Compound Interest

Percentage Accurate: 27.9% → 96.0%

Time: 15.3s

Alternatives: 20

Speedup: 24.3×

• could not determine a ground truth

  1. i = 1.7792049081855538e-190
  2. n = 1.2512310219322066e+39

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.9% 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: 96.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

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

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if -inf.0 < (/.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 24.6%

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

      1. pow-to-exp N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

      6. /-lowering-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

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

   1. Initial program 95.5%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. *-rgt-identity N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. /-lowering-/.f64 N/A

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

      15. distribute-neg-frac2 N/A

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

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

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

      17. neg-lowering-neg.f64 96.2

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

   4. Applied egg-rr 96.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

4. Final simplification 95.7%

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

Alternative 2: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], Infinity], N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.6%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 66.9

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

   5. Simplified 66.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 73.5

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

   7. Applied egg-rr 73.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

4. Final simplification 75.1%

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

Alternative 3: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -3.1 \cdot 10^{-208}:\\ \;\;\;\;100 \cdot \left(n \cdot \mathsf{fma}\left(i, \frac{e^{i}}{n} \cdot -0.5, t\_0\right)\right)\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-211}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-99}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -3.1e-208)
     (* 100.0 (* n (fma i (* (/ (exp i) n) -0.5) t_0)))
     (if (<= n 1.25e-211)
       (/ 0.0 i)
       (if (<= n 1.45e-99)
         (* 100.0 (/ (* n (- (log i) (log n))) (/ i n)))
         (* n (* 100.0 t_0)))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -3.1e-208) {
		tmp = 100.0 * (n * fma(i, ((exp(i) / n) * -0.5), t_0));
	} else if (n <= 1.25e-211) {
		tmp = 0.0 / i;
	} else if (n <= 1.45e-99) {
		tmp = 100.0 * ((n * (log(i) - log(n))) / (i / n));
	} else {
		tmp = n * (100.0 * t_0);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -3.1e-208)
		tmp = Float64(100.0 * Float64(n * fma(i, Float64(Float64(exp(i) / n) * -0.5), t_0)));
	elseif (n <= 1.25e-211)
		tmp = Float64(0.0 / i);
	elseif (n <= 1.45e-99)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(log(i) - log(n))) / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 * t_0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -3.1e-208], N[(100.0 * N[(n * N[(i * N[(N[(N[Exp[i], $MachinePrecision] / n), $MachinePrecision] * -0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.25e-211], N[(0.0 / i), $MachinePrecision], If[LessEqual[n, 1.45e-99], N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.0999999999999998e-208

   1. Initial program 27.6%

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

   3. Taylor expanded in n around inf

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

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

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. div-sub N/A

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

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

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

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

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

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

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

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

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

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

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

      12. accelerator-lowering-expm1.f64 83.3

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

   5. Simplified 83.3%

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

   if -3.0999999999999998e-208 < n < 1.2500000000000001e-211

   1. Initial program 72.3%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 20.8

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

   4. Applied egg-rr 20.8%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 80.7

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

   7. Simplified 80.7%

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

   if 1.2500000000000001e-211 < n < 1.44999999999999993e-99

   1. Initial program 25.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. log-lowering-log.f64 77.5

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

   5. Simplified 77.5%

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

   if 1.44999999999999993e-99 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 84.4

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

   5. Simplified 84.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 89.5

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

   7. Applied egg-rr 89.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{-208}:\\ \;\;\;\;100 \cdot \left(n \cdot \mathsf{fma}\left(i, \frac{e^{i}}{n} \cdot -0.5, \frac{\mathsf{expm1}\left(i\right)}{i}\right)\right)\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-211}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-99}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.1 \cdot 10^{-209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-213}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-99}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -1.1e-209)
     t_0
     (if (<= n 8e-213)
       (/ 0.0 i)
       (if (<= n 3.6e-99)
         (* 100.0 (/ (* n (- (log i) (log n))) (/ i n)))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -1.1e-209) {
		tmp = t_0;
	} else if (n <= 8e-213) {
		tmp = 0.0 / i;
	} else if (n <= 3.6e-99) {
		tmp = 100.0 * ((n * (log(i) - log(n))) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -1.1e-209) {
		tmp = t_0;
	} else if (n <= 8e-213) {
		tmp = 0.0 / i;
	} else if (n <= 3.6e-99) {
		tmp = 100.0 * ((n * (Math.log(i) - Math.log(n))) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -1.1e-209:
		tmp = t_0
	elif n <= 8e-213:
		tmp = 0.0 / i
	elif n <= 3.6e-99:
		tmp = 100.0 * ((n * (math.log(i) - math.log(n))) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -1.1e-209)
		tmp = t_0;
	elseif (n <= 8e-213)
		tmp = Float64(0.0 / i);
	elseif (n <= 3.6e-99)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(log(i) - log(n))) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.1e-209], t$95$0, If[LessEqual[n, 8e-213], N[(0.0 / i), $MachinePrecision], If[LessEqual[n, 3.6e-99], N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.10000000000000005e-209 or 3.6000000000000001e-99 < n

