Compound Interest

Percentage Accurate: 27.7% → 97.4%

Time: 12.5s

Alternatives: 16

Speedup: 8.1×

• could not determine a ground truth

  1. i = 9.78358562509778e-264
  2. n = 2.6437588831251583e+39

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n} - 1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \cdot 100\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ (/ i n) 1.0) n) 1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 0.0)
     (* (/ n (/ i (expm1 (* (log1p (/ i n)) n)))) 100.0)
     (if (<= t_1 INFINITY)
       (* (/ (* 100.0 n) i) t_0)
       (* (/ n (fma (- (/ 0.5 n) 0.5) i 1.0)) 100.0)))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n) - 1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (n / (i / expm1((log1p((i / n)) * n)))) * 100.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((100.0 * n) / i) * t_0;
	} else {
		tmp = (n / fma(((0.5 / n) - 0.5), i, 1.0)) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(n / Float64(i / expm1(Float64(log1p(Float64(i / n)) * n)))) * 100.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(100.0 * n) / i) * t_0);
	else
		tmp = Float64(Float64(n / fma(Float64(Float64(0.5 / n) - 0.5), i, 1.0)) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(n / N[(i / N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(n / N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 25.9

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 96.8

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

   4. Applied rewrites 96.8%

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

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

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 52.4

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

   4. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-expm1.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log1p.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 52.4%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log1p.f64 N/A

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

      4. pow-to-exp N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lower--.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 1.9

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n} - 1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ (/ i n) 1.0) n) 1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 0.0)
     (* (/ (expm1 (* (log1p (/ i n)) n)) i) (* 100.0 n))
     (if (<= t_1 INFINITY)
       (* (/ (* 100.0 n) i) t_0)
       (* (/ n (fma (- (/ 0.5 n) 0.5) i 1.0)) 100.0)))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n) - 1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) / i) * (100.0 * n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((100.0 * n) / i) * t_0;
	} else {
		tmp = (n / fma(((0.5 / n) - 0.5), i, 1.0)) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * Float64(100.0 * n));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(100.0 * n) / i) * t_0);
	else
		tmp = Float64(Float64(n / fma(Float64(Float64(0.5 / n) - 0.5), i, 1.0)) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(n / N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 96.3

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

   4. Applied rewrites 96.3%

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

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

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 52.4

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

   4. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-expm1.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log1p.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 52.4%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log1p.f64 N/A

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

      4. pow-to-exp N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lower--.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 1.9

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n} - 1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ (/ i n) 1.0) n) 1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 0.0)
     (* (* (expm1 (* (log1p (/ i n)) n)) 100.0) (/ n i))
     (if (<= t_1 INFINITY)
       (* (/ (* 100.0 n) i) t_0)
       (* (/ n (fma (- (/ 0.5 n) 0.5) i 1.0)) 100.0)))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n) - 1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) * 100.0) * (n / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((100.0 * n) / i) * t_0;
	} else {
		tmp = (n / fma(((0.5 / n) - 0.5), i, 1.0)) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) * Float64(n / i));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(100.0 * n) / i) * t_0);
	else
		tmp = Float64(Float64(n / fma(Float64(Float64(0.5 / n) - 0.5), i, 1.0)) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(n / N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 25.9

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 95.5

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

   4. Applied rewrites 95.5%

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

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

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 52.4

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

   4. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-expm1.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log1p.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 52.4%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log1p.f64 N/A

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

      4. pow-to-exp N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lower--.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 1.9

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{100 \cdot n}{i} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -22:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot n}{i} \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -22.0)
   (/ (* (expm1 i) (* 100.0 n)) i)
   (if (<= n 3.8e+15)
     (* (/ n (fma (- (/ 0.5 n) 0.5) i 1.0)) 100.0)
     (* (/ (* (expm1 i) n) i) 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -22.0) {
		tmp = (expm1(i) * (100.0 * n)) / i;
	} else if (n <= 3.8e+15) {
		tmp = (n / fma(((0.5 / n) - 0.5), i, 1.0)) * 100.0;
	} else {
		tmp = ((expm1(i) * n) / i) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -22.0)
		tmp = Float64(Float64(expm1(i) * Float64(100.0 * n)) / i);
	elseif (n <= 3.8e+15)
		tmp = Float64(Float64(n / fma(Float64(Float64(0.5 / n) - 0.5), i, 1.0)) * 100.0);
	else
		tmp = Float64(Float64(Float64(expm1(i) * n) / i) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -22.0], N[(N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 3.8e+15], N[(N[(n / N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -22

