Compound Interest

Percentage Accurate: 28.5% → 99.3%

Time: 10.4s

Alternatives: 19

Speedup: 24.3×

• could not determine a ground truth

  1. i = 2.4040977205691911e-147
  2. n = -2.9987068741543776e+37

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.5% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ t_1 := 100 \cdot \frac{{t\_0}^{n} - 1}{\frac{i}{n}}\\ t_2 := \mathsf{expm1}\left(\frac{n}{2} \cdot \log \left({t\_0}^{2}\right)\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{if}\;t\_1 \leq -20000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n)))
        (t_1 (* 100.0 (/ (- (pow t_0 n) 1.0) (/ i n))))
        (t_2 (* (expm1 (* (/ n 2.0) (log (pow t_0 2.0)))) (/ (* 100.0 n) i))))
   (if (<= t_1 -20000000.0)
     t_2
     (if (<= t_1 0.0)
       (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) (/ i n))
       (if (<= t_1 INFINITY)
         t_2
         (* (/ (expm1 (fma (/ (* i i) n) -0.5 i)) i) (* n 100.0)))))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double t_1 = 100.0 * ((pow(t_0, n) - 1.0) / (i / n));
	double t_2 = expm1(((n / 2.0) * log(pow(t_0, 2.0)))) * ((100.0 * n) / i);
	double tmp;
	if (t_1 <= -20000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (expm1(fma(((i * i) / n), -0.5, i)) / i) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	t_1 = Float64(100.0 * Float64(Float64((t_0 ^ n) - 1.0) / Float64(i / n)))
	t_2 = Float64(expm1(Float64(Float64(n / 2.0) * log((t_0 ^ 2.0)))) * Float64(Float64(100.0 * n) / i))
	tmp = 0.0
	if (t_1 <= -20000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(expm1(fma(Float64(Float64(i * i) / n), -0.5, i)) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(N[(N[Power[t$95$0, n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Exp[N[(N[(n / 2.0), $MachinePrecision] * N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000.0], t$95$2, 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[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(N[(Exp[N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * -0.5 + i), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -2e7 or -0.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.3

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 45.1

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

   4. Applied rewrites 45.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 45.1

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

   6. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log1p.f64 N/A

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

      4. log-pow-rev N/A

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

      5. sqr-pow N/A

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)} \cdot {\left(1 + \frac{i}{n}\right)}^{\left(\frac{n}{2}\right)}\right)}\right) \cdot \frac{100 \cdot n}{i} \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left({\left(\left(1 + \frac{i}{n}\right) \cdot \left(1 + \frac{i}{n}\right)\right)}^{\left(\frac{n}{2}\right)}\right)}\right) \cdot \frac{100 \cdot n}{i} \]

      7. log-pow N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{n}{2} \cdot \log \left(\left(1 + \frac{i}{n}\right) \cdot \left(1 + \frac{i}{n}\right)\right)}\right) \cdot \frac{100 \cdot n}{i} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{n}{2} \cdot \log \left(\left(1 + \frac{i}{n}\right) \cdot \left(1 + \frac{i}{n}\right)\right)}\right) \cdot \frac{100 \cdot n}{i} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{n}{2}} \cdot \log \left(\left(1 + \frac{i}{n}\right) \cdot \left(1 + \frac{i}{n}\right)\right)\right) \cdot \frac{100 \cdot n}{i} \]

      10. lower-log.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\frac{n}{2} \cdot \color{blue}{\log \left(\left(1 + \frac{i}{n}\right) \cdot \left(1 + \frac{i}{n}\right)\right)}\right) \cdot \frac{100 \cdot n}{i} \]

      11. pow2 N/A

          \[\leadsto \mathsf{expm1}\left(\frac{n}{2} \cdot \log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{2}\right)}\right) \cdot \frac{100 \cdot n}{i} \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\frac{n}{2} \cdot \log \color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{2}\right)}\right) \cdot \frac{100 \cdot n}{i} \]

      13. lower-+.f64 96.0

          \[\leadsto \mathsf{expm1}\left(\frac{n}{2} \cdot \log \left({\color{blue}{\left(1 + \frac{i}{n}\right)}}^{2}\right)\right) \cdot \frac{100 \cdot n}{i} \]

   8. Applied rewrites 96.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{n}{2} \cdot \log \left({\left(1 + \frac{i}{n}\right)}^{2}\right)}\right) \cdot \frac{100 \cdot n}{i} \]

   if -2e7 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 30.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 30.5