   1. Initial program 25.7%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 79.2

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

   5. Simplified 79.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 85.7

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

   7. Applied egg-rr 85.7%

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

   if -1.10000000000000005e-209 < n < 7.9999999999999996e-213

   1. Initial program 72.3%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 20.8

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

   4. Applied egg-rr 20.8%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 80.7

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

   7. Simplified 80.7%

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

   if 7.9999999999999996e-213 < n < 3.6000000000000001e-99

   1. Initial program 25.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. log-lowering-log.f64 77.5

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

   5. Simplified 77.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-209}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-213}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-99}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.4 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-208}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.95:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -1.4e+26)
     t_0
     (if (<= n -8.5e-208)
       (* 100.0 (* (/ n i) (expm1 i)))
       (if (<= n 1.5e-173)
         (/ 0.0 i)
         (if (<= n 1.95) (* 100.0 (/ i (/ i n))) t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -1.4e+26) {
		tmp = t_0;
	} else if (n <= -8.5e-208) {
		tmp = 100.0 * ((n / i) * expm1(i));
	} else if (n <= 1.5e-173) {
		tmp = 0.0 / i;
	} else if (n <= 1.95) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -1.4e+26) {
		tmp = t_0;
	} else if (n <= -8.5e-208) {
		tmp = 100.0 * ((n / i) * Math.expm1(i));
	} else if (n <= 1.5e-173) {
		tmp = 0.0 / i;
	} else if (n <= 1.95) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -1.4e+26:
		tmp = t_0
	elif n <= -8.5e-208:
		tmp = 100.0 * ((n / i) * math.expm1(i))
	elif n <= 1.5e-173:
		tmp = 0.0 / i
	elif n <= 1.95:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -1.4e+26)
		tmp = t_0;
	elseif (n <= -8.5e-208)
		tmp = Float64(100.0 * Float64(Float64(n / i) * expm1(i)));
	elseif (n <= 1.5e-173)
		tmp = Float64(0.0 / i);
	elseif (n <= 1.95)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.4e+26], t$95$0, If[LessEqual[n, -8.5e-208], N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5e-173], N[(0.0 / i), $MachinePrecision], If[LessEqual[n, 1.95], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.4e26 or 1.94999999999999996 < n

   1. Initial program 27.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 90.0

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

   5. Simplified 90.0%

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

   if -1.4e26 < n < -8.49999999999999997e-208

   1. Initial program 30.4%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 54.9

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

   5. Simplified 54.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 77.4

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

   7. Applied egg-rr 77.4%

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

   if -8.49999999999999997e-208 < n < 1.5000000000000001e-173

   1. Initial program 62.5%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 16.1

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

   4. Applied egg-rr 16.1%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 74.7

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

   7. Simplified 74.7%

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

   if 1.5000000000000001e-173 < n < 1.94999999999999996

   1. Initial program 11.3%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 74.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+26}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq -8.5 \cdot 10^{-208}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.95:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -4.5 \cdot 10^{-207}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -4.5e-207) t_0 (if (<= n 1.4e-99) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -4.5e-207) {
		tmp = t_0;
	} else if (n <= 1.4e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -4.5e-207) {
		tmp = t_0;
	} else if (n <= 1.4e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -4.5e-207:
		tmp = t_0
	elif n <= 1.4e-99:
		tmp = 0.0 / i
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -4.5e-207)
		tmp = t_0;
	elseif (n <= 1.4e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.5e-207], t$95$0, If[LessEqual[n, 1.4e-99], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.49999999999999992e-207 or 1.4e-99 < n

   1. Initial program 25.7%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 79.2

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

   5. Simplified 79.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 85.7

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

   7. Applied egg-rr 85.7%

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

   if -4.49999999999999992e-207 < n < 1.4e-99

   1. Initial program 50.0%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 14.0

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.4

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

   7. Simplified 67.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-207}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right)\\ \mathbf{if}\;n \leq -9.2 \cdot 10^{-209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* (expm1 i) (/ 100.0 i)))))
   (if (<= n -9.2e-209) t_0 (if (<= n 1.6e-99) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (expm1(i) * (100.0 / i));
	double tmp;
	if (n <= -9.2e-209) {
		tmp = t_0;
	} else if (n <= 1.6e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (Math.expm1(i) * (100.0 / i));
	double tmp;
	if (n <= -9.2e-209) {
		tmp = t_0;
	} else if (n <= 1.6e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (math.expm1(i) * (100.0 / i))
	tmp = 0
	if n <= -9.2e-209:
		tmp = t_0
	elif n <= 1.6e-99:
		tmp = 0.0 / i
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(expm1(i) * Float64(100.0 / i)))
	tmp = 0.0
	if (n <= -9.2e-209)
		tmp = t_0;
	elseif (n <= 1.6e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.2e-209], t$95$0, If[LessEqual[n, 1.6e-99], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.1999999999999999e-209 or 1.6e-99 < n

