqual[n, 1.44e-251], 0.0, If[LessEqual[n, 5.1e-14], t$95$0, N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.69999999999999978e124

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.4%

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

   if -2.69999999999999978e124 < n < -1.26e-194 or 1.44000000000000009e-251 < n < 5.0999999999999997e-14

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 63.7

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

   5. Applied rewrites 63.7%

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 66.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n \]

   if -1.26e-194 < n < 1.44000000000000009e-251

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 30.8%

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

   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. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto 0 \]

   if 5.0999999999999997e-14 < n

   1. Initial program 18.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 81.4%

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

Alternative 9: 69.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{+124}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.7e+124)
   (*
    (*
     (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
     100.0)
    n)
   (if (<= n 5e-48)
     (*
      (/
       1.0
       (fma
        (fma
         (fma (* i i) -1.388888888888889e-5 0.0008333333333333334)
         i
         -0.005)
        i
        0.01))
      n)
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.7e+124) {
		tmp = (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n;
	} else if (n <= 5e-48) {
		tmp = (1.0 / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n;
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.7e+124)
		tmp = Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n);
	elseif (n <= 5e-48)
		tmp = Float64(Float64(1.0 / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n);
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.7e+124], N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 5e-48], N[(N[(1.0 / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.69999999999999978e124

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.4%

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

   if -2.69999999999999978e124 < n < 4.9999999999999999e-48

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 59.3

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 64.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n \]

   if 4.9999999999999999e-48 < n

   1. Initial program 17.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 80.3%

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

Alternative 10: 70.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \mathbf{if}\;n \leq -2.1 \cdot 10^{+124}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq -1.26 \cdot 10^{-194}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01))))
   (if (<= n -2.1e+124)
     (*
      (*
       (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
       100.0)
      n)
     (if (<= n -1.26e-194)
       t_0
       (if (<= n 1.44e-251)
         0.0
         (if (<= n 1.55e-15)
           t_0
           (*
            (fma
             (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
             i
             100.0)
            n)))))))

C

double code(double i, double n) {
	double t_0 = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	double tmp;
	if (n <= -2.1e+124) {
		tmp = (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n;
	} else if (n <= -1.26e-194) {
		tmp = t_0;
	} else if (n <= 1.44e-251) {
		tmp = 0.0;
	} else if (n <= 1.55e-15) {
		tmp = t_0;
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01))
	tmp = 0.0
	if (n <= -2.1e+124)
		tmp = Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n);
	elseif (n <= -1.26e-194)
		tmp = t_0;
	elseif (n <= 1.44e-251)
		tmp = 0.0;
	elseif (n <= 1.55e-15)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.1e+124], N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -1.26e-194], t$95$0, If[LessEqual[n, 1.44e-251], 0.0, If[LessEqual[n, 1.55e-15], t$95$0, N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.10000000000000011e124

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.4%

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

   if -2.10000000000000011e124 < n < -1.26e-194 or 1.44000000000000009e-251 < n < 1.5499999999999999e-15

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 63.7

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

   5. Applied rewrites 63.7%

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

   7. Applied rewrites 63.7%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]
   8. Step-by-step derivation

   9. Applied rewrites 63.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 66.2%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]

   if -1.26e-194 < n < 1.44000000000000009e-251

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 30.8%

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

   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. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto 0 \]

   if 1.5499999999999999e-15 < n

   1. Initial program 18.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 81.4%

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

Alternative 11: 70.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.1 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -1.26 \cdot 10^{-194}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01)))
        (t_1
         (*
          (fma
           (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
           i
           100.0)
          n)))
   (if (<= n -2.1e+124)
     t_1
     (if (<= n -1.26e-194)
       t_0
       (if (<= n 1.44e-251) 0.0 (if (<= n 1.55e-15) t_0 t_1))))))

C

double code(double i, double n) {
	double t_0 = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	double t_1 = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -2.1e+124) {
		tmp = t_1;
	} else if (n <= -1.26e-194) {
		tmp = t_0;
	} else if (n <= 1.44e-251) {
		tmp = 0.0;
	} else if (n <= 1.55e-15) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01))
	t_1 = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -2.1e+124)
		tmp = t_1;
	elseif (n <= -1.26e-194)
		tmp = t_0;
	elseif (n <= 1.44e-251)
		tmp = 0.0;
	elseif (n <= 1.55e-15)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.1e+124], t$95$1, If[LessEqual[n, -1.26e-194], t$95$0, If[LessEqual[n, 1.44e-251], 0.0, If[LessEqual[n, 1.55e-15], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.10000000000000011e124 or 1.5499999999999999e-15 < n

   1. Initial program 17.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 79.7%

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

   if -2.10000000000000011e124 < n < -1.26e-194 or 1.44000000000000009e-251 < n < 1.5499999999999999e-15