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + i}\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{{i}^{2}}{n} \cdot \frac{-1}{2}} + i\right)}{i} \cdot \left(n \cdot 100\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{i}^{2}}{n}}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{i \cdot i}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.0%

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

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

   1. Initial program 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.3

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + i}\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{{i}^{2}}{n} \cdot \frac{-1}{2}} + i\right)}{i} \cdot \left(n \cdot 100\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{i}^{2}}{n}}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{i \cdot i}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.0%

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

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

   1. Initial program 31.2%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-log1p.f64 99.5

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

   4. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + i}\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{{i}^{2}}{n} \cdot \frac{-1}{2}} + i\right)}{i} \cdot \left(n \cdot 100\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{i}^{2}}{n}}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{i \cdot i}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.0%

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

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

   1. Initial program 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 98.8%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + i}\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{{i}^{2}}{n} \cdot \frac{-1}{2}} + i\right)}{i} \cdot \left(n \cdot 100\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{i}^{2}}{n}}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{i \cdot i}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

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

Alternative 5: 98.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.0%

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

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

   1. Initial program 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + i}\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{{i}^{2}}{n} \cdot \frac{-1}{2}} + i\right)}{i} \cdot \left(n \cdot 100\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{i}^{2}}{n}}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{i \cdot i}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 95.0%

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

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

   1. Initial program 31.2%

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

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

      5. lower-/.f64 31.0

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 98.7

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

   4. Applied rewrites 98.7%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + i}\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{{i}^{2}}{n} \cdot \frac{-1}{2}} + i\right)}{i} \cdot \left(n \cdot 100\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{i}^{2}}{n}}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{i \cdot i}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-1}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -4e-304)
     (* 100.0 (fma (/ (pow (+ (/ i n) 1.0) n) i) n (* (/ -1.0 i) n)))
     (if (<= t_0 0.0)
       (* (* (/ (expm1 i) i) 100.0) n)
       (if (<= t_0 INFINITY)
         t_0
         (* (/ (expm1 (fma (/ (* i i) n) -0.5 i)) i) (* n 100.0)))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -4e-304) {
		tmp = 100.0 * fma((pow(((i / n) + 1.0), n) / i), n, ((-1.0 / i) * n));
	} else if (t_0 <= 0.0) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (expm1(fma(((i * i) / n), -0.5, i)) / i) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -4e-304)
		tmp = Float64(100.0 * fma(Float64((Float64(Float64(i / n) + 1.0) ^ n) / i), n, Float64(Float64(-1.0 / i) * n)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(expm1(fma(Float64(Float64(i * i) / n), -0.5, i)) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -4e-304], N[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision] * n + N[(N[(-1.0 / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(Exp[N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * -0.5 + i), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -3.99999999999999988e-304

   1. Initial program 95.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

   if -3.99999999999999988e-304 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 73.3

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

   5. Applied rewrites 73.3%

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

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

   1. Initial program 92.7%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + i}\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{{i}^{2}}{n} \cdot \frac{-1}{2}} + i\right)}{i} \cdot \left(n \cdot 100\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{i}^{2}}{n}}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{i \cdot i}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -4e-304)
     t_0
     (if (<= t_0 0.0)
       (* (* (/ (expm1 i) i) 100.0) n)
       (if (<= t_0 INFINITY)
         t_0
         (* (/ (expm1 (fma (/ (* i i) n) -0.5 i)) i) (* n 100.0)))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -4e-304) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (expm1(fma(((i * i) / n), -0.5, i)) / i) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -4e-304)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(expm1(fma(Float64(Float64(i * i) / n), -0.5, i)) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -4e-304], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(Exp[N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * -0.5 + i), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -3.99999999999999988e-304 or -0.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 94.4%

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

   if -3.99999999999999988e-304 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 73.3

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

   5. Applied rewrites 73.3%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot \left(n \cdot 100\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + i}\right)}{i} \cdot \left(n \cdot 100\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{{i}^{2}}{n} \cdot \frac{-1}{2}} + i\right)}{i} \cdot \left(n \cdot 100\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{\frac{{i}^{2}}{n}}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{\color{blue}{i \cdot i}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot \left(n \cdot 100\right) \]

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-158} \lor \neg \left(n \leq 1.22 \cdot 10^{-132}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.85e-158) (not (<= n 1.22e-132)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.85e-158) || !(n <= 1.22e-132)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.85e-158) || !(n <= 1.22e-132)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.85e-158) or not (n <= 1.22e-132):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.85e-158) || !(n <= 1.22e-132))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.85e-158], N[Not[LessEqual[n, 1.22e-132]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.85e-158 or 1.2200000000000001e-132 < n