   1. Initial program 25.7%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 79.2

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

   5. Simplified 79.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 85.7

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

   7. Applied egg-rr 85.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 85.2

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

   9. Applied egg-rr 85.2%

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

   if -9.1999999999999999e-209 < n < 1.6e-99

   1. Initial program 50.0%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 14.0

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.4

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

   7. Simplified 67.4%

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

Alternative 8: 69.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ t_1 := 100 \cdot \frac{n \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i\right)}{i}\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-210}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.36 \cdot 10^{-173}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.7:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n))))
        (t_1
         (*
          100.0
          (/
           (*
            n
            (fma
             (* i i)
             (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5)
             i))
           i))))
   (if (<= n -6.5e+48)
     t_1
     (if (<= n -9e-210)
       t_0
       (if (<= n 1.36e-173) (/ 0.0 i) (if (<= n 1.7) t_0 t_1))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double t_1 = 100.0 * ((n * fma((i * i), fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), i)) / i);
	double tmp;
	if (n <= -6.5e+48) {
		tmp = t_1;
	} else if (n <= -9e-210) {
		tmp = t_0;
	} else if (n <= 1.36e-173) {
		tmp = 0.0 / i;
	} else if (n <= 1.7) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i / Float64(i / n)))
	t_1 = Float64(100.0 * Float64(Float64(n * fma(Float64(i * i), fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), i)) / i))
	tmp = 0.0
	if (n <= -6.5e+48)
		tmp = t_1;
	elseif (n <= -9e-210)
		tmp = t_0;
	elseif (n <= 1.36e-173)
		tmp = Float64(0.0 / i);
	elseif (n <= 1.7)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(N[(n * N[(N[(i * i), $MachinePrecision] * N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.5e+48], t$95$1, If[LessEqual[n, -9e-210], t$95$0, If[LessEqual[n, 1.36e-173], N[(0.0 / i), $MachinePrecision], If[LessEqual[n, 1.7], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.49999999999999972e48 or 1.69999999999999996 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{i \cdot \left(1 + i \cdot \left(\left(\frac{1}{2} + i \cdot \left(\left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{n}^{2}} + i \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}}{\frac{i}{n}} \]

   5. Simplified 43.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(i \cdot i, 0.5 + \mathsf{fma}\left(i, \left(0.16666666666666666 + \frac{0.3333333333333333}{n \cdot n}\right) + \mathsf{fma}\left(i, \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right) + \left(\frac{-0.25}{n \cdot \left(n \cdot n\right)} + \frac{-0.25}{n}\right), \frac{-0.5}{n}\right), \frac{-0.5}{n}\right), i\right)}}{\frac{i}{n}} \]

   6. Taylor expanded in n around inf

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

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

         \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(i + {i}^{2} \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}{i}} \]

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

         \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(i + {i}^{2} \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}}{i} \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\left({i}^{2} \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right) + i\right)}}{i} \]

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

         \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\mathsf{fma}\left({i}^{2}, \frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right), i\right)}}{i} \]

      5. unpow2 N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{fma}\left(\color{blue}{i \cdot i}, \frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right), i\right)}{i} \]

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

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{fma}\left(\color{blue}{i \cdot i}, \frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right), i\right)}{i} \]

      7. +-commutative N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{fma}\left(i \cdot i, \color{blue}{i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right) + \frac{1}{2}}, i\right)}{i} \]

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

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{fma}\left(i \cdot i, \color{blue}{\mathsf{fma}\left(i, \frac{1}{6} + \frac{1}{24} \cdot i, \frac{1}{2}\right)}, i\right)}{i} \]

      9. +-commutative N/A

         \[\leadsto 100 \cdot \frac{n \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, \color{blue}{\frac{1}{24} \cdot i + \frac{1}{6}}, \frac{1}{2}\right), i\right)}{i} \]

      10. *-commutative N/A

          \[\leadsto 100 \cdot \frac{n \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), i\right)}{i} \]

      11. accelerator-lowering-fma.f64 73.5

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

   8. Simplified 73.5%

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

   if -6.49999999999999972e48 < n < -9.00000000000000039e-210 or 1.3600000000000001e-173 < n < 1.69999999999999996

   1. Initial program 25.3%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 67.0%

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

   if -9.00000000000000039e-210 < n < 1.3600000000000001e-173

   1. Initial program 62.5%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 16.1

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

   4. Applied egg-rr 16.1%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 74.7

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

   7. Simplified 74.7%

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

Alternative 9: 66.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{+48}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{elif}\;n \leq -3.1 \cdot 10^{-209}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(i, 50, 100\right), i \cdot \left(i \cdot \left(n \cdot \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.6e+48)
   (*
    n
    (fma i (fma i (fma i 4.166666666666667 16.666666666666668) 50.0) 100.0))
   (if (<= n -3.1e-209)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 1.4e-99)
       (/ 0.0 i)
       (fma
        n
        (fma i 50.0 100.0)
        (* i (* i (* n (fma i 4.166666666666667 16.666666666666668)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.6e+48) {
		tmp = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	} else if (n <= -3.1e-209) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.4e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = fma(n, fma(i, 50.0, 100.0), (i * (i * (n * fma(i, 4.166666666666667, 16.666666666666668)))));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.6e+48)
		tmp = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0));
	elseif (n <= -3.1e-209)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.4e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = fma(n, fma(i, 50.0, 100.0), Float64(i * Float64(i * Float64(n * fma(i, 4.166666666666667, 16.666666666666668)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.6e+48], N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -3.1e-209], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.4e-99], N[(0.0 / i), $MachinePrecision], N[(n * N[(i * 50.0 + 100.0), $MachinePrecision] + N[(i * N[(i * N[(n * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.6e48

   1. Initial program 22.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 89.3

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

   5. Simplified 89.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 89.3

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

   7. Applied egg-rr 89.3%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 60.8

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

   10. Simplified 60.8%

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

   if -4.6e48 < n < -3.1e-209

   1. Initial program 34.6%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 62.3%

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

   if -3.1e-209 < n < 1.4e-99

   1. Initial program 50.0%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 14.0

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.4

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

   7. Simplified 67.4%

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

   if 1.4e-99 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 84.4

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

   5. Simplified 84.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 89.5

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

   7. Applied egg-rr 89.5%

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

   8. Taylor expanded in i around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. associate-*r* N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

