Compound Interest

Percentage Accurate: 27.7% → 95.9%

Time: 13.2s

Alternatives: 9

Speedup: 8.1×

• could not determine a ground truth

  1. i = 7.980705812877371e-113
  2. n = -4.816527700358104e+38

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: 27.7% 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: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-77}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(t\_0, 100, -100\right)}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(n \cdot t\_0, \frac{1}{i}, \frac{-n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1e-77)
     (* n (/ (fma t_0 100.0 -100.0) i))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_1 INFINITY)
         (* 100.0 (fma (* n t_0) (/ 1.0 i) (/ (- n) i)))
         (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-77) {
		tmp = n * (fma(t_0, 100.0, -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * fma((n * t_0), (1.0 / i), (-n / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -1e-77)
		tmp = Float64(n * Float64(fma(t_0, 100.0, -100.0) / i));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * fma(Float64(n * t_0), Float64(1.0 / i), Float64(Float64(-n) / i)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-77], N[(n * N[(N[(t$95$0 * 100.0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(n * t$95$0), $MachinePrecision] * N[(1.0 / i), $MachinePrecision] + N[((-n) / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -9.9999999999999993e-78

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   if -9.9999999999999993e-78 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 19.9%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 99.8

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

   4. Applied rewrites 99.8%

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

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

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 98.4

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

   4. Applied rewrites 98.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.9

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

   5. Applied rewrites 76.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-77}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(n \cdot {\left(1 + \frac{i}{n}\right)}^{n}, \frac{1}{i}, \frac{-n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -8.5 \cdot 10^{-195}:\\ \;\;\;\;100 \cdot \left(n \cdot \mathsf{fma}\left(i, \frac{e^{i}}{n} \cdot -0.5, t\_0\right)\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-176}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -8.5e-195)
     (* 100.0 (* n (fma i (* (/ (exp i) n) -0.5) t_0)))
     (if (<= n 2.1e-176) 0.0 (* (* n 100.0) t_0)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -8.5e-195) {
		tmp = 100.0 * (n * fma(i, ((exp(i) / n) * -0.5), t_0));
	} else if (n <= 2.1e-176) {
		tmp = 0.0;
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -8.5e-195)
		tmp = Float64(100.0 * Float64(n * fma(i, Float64(Float64(exp(i) / n) * -0.5), t_0)));
	elseif (n <= 2.1e-176)
		tmp = 0.0;
	else
		tmp = Float64(Float64(n * 100.0) * t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -8.5e-195], N[(100.0 * N[(n * N[(i * N[(N[(N[Exp[i], $MachinePrecision] / n), $MachinePrecision] * -0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-176], 0.0, N[(N[(n * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.50000000000000023e-195

   1. Initial program 28.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. div-sub N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-expm1.f64 79.5

          \[\leadsto 100 \cdot \left(n \cdot \mathsf{fma}\left(i, \frac{e^{i}}{n} \cdot -0.5, \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i}\right)\right) \]

   5. Applied rewrites 79.5%

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \mathsf{fma}\left(i, \frac{e^{i}}{n} \cdot -0.5, \frac{\mathsf{expm1}\left(i\right)}{i}\right)\right)} \]

   if -8.50000000000000023e-195 < n < 2.09999999999999992e-176

   1. Initial program 59.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 19.2

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

   4. Applied rewrites 19.2%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 83.4%

       \[\leadsto 0 \]

   if 2.09999999999999992e-176 < n

   1. Initial program 22.2%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 85.9

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

   8. Applied rewrites 85.9%

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

Alternative 3: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -8.5 \cdot 10^{-195}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-176}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n 100.0) (/ (expm1 i) i))))
   (if (<= n -8.5e-195) t_0 (if (<= n 2.1e-176) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * (expm1(i) / i);
	double tmp;
	if (n <= -8.5e-195) {
		tmp = t_0;
	} else if (n <= 2.1e-176) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) * (Math.expm1(i) / i);
	double tmp;
	if (n <= -8.5e-195) {
		tmp = t_0;
	} else if (n <= 2.1e-176) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) * (math.expm1(i) / i)
	tmp = 0
	if n <= -8.5e-195:
		tmp = t_0
	elif n <= 2.1e-176:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * Float64(expm1(i) / i))
	tmp = 0.0
	if (n <= -8.5e-195)
		tmp = t_0;
	elseif (n <= 2.1e-176)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -8.5e-195], t$95$0, If[LessEqual[n, 2.1e-176], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.50000000000000023e-195 or 2.09999999999999992e-176 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 82.9

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

   8. Applied rewrites 82.9%

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

   if -8.50000000000000023e-195 < n < 2.09999999999999992e-176

   1. Initial program 59.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 19.2

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

   4. Applied rewrites 19.2%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 83.4%

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

Alternative 4: 65.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-164}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-176}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), n \cdot 50\right), n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.5e-164)
   (*
    (* n 100.0)
    (fma i (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5) 1.0))
   (if (<= n 2.1e-176)
     0.0
     (fma
      i
      (fma i (* n (fma 4.166666666666667 i 16.666666666666668)) (* n 50.0))
      (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.5e-164) {
		tmp = (n * 100.0) * fma(i, fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0);
	} else if (n <= 2.1e-176) {
		tmp = 0.0;
	} else {
		tmp = fma(i, fma(i, (n * fma(4.166666666666667, i, 16.666666666666668)), (n * 50.0)), (n * 100.0));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.5e-164)
		tmp = Float64(Float64(n * 100.0) * fma(i, fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0));
	elseif (n <= 2.1e-176)
		tmp = 0.0;
	else
		tmp = fma(i, fma(i, Float64(n * fma(4.166666666666667, i, 16.666666666666668)), Float64(n * 50.0)), Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.5e-164], N[(N[(n * 100.0), $MachinePrecision] * N[(i * N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-176], 0.0, N[(i * N[(i * N[(n * N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision]), $MachinePrecision] + N[(n * 50.0), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.49999999999999981e-164