   1. Initial program 28.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

   if -1.85e-158 < n < 1.2200000000000001e-132

   1. Initial program 50.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 74.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-158} \lor \neg \left(n \leq 1.22 \cdot 10^{-132}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.85 \cdot 10^{-158}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.22 \cdot 10^{-132}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -1.85e-158)
     (* 100.0 (* t_0 n))
     (if (<= n 1.22e-132)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -1.85e-158) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= 1.22e-132) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -1.85e-158) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= 1.22e-132) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -1.85e-158:
		tmp = 100.0 * (t_0 * n)
	elif n <= 1.22e-132:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -1.85e-158)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= 1.22e-132)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.85e-158], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.22e-132], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.85e-158

   1. Initial program 32.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -1.85e-158 < n < 1.2200000000000001e-132

   1. Initial program 50.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 74.5%

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

   if 1.2200000000000001e-132 < n

   1. Initial program 23.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

Alternative 11: 64.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{10000 \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.4e-159)
   (* (fma (* (fma 0.16666666666666666 i 0.5) i) 100.0 100.0) n)
   (if (<= n 6e-224)
     (sqrt (* 10000.0 (* n n)))
     (if (<= n 2.8e-11)
       (/ (* 100.0 i) (/ i n))
       (* 100.0 (fma (* (- 0.5 (/ 0.5 n)) n) i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.4e-159) {
		tmp = fma((fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n;
	} else if (n <= 6e-224) {
		tmp = sqrt((10000.0 * (n * n)));
	} else if (n <= 2.8e-11) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = 100.0 * fma(((0.5 - (0.5 / n)) * n), i, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.4e-159)
		tmp = Float64(fma(Float64(fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n);
	elseif (n <= 6e-224)
		tmp = sqrt(Float64(10000.0 * Float64(n * n)));
	elseif (n <= 2.8e-11)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = Float64(100.0 * fma(Float64(Float64(0.5 - Float64(0.5 / n)) * n), i, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.4e-159], N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 6e-224], N[Sqrt[N[(10000.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.8e-11], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.3999999999999999e-159

   1. Initial program 32.7%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.9%

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

   if -6.3999999999999999e-159 < n < 5.99999999999999963e-224

   1. Initial program 60.6%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 14.8

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

   5. Applied rewrites 14.8%

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

   7. Applied rewrites 75.4%

      \[\leadsto \sqrt{10000 \cdot \left(n \cdot n\right)} \]

   if 5.99999999999999963e-224 < n < 2.8e-11

   1. Initial program 19.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 19.3

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 79.6

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 64.4

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

   7. Applied rewrites 64.4%

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

   if 2.8e-11 < n

   1. Initial program 26.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 64.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{10000 \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.4e-159)
   (* (fma (* (fma 0.16666666666666666 i 0.5) i) 100.0 100.0) n)
   (if (<= n 6e-224)
     (sqrt (* 10000.0 (* n n)))
     (if (<= n 2.8e-11)
       (/ (* 100.0 i) (/ i n))
       (fma (- (* 50.0 n) 50.0) i (* 100.0 n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.4e-159) {
		tmp = fma((fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n;
	} else if (n <= 6e-224) {
		tmp = sqrt((10000.0 * (n * n)));
	} else if (n <= 2.8e-11) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = fma(((50.0 * n) - 50.0), i, (100.0 * n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.4e-159)
		tmp = Float64(fma(Float64(fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n);
	elseif (n <= 6e-224)
		tmp = sqrt(Float64(10000.0 * Float64(n * n)));
	elseif (n <= 2.8e-11)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = fma(Float64(Float64(50.0 * n) - 50.0), i, Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.4e-159], N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 6e-224], N[Sqrt[N[(10000.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.8e-11], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(50.0 * n), $MachinePrecision] - 50.0), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.3999999999999999e-159

   1. Initial program 32.7%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.9%

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

   if -6.3999999999999999e-159 < n < 5.99999999999999963e-224

   1. Initial program 60.6%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 14.8

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

   5. Applied rewrites 14.8%

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

   7. Applied rewrites 75.4%

      \[\leadsto \sqrt{10000 \cdot \left(n \cdot n\right)} \]

   if 5.99999999999999963e-224 < n < 2.8e-11

   1. Initial program 19.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 19.3

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 79.6

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 64.4

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

   7. Applied rewrites 64.4%

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

   if 2.8e-11 < n

   1. Initial program 26.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 35.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right), n, \left(\left(-0.5 \cdot i - 0.5\right) \cdot i\right) \cdot 100\right), n, 33.333333333333336 \cdot \left(i \cdot i\right)\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 74.6%