   10. Simplified 79.9%

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

Alternative 10: 65.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-157}:\\ \;\;\;\;n \cdot \left(100 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right) + \frac{\mathsf{fma}\left(i, -0.5, -0.5\right)}{n}, 1\right)\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(i, 50, 100\right), i \cdot \left(i \cdot \left(n \cdot \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.4e-157)
   (*
    n
    (*
     100.0
     (fma i (+ (fma i 0.16666666666666666 0.5) (/ (fma i -0.5 -0.5) n)) 1.0)))
   (if (<= n 1.4e-99)
     (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
     (fma
      n
      (fma i 50.0 100.0)
      (* i (* i (* n (fma i 4.166666666666667 16.666666666666668))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.4e-157) {
		tmp = n * (100.0 * fma(i, (fma(i, 0.16666666666666666, 0.5) + (fma(i, -0.5, -0.5) / n)), 1.0));
	} else if (n <= 1.4e-99) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = fma(n, fma(i, 50.0, 100.0), (i * (i * (n * fma(i, 4.166666666666667, 16.666666666666668)))));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.4e-157)
		tmp = Float64(n * Float64(100.0 * fma(i, Float64(fma(i, 0.16666666666666666, 0.5) + Float64(fma(i, -0.5, -0.5) / n)), 1.0)));
	elseif (n <= 1.4e-99)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = fma(n, fma(i, 50.0, 100.0), Float64(i * Float64(i * Float64(n * fma(i, 4.166666666666667, 16.666666666666668)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.4e-157], N[(n * N[(100.0 * N[(i * N[(N[(i * 0.16666666666666666 + 0.5), $MachinePrecision] + N[(N[(i * -0.5 + -0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.4e-99], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(i * 50.0 + 100.0), $MachinePrecision] + N[(i * N[(i * N[(n * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.4e-157

   1. Initial program 25.8%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Simplified 59.6%

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

   6. Taylor expanded in n around inf

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

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

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

      2. distribute-lft-out N/A

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

      3. associate-+r+ N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out N/A

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

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

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

   8. Simplified 59.1%

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

   if -5.4e-157 < n < 1.4e-99

   1. Initial program 49.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. Simplified 64.2%

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

   if 1.4e-99 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 84.4

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

   5. Simplified 84.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 89.5

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

   7. Applied egg-rr 89.5%

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

   8. Taylor expanded in i around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

      15. associate-*r* N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

   10. Simplified 79.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{-157}:\\ \;\;\;\;n \cdot \left(100 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right) + \frac{\mathsf{fma}\left(i, -0.5, -0.5\right)}{n}, 1\right)\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(i, 50, 100\right), i \cdot \left(i \cdot \left(n \cdot \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{+48}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{elif}\;n \leq -1.45 \cdot 10^{-208}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.6e+48)
   (*
    n
    (fma i (fma i (fma i 4.166666666666667 16.666666666666668) 50.0) 100.0))
   (if (<= n -1.45e-208)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 1.6e-99)
       (/ 0.0 i)
       (*
        100.0
        (fma
         i
         (* n (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5))
         n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.6e+48) {
		tmp = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	} else if (n <= -1.45e-208) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.6e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = 100.0 * fma(i, (n * fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5)), n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.6e+48)
		tmp = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0));
	elseif (n <= -1.45e-208)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.6e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = Float64(100.0 * fma(i, Float64(n * fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5)), n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.6e+48], N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.45e-208], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.6e-99], N[(0.0 / i), $MachinePrecision], N[(100.0 * N[(i * N[(n * N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.6e48

   1. Initial program 22.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 89.3

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

   5. Simplified 89.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 89.3

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

   7. Applied egg-rr 89.3%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 60.8

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

   10. Simplified 60.8%

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

   if -4.6e48 < n < -1.45e-208

   1. Initial program 34.6%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 62.3%

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

   if -1.45e-208 < n < 1.6e-99

   1. Initial program 50.0%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 14.0

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.4

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

   7. Simplified 67.4%

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

   if 1.6e-99 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right) + n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]

   5. Simplified 79.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(n, 0.5 - \frac{0.5}{n}, \left(n \cdot i\right) \cdot \mathsf{fma}\left(i, \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right) + \left(\frac{-0.25}{n \cdot \left(n \cdot n\right)} + \frac{-0.25}{n}\right), \frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right)\right), n\right)} \]

   6. Taylor expanded in n around inf

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

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

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{n \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)}, n\right) \]

      2. +-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, n \cdot \color{blue}{\left(i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right) + \frac{1}{2}\right)}, n\right) \]

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

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, n \cdot \color{blue}{\mathsf{fma}\left(i, \frac{1}{6} + \frac{1}{24} \cdot i, \frac{1}{2}\right)}, n\right) \]

      4. +-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(i, \color{blue}{\frac{1}{24} \cdot i + \frac{1}{6}}, \frac{1}{2}\right), n\right) \]

      5. *-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{1}{24}} + \frac{1}{6}, \frac{1}{2}\right), n\right) \]

      6. accelerator-lowering-fma.f64 79.9

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

   8. Simplified 79.9%

      \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)}, n\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 66.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{if}\;n \leq -4.6 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.2 \cdot 10^{-209}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (fma
           i
           (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
           100.0))))
   (if (<= n -4.6e+48)
     t_0
     (if (<= n -2.2e-209)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 1.4e-99) (/ 0.0 i) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	double tmp;
	if (n <= -4.6e+48) {
		tmp = t_0;
	} else if (n <= -2.2e-209) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.4e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0))
	tmp = 0.0
	if (n <= -4.6e+48)
		tmp = t_0;
	elseif (n <= -2.2e-209)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.4e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.6e+48], t$95$0, If[LessEqual[n, -2.2e-209], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.4e-99], N[(0.0 / i), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.6e48 or 1.4e-99 < n