   1. Initial program 26.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 89.5%

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

   if -22 < n < 3.8e15

   1. Initial program 32.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 32.7

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if 3.8e15 < n

   1. Initial program 20.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -22:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot n}{i} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -22:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -22.0)
   (/ (* (expm1 i) (* 100.0 n)) i)
   (if (<= n 3.8e+15)
     (* (/ n (fma (- (/ 0.5 n) 0.5) i 1.0)) 100.0)
     (* (* (/ (expm1 i) i) 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -22.0) {
		tmp = (expm1(i) * (100.0 * n)) / i;
	} else if (n <= 3.8e+15) {
		tmp = (n / fma(((0.5 / n) - 0.5), i, 1.0)) * 100.0;
	} else {
		tmp = ((expm1(i) / i) * 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -22.0)
		tmp = Float64(Float64(expm1(i) * Float64(100.0 * n)) / i);
	elseif (n <= 3.8e+15)
		tmp = Float64(Float64(n / fma(Float64(Float64(0.5 / n) - 0.5), i, 1.0)) * 100.0);
	else
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -22.0], N[(N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 3.8e+15], N[(N[(n / N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -22

   1. Initial program 26.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 89.5%

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

   if -22 < n < 3.8e15

   1. Initial program 32.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 32.7

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if 3.8e15 < n

   1. Initial program 20.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -22:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot \left(100 \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -22:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -22.0)
     t_0
     (if (<= n 3.8e+15) (* (/ n (fma (- (/ 0.5 n) 0.5) i 1.0)) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -22.0) {
		tmp = t_0;
	} else if (n <= 3.8e+15) {
		tmp = (n / fma(((0.5 / n) - 0.5), i, 1.0)) * 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 <= -22.0)
		tmp = t_0;
	elseif (n <= 3.8e+15)
		tmp = Float64(Float64(n / fma(Float64(Float64(0.5 / n) - 0.5), i, 1.0)) * 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, -22.0], t$95$0, If[LessEqual[n, 3.8e+15], N[(N[(n / N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -22 or 3.8e15 < n

   1. Initial program 23.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-/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 92.4

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

   5. Applied rewrites 92.4%

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

   if -22 < n < 3.8e15

   1. Initial program 32.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 32.7

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

      \[\leadsto 100 \cdot \frac{n}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -22:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.8e+121)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   (if (<= n 3.8e+15)
     (* (/ n (fma (- (/ 0.5 n) 0.5) i 1.0)) 100.0)
     (fma
      n
      100.0
      (* (* (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) n) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.8e+121) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= 3.8e+15) {
		tmp = (n / fma(((0.5 / n) - 0.5), i, 1.0)) * 100.0;
	} else {
		tmp = fma(n, 100.0, ((fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.8e+121)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= 3.8e+15)
		tmp = Float64(Float64(n / fma(Float64(Float64(0.5 / n) - 0.5), i, 1.0)) * 100.0);
	else
		tmp = fma(n, 100.0, Float64(Float64(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.8e+121], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 3.8e+15], N[(N[(n / N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(n * 100.0 + N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * n), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.7999999999999998e121

   1. Initial program 7.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   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 76.9%

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

   if -5.7999999999999998e121 < n < 3.8e15

   1. Initial program 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 37.4

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 85.2

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if 3.8e15 < n

   1. Initial program 20.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 80.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50 \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, i, 1\right)} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6200:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-244}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6200.0)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   (if (<= n -6e-244)
     (* (* 100.0 i) (/ n i))
     (if (<= n 1.75e-202)
       0.0
       (fma
        n
        100.0
        (*
         (* (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) n)
         i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6200.0) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= -6e-244) {
		tmp = (100.0 * i) * (n / i);
	} else if (n <= 1.75e-202) {
		tmp = 0.0;
	} else {
		tmp = fma(n, 100.0, ((fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6200.0)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= -6e-244)
		tmp = Float64(Float64(100.0 * i) * Float64(n / i));
	elseif (n <= 1.75e-202)
		tmp = 0.0;
	else
		tmp = fma(n, 100.0, Float64(Float64(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6200.0], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -6e-244], N[(N[(100.0 * i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-202], 0.0, N[(n * 100.0 + N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * n), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6200