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

Alternative 13: 62.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.25 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.22 \cdot 10^{-132}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - \frac{0.5}{n}, i, 1\right) \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.25e-98)
   (* (fma (* (fma 0.16666666666666666 i 0.5) i) 100.0 100.0) n)
   (if (<= n 1.22e-132)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (* (fma (- 0.5 (/ 0.5 n)) i 1.0) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.25e-98) {
		tmp = fma((fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n;
	} else if (n <= 1.22e-132) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = fma((0.5 - (0.5 / n)), i, 1.0) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.25e-98)
		tmp = Float64(fma(Float64(fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n);
	elseif (n <= 1.22e-132)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 / n)), i, 1.0) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.25e-98], N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.22e-132], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + 1.0), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.24999999999999998e-98

   1. Initial program 33.1%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 52.3%

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

   if -2.24999999999999998e-98 < n < 1.2200000000000001e-132

   1. Initial program 47.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.9%

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

   if 1.2200000000000001e-132 < n

   1. Initial program 23.6%

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

      1. lift-*.f64 N/A

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

      2. *-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. lower-*.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 70.1

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

   7. Applied rewrites 70.1%

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

Alternative 14: 62.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.25 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.22 \cdot 10^{-132}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.25e-98)
   (* (fma (* (fma 0.16666666666666666 i 0.5) i) 100.0 100.0) n)
   (if (<= n 1.22e-132)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (fma (- (* 50.0 n) 50.0) i (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.25e-98) {
		tmp = fma((fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n;
	} else if (n <= 1.22e-132) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = fma(((50.0 * n) - 50.0), i, (100.0 * n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.25e-98)
		tmp = Float64(fma(Float64(fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n);
	elseif (n <= 1.22e-132)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = fma(Float64(Float64(50.0 * n) - 50.0), i, Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.25e-98], N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.22e-132], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(50.0 * n), $MachinePrecision] - 50.0), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.24999999999999998e-98

   1. Initial program 33.1%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 52.3%

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

   if -2.24999999999999998e-98 < n < 1.2200000000000001e-132

   1. Initial program 47.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.9%

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

   if 1.2200000000000001e-132 < n

   1. Initial program 23.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 41.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right), n, \left(\left(-0.5 \cdot i - 0.5\right) \cdot i\right) \cdot 100\right), n, 33.333333333333336 \cdot \left(i \cdot i\right)\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 70.1%

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

Alternative 15: 62.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-159} \lor \neg \left(n \leq 1.05 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(10000 \cdot n\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -6.4e-159) (not (<= n 1.05e-10)))
   (fma (- (* 50.0 n) 50.0) i (* 100.0 n))
   (sqrt (* (* 10000.0 n) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -6.4e-159) || !(n <= 1.05e-10)) {
		tmp = fma(((50.0 * n) - 50.0), i, (100.0 * n));
	} else {
		tmp = sqrt(((10000.0 * n) * n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -6.4e-159) || !(n <= 1.05e-10))
		tmp = fma(Float64(Float64(50.0 * n) - 50.0), i, Float64(100.0 * n));
	else
		tmp = sqrt(Float64(Float64(10000.0 * n) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -6.4e-159], N[Not[LessEqual[n, 1.05e-10]], $MachinePrecision]], N[(N[(N[(50.0 * n), $MachinePrecision] - 50.0), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(10000.0 * n), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.3999999999999999e-159 or 1.05e-10 < n

   1. Initial program 30.1%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 36.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right), n, \left(\left(-0.5 \cdot i - 0.5\right) \cdot i\right) \cdot 100\right), n, 33.333333333333336 \cdot \left(i \cdot i\right)\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 59.9%