   1. Initial program 23.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-expm1.f64 86.4

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

   5. Simplified 86.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 89.4

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

   7. Applied egg-rr 89.4%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 72.0

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

   10. Simplified 72.0%

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

   if -4.6e48 < n < -2.2000000000000001e-209

   1. Initial program 34.6%

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

   3. Taylor expanded in i around 0

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

   5. Simplified 62.3%

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

   if -2.2000000000000001e-209 < n < 1.4e-99

   1. Initial program 50.0%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 14.0

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

   4. Applied egg-rr 14.0%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.4

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

   7. Simplified 67.4%

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

Alternative 13: 65.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{if}\;n \leq -9.4 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (fma
           i
           (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
           100.0))))
   (if (<= n -9.4e-153) t_0 (if (<= n 4e-99) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	double tmp;
	if (n <= -9.4e-153) {
		tmp = t_0;
	} else if (n <= 4e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0))
	tmp = 0.0
	if (n <= -9.4e-153)
		tmp = t_0;
	elseif (n <= 4e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.4e-153], t$95$0, If[LessEqual[n, 4e-99], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.3999999999999998e-153 or 4.0000000000000001e-99 < n

   1. Initial program 24.7%

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

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

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

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

      3. accelerator-lowering-expm1.f64 80.6

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

   5. Simplified 80.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-expm1.f64 86.0

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

   7. Applied egg-rr 86.0%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 68.6

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

   10. Simplified 68.6%

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

   if -9.3999999999999998e-153 < n < 4.0000000000000001e-99

   1. Initial program 49.7%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 19.3

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

   4. Applied egg-rr 19.3%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 64.2

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

   7. Simplified 64.2%

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

Alternative 14: 65.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-156}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i \cdot n, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), n\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(4.166666666666667, i \cdot \left(i \cdot i\right), 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.7e-156)
   (* 100.0 (fma (* i n) (fma i 0.16666666666666666 0.5) n))
   (if (<= n 1.4e-99)
     (/ 0.0 i)
     (* n (fma 4.166666666666667 (* i (* i i)) 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.7e-156) {
		tmp = 100.0 * fma((i * n), fma(i, 0.16666666666666666, 0.5), n);
	} else if (n <= 1.4e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = n * fma(4.166666666666667, (i * (i * i)), 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.7e-156)
		tmp = Float64(100.0 * fma(Float64(i * n), fma(i, 0.16666666666666666, 0.5), n));
	elseif (n <= 1.4e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = Float64(n * fma(4.166666666666667, Float64(i * Float64(i * i)), 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.7e-156], N[(100.0 * N[(N[(i * n), $MachinePrecision] * N[(i * 0.16666666666666666 + 0.5), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.4e-99], N[(0.0 / i), $MachinePrecision], N[(n * N[(4.166666666666667 * N[(i * N[(i * i), $MachinePrecision]), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.70000000000000012e-156

   1. Initial program 25.8%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Simplified 59.6%

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

   6. Taylor expanded in n around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 58.4

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

   8. Simplified 58.4%

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

   if -2.70000000000000012e-156 < n < 1.4e-99

   1. Initial program 49.7%

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

      1. div-sub N/A

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

      2. clear-num N/A

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

      3. sub-neg N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      10. distribute-neg-frac2 N/A

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

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

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

      12. neg-lowering-neg.f64 19.3

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

   4. Applied egg-rr 19.3%

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

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. /-lowering-/.f64 64.2

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   if 1.4e-99 < n

   1. Initial program 23.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right) + n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]

   5. Simplified 79.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(n, 0.5 - \frac{0.5}{n}, \left(n \cdot i\right) \cdot \mathsf{fma}\left(i, \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right) + \left(\frac{-0.25}{n \cdot \left(n \cdot n\right)} + \frac{-0.25}{n}\right), \frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right)\right), n\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{{i}^{2} \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)}, n\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left({i}^{2} \cdot n\right) \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}, n\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left({i}^{2} \cdot n\right) \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}, n\right) \]

      3. *-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left(n \cdot {i}^{2}\right)} \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left(n \cdot {i}^{2}\right)} \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      5. unpow2 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \color{blue}{\left(i \cdot i\right)}\right) \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \color{blue}{\left(i \cdot i\right)}\right) \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}, n\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)} - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      9. associate-*r/ N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \color{blue}{\frac{\frac{11}{24} \cdot 1}{{n}^{2}}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      10. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\color{blue}{\frac{11}{24}}}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \color{blue}{\frac{\frac{11}{24}}{{n}^{2}}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      12. unpow2 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{\color{blue}{n \cdot n}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{\color{blue}{n \cdot n}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \color{blue}{\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)}\right), n\right) \]

      15. associate-*r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{n}} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\color{blue}{\frac{1}{4}}}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\color{blue}{\frac{\frac{1}{4}}{n}} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      18. associate-*r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \color{blue}{\frac{\frac{1}{4} \cdot 1}{{n}^{3}}}\right)\right), n\right) \]