   1. Initial program 25.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

   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 65.9%

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

   if -6200 < n < -6.0000000000000002e-244

   1. Initial program 33.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 33.8

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -6.0000000000000002e-244 < n < 1.75e-202

   1. Initial program 55.7%

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

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 15.6

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

   4. Applied rewrites 15.6%

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

   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 83.7

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 83.7%

       \[\leadsto 0 \]

   if 1.75e-202 < n

   1. Initial program 18.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50 \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6200:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-244}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6200:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-244}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6200.0)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   (if (<= n -6e-244)
     (* (* 100.0 i) (/ n i))
     (if (<= n 1.75e-202)
       0.0
       (*
        (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
        n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6200.0) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= -6e-244) {
		tmp = (100.0 * i) * (n / i);
	} else if (n <= 1.75e-202) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6200.0)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= -6e-244)
		tmp = Float64(Float64(100.0 * i) * Float64(n / i));
	elseif (n <= 1.75e-202)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6200.0], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -6e-244], N[(N[(100.0 * i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-202], 0.0, N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6200

   1. Initial program 25.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

   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 65.9%

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

   if -6200 < n < -6.0000000000000002e-244

   1. Initial program 33.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 33.8

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -6.0000000000000002e-244 < n < 1.75e-202

   1. Initial program 55.7%

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

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 15.6

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

   4. Applied rewrites 15.6%

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

   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 83.7

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 83.7%

       \[\leadsto 0 \]

   if 1.75e-202 < n

   1. Initial program 18.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 88.1

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

   5. Applied rewrites 88.1%

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

   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 76.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6200:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-244}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6200:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-244}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right) \cdot n, i, 100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6200.0)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   (if (<= n -6e-244)
     (* (* 100.0 i) (/ n i))
     (if (<= n 1.75e-202)
       0.0
       (fma (* (fma 16.666666666666668 i 50.0) n) i (* 100.0 n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6200.0) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= -6e-244) {
		tmp = (100.0 * i) * (n / i);
	} else if (n <= 1.75e-202) {
		tmp = 0.0;
	} else {
		tmp = fma((fma(16.666666666666668, i, 50.0) * n), i, (100.0 * n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6200.0)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= -6e-244)
		tmp = Float64(Float64(100.0 * i) * Float64(n / i));
	elseif (n <= 1.75e-202)
		tmp = 0.0;
	else
		tmp = fma(Float64(fma(16.666666666666668, i, 50.0) * n), i, Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6200.0], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -6e-244], N[(N[(100.0 * i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-202], 0.0, N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * n), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6200

   1. Initial program 25.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

   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 65.9%

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

   if -6200 < n < -6.0000000000000002e-244

   1. Initial program 33.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 33.8

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -6.0000000000000002e-244 < n < 1.75e-202

   1. Initial program 55.7%

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

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 15.6

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

   4. Applied rewrites 15.6%

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

   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 83.7

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 83.7%

       \[\leadsto 0 \]

   if 1.75e-202 < n

   1. Initial program 18.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 73.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6200:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-244}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right) \cdot n, i, 100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -6200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-244}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -6200.0)
     t_0
     (if (<= n -6e-244)
       (* (* 100.0 i) (/ n i))
       (if (<= n 1.75e-202) 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 <= -6200.0) {
		tmp = t_0;
	} else if (n <= -6e-244) {
		tmp = (100.0 * i) * (n / i);
	} else if (n <= 1.75e-202) {
		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 <= -6200.0)
		tmp = t_0;
	elseif (n <= -6e-244)
		tmp = Float64(Float64(100.0 * i) * Float64(n / i));
	elseif (n <= 1.75e-202)
		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, -6200.0], t$95$0, If[LessEqual[n, -6e-244], N[(N[(100.0 * i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-202], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6200 or 1.75e-202 < n

   1. Initial program 21.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   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 69.9%

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

   if -6200 < n < -6.0000000000000002e-244

   1. Initial program 33.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 33.8

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -6.0000000000000002e-244 < n < 1.75e-202

   1. Initial program 55.7%

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

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 15.6

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

   4. Applied rewrites 15.6%

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

   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 83.7

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 83.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6200:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-244}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -8.8 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -8.8e-190) t_0 (if (<= n 1.75e-202) 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 <= -8.8e-190) {
		tmp = t_0;
	} else if (n <= 1.75e-202) {
		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 <= -8.8e-190)
		tmp = t_0;
	elseif (n <= 1.75e-202)
		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, -8.8e-190], t$95$0, If[LessEqual[n, 1.75e-202], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.80000000000000017e-190 or 1.75e-202 < n