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

   if -6.3999999999999999e-159 < n < 1.05e-10

   1. Initial program 38.3%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 63.1%

      \[\leadsto \sqrt{10000 \cdot \left(n \cdot n\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 63.1%

      \[\leadsto \sqrt{\left(10000 \cdot n\right) \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-159} \lor \neg \left(n \leq 1.05 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(10000 \cdot n\right) \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-159} \lor \neg \left(n \leq 1.05 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{10000 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -6.4e-159) (not (<= n 1.05e-10)))
   (fma (- (* 50.0 n) 50.0) i (* 100.0 n))
   (sqrt (* 10000.0 (* n n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -6.4e-159) || !(n <= 1.05e-10)) {
		tmp = fma(((50.0 * n) - 50.0), i, (100.0 * n));
	} else {
		tmp = sqrt((10000.0 * (n * n)));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -6.4e-159) || !(n <= 1.05e-10))
		tmp = fma(Float64(Float64(50.0 * n) - 50.0), i, Float64(100.0 * n));
	else
		tmp = sqrt(Float64(10000.0 * Float64(n * n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -6.4e-159], N[Not[LessEqual[n, 1.05e-10]], $MachinePrecision]], N[(N[(N[(50.0 * n), $MachinePrecision] - 50.0), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(10000.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.3999999999999999e-159 or 1.05e-10 < n

   1. Initial program 30.1%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 36.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right), n, \left(\left(-0.5 \cdot i - 0.5\right) \cdot i\right) \cdot 100\right), n, 33.333333333333336 \cdot \left(i \cdot i\right)\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 59.9%

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

   if -6.3999999999999999e-159 < n < 1.05e-10

   1. Initial program 38.3%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 63.1%

      \[\leadsto \sqrt{10000 \cdot \left(n \cdot n\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-159} \lor \neg \left(n \leq 1.05 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{10000 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(10000 \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.4e-159)
   (* (fma (* (fma 0.16666666666666666 i 0.5) i) 100.0 100.0) n)
   (if (<= n 1.05e-10)
     (sqrt (* (* 10000.0 n) n))
     (fma (- (* 50.0 n) 50.0) i (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.4e-159) {
		tmp = fma((fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n;
	} else if (n <= 1.05e-10) {
		tmp = sqrt(((10000.0 * n) * n));
	} else {
		tmp = fma(((50.0 * n) - 50.0), i, (100.0 * n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.4e-159)
		tmp = Float64(fma(Float64(fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n);
	elseif (n <= 1.05e-10)
		tmp = sqrt(Float64(Float64(10000.0 * n) * n));
	else
		tmp = fma(Float64(Float64(50.0 * n) - 50.0), i, Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.4e-159], N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.05e-10], N[Sqrt[N[(N[(10000.0 * n), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[(N[(N[(50.0 * n), $MachinePrecision] - 50.0), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.3999999999999999e-159

   1. Initial program 32.7%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.9%

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

   if -6.3999999999999999e-159 < n < 1.05e-10

   1. Initial program 38.3%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 63.1%

      \[\leadsto \sqrt{10000 \cdot \left(n \cdot n\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 63.1%

      \[\leadsto \sqrt{\left(10000 \cdot n\right) \cdot n} \]

   if 1.05e-10 < n

   1. Initial program 26.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 35.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right), n, \left(\left(-0.5 \cdot i - 0.5\right) \cdot i\right) \cdot 100\right), n, 33.333333333333336 \cdot \left(i \cdot i\right)\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 74.6%

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

Alternative 18: 54.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (fma (- (* 50.0 n) 50.0) i (* 100.0 n)))

C

double code(double i, double n) {
	return fma(((50.0 * n) - 50.0), i, (100.0 * n));
}

Julia

function code(i, n)
	return fma(Float64(Float64(50.0 * n) - 50.0), i, Float64(100.0 * n))
end

Wolfram

code[i_, n_] := N[(N[(N[(50.0 * n), $MachinePrecision] - 50.0), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5%

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

3. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 48.1%

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

6. Taylor expanded in n around 0

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

8. Applied rewrites 33.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right), n, \left(\left(-0.5 \cdot i - 0.5\right) \cdot i\right) \cdot 100\right), n, 33.333333333333336 \cdot \left(i \cdot i\right)\right)}{\color{blue}{n}} \]

9. Taylor expanded in i around 0

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

11. Applied rewrites 51.0%

    \[\leadsto \mathsf{fma}\left(50 \cdot n - 50, i, 100 \cdot n\right) \]
12. Add Preprocessing

Alternative 19: 49.4% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.5%

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

3. Taylor expanded in i around 0

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

   1. lower-*.f64 44.5

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

5. Applied rewrites 44.5%

   \[\leadsto \color{blue}{100 \cdot n} \]
6. Add Preprocessing

Developer Target 1: 33.7% 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 2024337 
(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))))