      19. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \frac{\color{blue}{\frac{1}{4}}}{{n}^{3}}\right)\right), n\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \color{blue}{\frac{\frac{1}{4}}{{n}^{3}}}\right)\right), n\right) \]

      21. cube-mult N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \frac{\frac{1}{4}}{\color{blue}{n \cdot \left(n \cdot n\right)}}\right)\right), n\right) \]

      22. unpow2 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \frac{\frac{1}{4}}{n \cdot \color{blue}{{n}^{2}}}\right)\right), n\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \frac{\frac{1}{4}}{\color{blue}{n \cdot {n}^{2}}}\right)\right), n\right) \]

   8. Simplified 79.0%

      \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right) - \left(\frac{0.25}{n} + \frac{0.25}{n \cdot \left(n \cdot n\right)}\right)\right)}, n\right) \]

   9. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(n \cdot 100\right)} \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto n \cdot \left(100 \cdot \color{blue}{\left(\frac{1}{24} \cdot {i}^{3} + 1\right)}\right) \]

       6. distribute-lft-in N/A

          \[\leadsto n \cdot \color{blue}{\left(100 \cdot \left(\frac{1}{24} \cdot {i}^{3}\right) + 100 \cdot 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto n \cdot \left(\color{blue}{\left(100 \cdot \frac{1}{24}\right) \cdot {i}^{3}} + 100 \cdot 1\right) \]

       8. metadata-eval N/A

          \[\leadsto n \cdot \left(\color{blue}{\frac{25}{6}} \cdot {i}^{3} + 100 \cdot 1\right) \]

       9. metadata-eval N/A

          \[\leadsto n \cdot \left(\frac{25}{6} \cdot {i}^{3} + \color{blue}{100}\right) \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(\frac{25}{6}, {i}^{3}, 100\right)} \]

       11. cube-mult N/A

           \[\leadsto n \cdot \mathsf{fma}\left(\frac{25}{6}, \color{blue}{i \cdot \left(i \cdot i\right)}, 100\right) \]

       12. unpow2 N/A

           \[\leadsto n \cdot \mathsf{fma}\left(\frac{25}{6}, i \cdot \color{blue}{{i}^{2}}, 100\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto n \cdot \mathsf{fma}\left(\frac{25}{6}, \color{blue}{i \cdot {i}^{2}}, 100\right) \]

       14. unpow2 N/A

           \[\leadsto n \cdot \mathsf{fma}\left(\frac{25}{6}, i \cdot \color{blue}{\left(i \cdot i\right)}, 100\right) \]

       15. *-lowering-*.f64 79.1

           \[\leadsto n \cdot \mathsf{fma}\left(4.166666666666667, i \cdot \color{blue}{\left(i \cdot i\right)}, 100\right) \]

   11. Simplified 79.1%

       \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(4.166666666666667, i \cdot \left(i \cdot i\right), 100\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-156}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i \cdot n, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), n\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(4.166666666666667, i \cdot \left(i \cdot i\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{-155}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(4.166666666666667, i \cdot \left(i \cdot i\right), 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.2e-155)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 1.9e-99)
     (/ 0.0 i)
     (* n (fma 4.166666666666667 (* i (* i i)) 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.2e-155) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 1.9e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = n * fma(4.166666666666667, (i * (i * i)), 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.2e-155)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 1.9e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = Float64(n * fma(4.166666666666667, Float64(i * Float64(i * i)), 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.2e-155], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.9e-99], N[(0.0 / i), $MachinePrecision], N[(n * N[(4.166666666666667 * N[(i * N[(i * i), $MachinePrecision]), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.1999999999999999e-155

   1. Initial program 25.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(e^{i} - 1\right)}}{i} \]

      3. accelerator-lowering-expm1.f64 77.3

         \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   5. Simplified 77.3%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \cdot 100 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{n \cdot \left(\frac{e^{i} - 1}{i} \cdot 100\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{n \cdot \left(\frac{e^{i} - 1}{i} \cdot 100\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto n \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto n \cdot \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \]

      7. accelerator-lowering-expm1.f64 83.0

         \[\leadsto n \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \]

   7. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{n \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50 + \frac{50}{3} \cdot i, 100\right)} \]

      3. +-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\frac{50}{3} \cdot i + 50}, 100\right) \]

      4. *-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{50}{3}} + 50, 100\right) \]

      5. accelerator-lowering-fma.f64 58.4

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(i, 16.666666666666668, 50\right)}, 100\right) \]

   10. Simplified 58.4%

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)} \]

   if -2.1999999999999999e-155 < n < 1.8999999999999998e-99

   1. Initial program 49.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      2. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      3. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      4. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{n}{\mathsf{neg}\left(i\right)}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{n}{\mathsf{neg}\left(i\right)}}\right) \]

      12. neg-lowering-neg.f64 19.3

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{n}{\color{blue}{-i}}\right) \]

   4. Applied egg-rr 19.3%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{n}{-i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. /-lowering-/.f64 64.2

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   if 1.8999999999999998e-99 < n

   1. Initial program 23.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right) + n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]

   5. Simplified 79.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(n, 0.5 - \frac{0.5}{n}, \left(n \cdot i\right) \cdot \mathsf{fma}\left(i, \left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right) + \left(\frac{-0.25}{n \cdot \left(n \cdot n\right)} + \frac{-0.25}{n}\right), \frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right)\right), n\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{{i}^{2} \cdot \left(n \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)\right)}, n\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left({i}^{2} \cdot n\right) \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}, n\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left({i}^{2} \cdot n\right) \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}, n\right) \]