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 63.7

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

   5. Applied rewrites 63.7%

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

   7. Applied rewrites 63.7%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]
   8. Step-by-step derivation

   9. Applied rewrites 63.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 66.2%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]

   if -1.26e-194 < n < 1.44000000000000009e-251

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 30.8%

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

   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. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto 0 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (fma -0.005 i 0.01)) n))
        (t_1
         (*
          (fma
           (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
           i
           100.0)
          n)))
   (if (<= n -2.9e+124)
     t_1
     (if (<= n -1.35e-192)
       t_0
       (if (<= n 1.44e-251) 0.0 (if (<= n 5e-48) t_0 t_1))))))

C

double code(double i, double n) {
	double t_0 = (1.0 / fma(-0.005, i, 0.01)) * n;
	double t_1 = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -2.9e+124) {
		tmp = t_1;
	} else if (n <= -1.35e-192) {
		tmp = t_0;
	} else if (n <= 1.44e-251) {
		tmp = 0.0;
	} else if (n <= 5e-48) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(1.0 / fma(-0.005, i, 0.01)) * n)
	t_1 = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -2.9e+124)
		tmp = t_1;
	elseif (n <= -1.35e-192)
		tmp = t_0;
	elseif (n <= 1.44e-251)
		tmp = 0.0;
	elseif (n <= 5e-48)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(1.0 / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.9e+124], t$95$1, If[LessEqual[n, -1.35e-192], t$95$0, If[LessEqual[n, 1.44e-251], 0.0, If[LessEqual[n, 5e-48], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.90000000000000021e124 or 4.9999999999999999e-48 < n

   1. Initial program 17.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 79.0%

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

   if -2.90000000000000021e124 < n < -1.34999999999999996e-192 or 1.44000000000000009e-251 < n < 4.9999999999999999e-48

   1. Initial program 26.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 63.8

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

   5. Applied rewrites 63.8%

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + \frac{-1}{200} \cdot i} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 63.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n \]

   if -1.34999999999999996e-192 < n < 1.44000000000000009e-251

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 30.8%

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

   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. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto 0 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{+124}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.1e+124)
   (*
    (*
     (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
     100.0)
    n)
   (if (<= n 5e-48)
     (/
      n
      (fma
       (fma (fma (* i i) -1.388888888888889e-5 0.0008333333333333334) i -0.005)
       i
       0.01))
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.1e+124) {
		tmp = (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n;
	} else if (n <= 5e-48) {
		tmp = n / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01);
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.1e+124)
		tmp = Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n);
	elseif (n <= 5e-48)
		tmp = Float64(n / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01));
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.1e+124], N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 5e-48], N[(n / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.10000000000000011e124

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.4%

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

   if -2.10000000000000011e124 < n < 4.9999999999999999e-48

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 59.3

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 59.2%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]
   8. Step-by-step derivation

   9. Applied rewrites 59.2%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 64.0%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), \color{blue}{i}, 0.01\right)} \]

   if 4.9999999999999999e-48 < n

   1. Initial program 17.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 80.3%

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

Alternative 14: 66.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (fma -0.005 i 0.01)) n)))
   (if (<= n -1.35e-192)
     t_0
     (if (<= n 1.44e-251)
       0.0
       (if (<= n 3.4e-48)
         t_0
         (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = (1.0 / fma(-0.005, i, 0.01)) * n;
	double tmp;
	if (n <= -1.35e-192) {
		tmp = t_0;
	} else if (n <= 1.44e-251) {
		tmp = 0.0;
	} else if (n <= 3.4e-48) {
		tmp = t_0;
	} else {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(1.0 / fma(-0.005, i, 0.01)) * n)
	tmp = 0.0
	if (n <= -1.35e-192)
		tmp = t_0;
	elseif (n <= 1.44e-251)
		tmp = 0.0;
	elseif (n <= 3.4e-48)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(1.0 / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.35e-192], t$95$0, If[LessEqual[n, 1.44e-251], 0.0, If[LessEqual[n, 3.4e-48], t$95$0, N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.34999999999999996e-192 or 1.44000000000000009e-251 < n < 3.40000000000000028e-48