   1. Initial program 28.4%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 79.8

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

   8. Applied rewrites 79.8%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 59.6%

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

   if -2.49999999999999981e-164 < n < 2.09999999999999992e-176

   1. Initial program 59.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 81.7

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 81.7%

       \[\leadsto 0 \]

   if 2.09999999999999992e-176 < n

   1. Initial program 22.2%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 85.9

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 74.7%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-164}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-176}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), n \cdot 50\right), n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 100\right) \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-176}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (* n 100.0)
          (fma
           i
           (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5)
           1.0))))
   (if (<= n -2.5e-164) t_0 (if (<= n 2.1e-176) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) * fma(i, fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0);
	double tmp;
	if (n <= -2.5e-164) {
		tmp = t_0;
	} else if (n <= 2.1e-176) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) * fma(i, fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0))
	tmp = 0.0
	if (n <= -2.5e-164)
		tmp = t_0;
	elseif (n <= 2.1e-176)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] * N[(i * N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.5e-164], t$95$0, If[LessEqual[n, 2.1e-176], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.49999999999999981e-164 or 2.09999999999999992e-176 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 83.1

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 67.8%

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

   if -2.49999999999999981e-164 < n < 2.09999999999999992e-176

   1. Initial program 59.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 81.7

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 81.7%

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

Alternative 6: 64.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(100, n, i \cdot \left(n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right)\right)\right)\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-176}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (fma 100.0 n (* i (* n (fma 16.666666666666668 i 50.0))))))
   (if (<= n -2.5e-164) t_0 (if (<= n 2.1e-176) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = fma(100.0, n, (i * (n * fma(16.666666666666668, i, 50.0))));
	double tmp;
	if (n <= -2.5e-164) {
		tmp = t_0;
	} else if (n <= 2.1e-176) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = fma(100.0, n, Float64(i * Float64(n * fma(16.666666666666668, i, 50.0))))
	tmp = 0.0
	if (n <= -2.5e-164)
		tmp = t_0;
	elseif (n <= 2.1e-176)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * n + N[(i * N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.5e-164], t$95$0, If[LessEqual[n, 2.1e-176], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.49999999999999981e-164 or 2.09999999999999992e-176 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 83.1

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 65.6%

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

   if -2.49999999999999981e-164 < n < 2.09999999999999992e-176

   1. Initial program 59.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 81.7

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 81.7%

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

Alternative 7: 62.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-195}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma 50.0 i 100.0))))
   (if (<= n -2.5e-164) t_0 (if (<= n 8.2e-195) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(50.0, i, 100.0);
	double tmp;
	if (n <= -2.5e-164) {
		tmp = t_0;
	} else if (n <= 8.2e-195) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(50.0, i, 100.0))
	tmp = 0.0
	if (n <= -2.5e-164)
		tmp = t_0;
	elseif (n <= 8.2e-195)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.5e-164], t$95$0, If[LessEqual[n, 8.2e-195], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.49999999999999981e-164 or 8.20000000000000024e-195 < n

   1. Initial program 24.6%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.5

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 82.2

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

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 60.9%

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

   if -2.49999999999999981e-164 < n < 8.20000000000000024e-195

   1. Initial program 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 23.9

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 87.9%

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

Alternative 8: 56.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-164}:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{-195}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.5e-164) (* n 100.0) (if (<= n 8.2e-195) 0.0 (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.5e-164) {
		tmp = n * 100.0;
	} else if (n <= 8.2e-195) {
		tmp = 0.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.5d-164)) then
        tmp = n * 100.0d0
    else if (n <= 8.2d-195) then
        tmp = 0.0d0
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.5e-164) {
		tmp = n * 100.0;
	} else if (n <= 8.2e-195) {
		tmp = 0.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.5e-164:
		tmp = n * 100.0
	elif n <= 8.2e-195:
		tmp = 0.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.5e-164)
		tmp = Float64(n * 100.0);
	elseif (n <= 8.2e-195)
		tmp = 0.0;
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.5e-164)
		tmp = n * 100.0;
	elseif (n <= 8.2e-195)
		tmp = 0.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.5e-164], N[(n * 100.0), $MachinePrecision], If[LessEqual[n, 8.2e-195], 0.0, N[(n * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.49999999999999981e-164 or 8.20000000000000024e-195 < n

   1. Initial program 24.6%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.5

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

   5. Applied rewrites 50.5%

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

   if -2.49999999999999981e-164 < n < 8.20000000000000024e-195

   1. Initial program 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 23.9

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 87.9%

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

Alternative 9: 17.7% 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 29.9%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. sub-neg N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. div-inv N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. lower-/.f64 N/A

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

   17. lower-neg.f64 25.3

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

4. Applied rewrites 25.3%

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

5. Taylor expanded in i around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 15.7

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

7. Applied rewrites 15.7%

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

8. Taylor expanded in i around 0

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

10. Applied rewrites 15.7%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 33.8% 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 2024233 
(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))))