      3. *-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left(n \cdot {i}^{2}\right)} \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left(n \cdot {i}^{2}\right)} \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      5. unpow2 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \color{blue}{\left(i \cdot i\right)}\right) \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \color{blue}{\left(i \cdot i\right)}\right) \cdot \left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right)}, n\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{24} + \frac{11}{24} \cdot \frac{1}{{n}^{2}}\right)} - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      9. associate-*r/ N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \color{blue}{\frac{\frac{11}{24} \cdot 1}{{n}^{2}}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      10. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\color{blue}{\frac{11}{24}}}{{n}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \color{blue}{\frac{\frac{11}{24}}{{n}^{2}}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      12. unpow2 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{\color{blue}{n \cdot n}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{\color{blue}{n \cdot n}}\right) - \left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \color{blue}{\left(\frac{1}{4} \cdot \frac{1}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)}\right), n\right) \]

      15. associate-*r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{n}} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\color{blue}{\frac{1}{4}}}{n} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\color{blue}{\frac{\frac{1}{4}}{n}} + \frac{1}{4} \cdot \frac{1}{{n}^{3}}\right)\right), n\right) \]

      18. associate-*r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \color{blue}{\frac{\frac{1}{4} \cdot 1}{{n}^{3}}}\right)\right), n\right) \]

      19. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \frac{\color{blue}{\frac{1}{4}}}{{n}^{3}}\right)\right), n\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \color{blue}{\frac{\frac{1}{4}}{{n}^{3}}}\right)\right), n\right) \]

      21. cube-mult N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \frac{\frac{1}{4}}{\color{blue}{n \cdot \left(n \cdot n\right)}}\right)\right), n\right) \]

      22. unpow2 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \frac{\frac{1}{4}}{n \cdot \color{blue}{{n}^{2}}}\right)\right), n\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(i, \left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(\frac{1}{24} + \frac{\frac{11}{24}}{n \cdot n}\right) - \left(\frac{\frac{1}{4}}{n} + \frac{\frac{1}{4}}{\color{blue}{n \cdot {n}^{2}}}\right)\right), n\right) \]

   8. Simplified 79.0%

      \[\leadsto 100 \cdot \mathsf{fma}\left(i, \color{blue}{\left(n \cdot \left(i \cdot i\right)\right) \cdot \left(\left(0.041666666666666664 + \frac{0.4583333333333333}{n \cdot n}\right) - \left(\frac{0.25}{n} + \frac{0.25}{n \cdot \left(n \cdot n\right)}\right)\right)}, n\right) \]

   9. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \left(n \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(100 \cdot n\right) \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(n \cdot 100\right)} \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + \frac{1}{24} \cdot {i}^{3}\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto n \cdot \left(100 \cdot \color{blue}{\left(\frac{1}{24} \cdot {i}^{3} + 1\right)}\right) \]

       6. distribute-lft-in N/A

          \[\leadsto n \cdot \color{blue}{\left(100 \cdot \left(\frac{1}{24} \cdot {i}^{3}\right) + 100 \cdot 1\right)} \]

       7. associate-*r* N/A

          \[\leadsto n \cdot \left(\color{blue}{\left(100 \cdot \frac{1}{24}\right) \cdot {i}^{3}} + 100 \cdot 1\right) \]

       8. metadata-eval N/A

          \[\leadsto n \cdot \left(\color{blue}{\frac{25}{6}} \cdot {i}^{3} + 100 \cdot 1\right) \]

       9. metadata-eval N/A

          \[\leadsto n \cdot \left(\frac{25}{6} \cdot {i}^{3} + \color{blue}{100}\right) \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(\frac{25}{6}, {i}^{3}, 100\right)} \]

       11. cube-mult N/A

           \[\leadsto n \cdot \mathsf{fma}\left(\frac{25}{6}, \color{blue}{i \cdot \left(i \cdot i\right)}, 100\right) \]

       12. unpow2 N/A

           \[\leadsto n \cdot \mathsf{fma}\left(\frac{25}{6}, i \cdot \color{blue}{{i}^{2}}, 100\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto n \cdot \mathsf{fma}\left(\frac{25}{6}, \color{blue}{i \cdot {i}^{2}}, 100\right) \]

       14. unpow2 N/A

           \[\leadsto n \cdot \mathsf{fma}\left(\frac{25}{6}, i \cdot \color{blue}{\left(i \cdot i\right)}, 100\right) \]

       15. *-lowering-*.f64 79.1

           \[\leadsto n \cdot \mathsf{fma}\left(4.166666666666667, i \cdot \color{blue}{\left(i \cdot i\right)}, 100\right) \]

   11. Simplified 79.1%

       \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(4.166666666666667, i \cdot \left(i \cdot i\right), 100\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 64.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{if}\;n \leq -2.6 \cdot 10^{-155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i (fma i 16.666666666666668 50.0) 100.0))))
   (if (<= n -2.6e-155) t_0 (if (<= n 1.75e-99) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	double tmp;
	if (n <= -2.6e-155) {
		tmp = t_0;
	} else if (n <= 1.75e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0))
	tmp = 0.0
	if (n <= -2.6e-155)
		tmp = t_0;
	elseif (n <= 1.75e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.6e-155], t$95$0, If[LessEqual[n, 1.75e-99], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.60000000000000008e-155 or 1.7499999999999999e-99 < n