   1. Initial program 23.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + \frac{-1}{200} \cdot i} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 63.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n \]

   if -1.34999999999999996e-192 < n < 1.44000000000000009e-251

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 30.8%

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

   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. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto 0 \]

   if 3.40000000000000028e-48 < n

   1. Initial program 17.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.3%

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

Alternative 15: 66.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{\mathsf{fma}\left(-0.005, i, 0.01\right)}\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.44 \cdot 10^{-251}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ n (fma -0.005 i 0.01))))
   (if (<= n -1.35e-192)
     t_0
     (if (<= n 1.44e-251)
       0.0
       (if (<= n 3.4e-48)
         t_0
         (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = n / fma(-0.005, i, 0.01);
	double tmp;
	if (n <= -1.35e-192) {
		tmp = t_0;
	} else if (n <= 1.44e-251) {
		tmp = 0.0;
	} else if (n <= 3.4e-48) {
		tmp = t_0;
	} else {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n / fma(-0.005, i, 0.01))
	tmp = 0.0
	if (n <= -1.35e-192)
		tmp = t_0;
	elseif (n <= 1.44e-251)
		tmp = 0.0;
	elseif (n <= 3.4e-48)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.35e-192], t$95$0, If[LessEqual[n, 1.44e-251], 0.0, If[LessEqual[n, 3.4e-48], t$95$0, N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.34999999999999996e-192 or 1.44000000000000009e-251 < n < 3.40000000000000028e-48

   1. Initial program 23.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.2%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]
   8. Step-by-step derivation

   9. Applied rewrites 73.2%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{\frac{-1}{200} \cdot i}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.8%

       \[\leadsto \frac{n}{\mathsf{fma}\left(-0.005, \color{blue}{i}, 0.01\right)} \]

   if -1.34999999999999996e-192 < n < 1.44000000000000009e-251

   1. Initial program 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 30.8%

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

   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. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto 0 \]

   if 3.40000000000000028e-48 < n

   1. Initial program 17.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.3%

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

Alternative 16: 64.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-101}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -2.9e-188) t_0 (if (<= n 3.2e-101) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -2.9e-188) {
		tmp = t_0;
	} else if (n <= 3.2e-101) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -2.9e-188)
		tmp = t_0;
	elseif (n <= 3.2e-101)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.9e-188], t$95$0, If[LessEqual[n, 3.2e-101], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.9000000000000001e-188 or 3.19999999999999978e-101 < n

   1. Initial program 21.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 68.0%

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

   if -2.9000000000000001e-188 < n < 3.19999999999999978e-101

   1. Initial program 47.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 16.8%

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

   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. lower-/.f64 61.8

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

   7. Applied rewrites 61.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 61.8%

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

Alternative 17: 60.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.95:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 4.6 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.95) 0.0 (if (<= i 4.6e+169) (* (fma 50.0 i 100.0) n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.95) {
		tmp = 0.0;
	} else if (i <= 4.6e+169) {
		tmp = fma(50.0, i, 100.0) * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.95)
		tmp = 0.0;
	elseif (i <= 4.6e+169)
		tmp = Float64(fma(50.0, i, 100.0) * n);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.95], 0.0, If[LessEqual[i, 4.6e+169], N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.94999999999999996 or 4.5999999999999999e169 < i

   1. Initial program 65.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 57.4%

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

   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. lower-/.f64 36.3

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

   7. Applied rewrites 36.3%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 36.3%

       \[\leadsto 0 \]

   if -1.94999999999999996 < i < 4.5999999999999999e169

   1. Initial program 11.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.1%

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

Alternative 18: 59.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.38 \cdot 10^{+14}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.38e+14) 0.0 (if (<= i 5.8e-7) (* 100.0 n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.38e+14) {
		tmp = 0.0;
	} else if (i <= 5.8e-7) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.38d+14)) then
        tmp = 0.0d0
    else if (i <= 5.8d-7) then
        tmp = 100.0d0 * n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.38e+14) {
		tmp = 0.0;
	} else if (i <= 5.8e-7) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.38e+14:
		tmp = 0.0
	elif i <= 5.8e-7:
		tmp = 100.0 * n
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.38e+14)
		tmp = 0.0;
	elseif (i <= 5.8e-7)
		tmp = Float64(100.0 * n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.38e+14)
		tmp = 0.0;
	elseif (i <= 5.8e-7)
		tmp = 100.0 * n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.38e+14], 0.0, If[LessEqual[i, 5.8e-7], N[(100.0 * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.38e14 or 5.7999999999999995e-7 < i

   1. Initial program 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 46.3%

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

   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. lower-/.f64 30.7

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

   7. Applied rewrites 30.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 30.7%

       \[\leadsto 0 \]

   if -1.38e14 < i < 5.7999999999999995e-7

   1. Initial program 8.5%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

Alternative 19: 18.1% 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 26.1%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

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

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

   11. distribute-frac-neg2 N/A

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

   12. lower-fma.f64 N/A

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

   13. frac-2neg N/A

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

   14. remove-double-neg N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-/.f64 N/A

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

   18. associate-/r/ N/A

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

   19. lower-*.f64 N/A

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

4. Applied rewrites 20.4%

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

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. lower-/.f64 16.8

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

7. Applied rewrites 16.8%

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

8. Taylor expanded in i around 0

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

10. Applied rewrites 16.8%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 33.9% 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 2024295 
(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))))