   1. Initial program 22.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 67.5%

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

   if -8.80000000000000017e-190 < n < 1.75e-202

   1. Initial program 53.7%

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

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 19.2

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

   4. Applied rewrites 19.2%

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

   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 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 72.1%

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

Alternative 13: 61.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n, i, 100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8.8e-190)
   (* (fma 50.0 i 100.0) n)
   (if (<= n 1.25e-202) 0.0 (fma (* 50.0 n) i (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8.8e-190) {
		tmp = fma(50.0, i, 100.0) * n;
	} else if (n <= 1.25e-202) {
		tmp = 0.0;
	} else {
		tmp = fma((50.0 * n), i, (100.0 * n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8.8e-190)
		tmp = Float64(fma(50.0, i, 100.0) * n);
	elseif (n <= 1.25e-202)
		tmp = 0.0;
	else
		tmp = fma(Float64(50.0 * n), i, Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8.8e-190], N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.25e-202], 0.0, N[(N[(50.0 * n), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.80000000000000017e-190

   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 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 60.6%

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

   if -8.80000000000000017e-190 < n < 1.24999999999999993e-202

   1. Initial program 53.7%

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

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 19.2

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

   4. Applied rewrites 19.2%

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

   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 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 72.1%

       \[\leadsto 0 \]

   if 1.24999999999999993e-202 < n

   1. Initial program 18.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.8%

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

   9. Taylor expanded in i around 0

      \[\leadsto \mathsf{fma}\left(50 \cdot n, i, n \cdot 100\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n, i, 100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -8.8 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-202}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 50.0 i 100.0) n)))
   (if (<= n -8.8e-190) t_0 (if (<= n 1.25e-202) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = fma(50.0, i, 100.0) * n;
	double tmp;
	if (n <= -8.8e-190) {
		tmp = t_0;
	} else if (n <= 1.25e-202) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(50.0, i, 100.0) * n)
	tmp = 0.0
	if (n <= -8.8e-190)
		tmp = t_0;
	elseif (n <= 1.25e-202)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -8.8e-190], t$95$0, If[LessEqual[n, 1.25e-202], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.80000000000000017e-190 or 1.24999999999999993e-202 < n

   1. Initial program 22.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 64.9%

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

   if -8.80000000000000017e-190 < n < 1.24999999999999993e-202

   1. Initial program 53.7%

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

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 19.2

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

   4. Applied rewrites 19.2%

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

   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 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 72.1%

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

Alternative 15: 59.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.8 \cdot 10^{-15}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 16:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.8e-15) 0.0 (if (<= i 16.0) (* 100.0 n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.8e-15) {
		tmp = 0.0;
	} else if (i <= 16.0) {
		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 <= (-4.8d-15)) then
        tmp = 0.0d0
    else if (i <= 16.0d0) 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 <= -4.8e-15) {
		tmp = 0.0;
	} else if (i <= 16.0) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -4.8e-15:
		tmp = 0.0
	elif i <= 16.0:
		tmp = 100.0 * n
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.8e-15)
		tmp = 0.0;
	elseif (i <= 16.0)
		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 <= -4.8e-15)
		tmp = 0.0;
	elseif (i <= 16.0)
		tmp = 100.0 * n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.8e-15], 0.0, If[LessEqual[i, 16.0], N[(100.0 * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -4.7999999999999999e-15 or 16 < i

   1. Initial program 54.8%

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

      1. lift-/.f64 N/A

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

      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. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 48.8

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

   4. Applied rewrites 48.8%

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

   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 27.8

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

   7. Applied rewrites 27.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 27.8%

       \[\leadsto 0 \]

   if -4.7999999999999999e-15 < i < 16

   1. Initial program 8.2%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 16: 17.4% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 27.1%

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

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

   8. associate-/r/ N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. distribute-neg-frac N/A

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

   15. lower-/.f64 N/A

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

   16. lower-neg.f64 21.8

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

4. Applied rewrites 21.8%

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

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 16.5

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

7. Applied rewrites 16.5%

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

8. Taylor expanded in i around 0

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

10. Applied rewrites 16.5%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 34.3% 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 2024276 
(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))))