   1. Initial program 24.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(e^{i} - 1\right)}}{i} \]

      3. accelerator-lowering-expm1.f64 80.6

         \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   5. Simplified 80.6%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i} \cdot 100} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \cdot 100 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{n \cdot \left(\frac{e^{i} - 1}{i} \cdot 100\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{n \cdot \left(\frac{e^{i} - 1}{i} \cdot 100\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto n \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto n \cdot \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \]

      7. accelerator-lowering-expm1.f64 86.0

         \[\leadsto n \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \]

   7. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{n \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto n \cdot \color{blue}{\left(i \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50 + \frac{50}{3} \cdot i, 100\right)} \]

      3. +-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\frac{50}{3} \cdot i + 50}, 100\right) \]

      4. *-commutative N/A

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{i \cdot \frac{50}{3}} + 50, 100\right) \]

      5. accelerator-lowering-fma.f64 66.2

         \[\leadsto n \cdot \mathsf{fma}\left(i, \color{blue}{\mathsf{fma}\left(i, 16.666666666666668, 50\right)}, 100\right) \]

   10. Simplified 66.2%

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)} \]

   if -2.60000000000000008e-155 < n < 1.7499999999999999e-99

   1. Initial program 49.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      2. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      3. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      4. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{n}{\mathsf{neg}\left(i\right)}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{n}{\mathsf{neg}\left(i\right)}}\right) \]

      12. neg-lowering-neg.f64 19.3

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{n}{\color{blue}{-i}}\right) \]

   4. Applied egg-rr 19.3%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{n}{-i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. /-lowering-/.f64 64.2

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i 50.0 100.0))))
   (if (<= n -1.15e-159) t_0 (if (<= n 1.4e-99) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, 50.0, 100.0);
	double tmp;
	if (n <= -1.15e-159) {
		tmp = t_0;
	} else if (n <= 1.4e-99) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, 50.0, 100.0))
	tmp = 0.0
	if (n <= -1.15e-159)
		tmp = t_0;
	elseif (n <= 1.4e-99)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.15e-159], t$95$0, If[LessEqual[n, 1.4e-99], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.14999999999999989e-159 or 1.4e-99 < n

   1. Initial program 24.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(e^{i} - 1\right)}}{i} \]

      3. accelerator-lowering-expm1.f64 80.6

         \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   5. Simplified 80.6%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

      2. distribute-rgt-out N/A

         \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

      3. metadata-eval N/A

         \[\leadsto n \cdot \left(\color{blue}{\left(100 \cdot \frac{1}{2}\right)} \cdot i + 100\right) \]

      4. associate-*r* N/A

         \[\leadsto n \cdot \left(\color{blue}{100 \cdot \left(\frac{1}{2} \cdot i\right)} + 100\right) \]

      5. metadata-eval N/A

         \[\leadsto n \cdot \left(100 \cdot \left(\frac{1}{2} \cdot i\right) + \color{blue}{100 \cdot 1}\right) \]

      6. distribute-lft-in N/A

         \[\leadsto n \cdot \color{blue}{\left(100 \cdot \left(\frac{1}{2} \cdot i + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto n \cdot \left(100 \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot i\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto n \cdot \left(100 \cdot \color{blue}{\left(\frac{1}{2} \cdot i + 1\right)}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto n \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot i\right) \cdot 100 + 1 \cdot 100\right)} \]

      11. *-commutative N/A

          \[\leadsto n \cdot \left(\color{blue}{\left(i \cdot \frac{1}{2}\right)} \cdot 100 + 1 \cdot 100\right) \]

      12. associate-*l* N/A

          \[\leadsto n \cdot \left(\color{blue}{i \cdot \left(\frac{1}{2} \cdot 100\right)} + 1 \cdot 100\right) \]

      13. metadata-eval N/A

          \[\leadsto n \cdot \left(i \cdot \color{blue}{50} + 1 \cdot 100\right) \]

      14. metadata-eval N/A

          \[\leadsto n \cdot \left(i \cdot 50 + \color{blue}{100}\right) \]

      15. accelerator-lowering-fma.f64 63.7

          \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50, 100\right)} \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(i, 50, 100\right)} \]

   if -1.14999999999999989e-159 < n < 1.4e-99

   1. Initial program 49.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      2. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      3. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      4. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{n}{\mathsf{neg}\left(i\right)}}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{n}{\mathsf{neg}\left(i\right)}}\right) \]

      12. neg-lowering-neg.f64 19.3

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{n}{\color{blue}{-i}}\right) \]

   4. Applied egg-rr 19.3%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{n}{-i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. /-lowering-/.f64 64.2

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 55.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ n \cdot \mathsf{fma}\left(i, 50, 100\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n (fma i 50.0 100.0)))

C

double code(double i, double n) {
	return n * fma(i, 50.0, 100.0);
}

Julia

function code(i, n)
	return Float64(n * fma(i, 50.0, 100.0))
end

Wolfram

code[i_, n_] := N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision]

Derivation

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 n around inf

   \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(e^{i} - 1\right)}}{i} \]

   3. accelerator-lowering-expm1.f64 69.6

      \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

5. Simplified 69.6%

   \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

6. Taylor expanded in i around 0

   \[\leadsto \color{blue}{50 \cdot \left(i \cdot n\right) + 100 \cdot n} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(50 \cdot i\right) \cdot n} + 100 \cdot n \]

   2. distribute-rgt-out N/A

      \[\leadsto \color{blue}{n \cdot \left(50 \cdot i + 100\right)} \]

   3. metadata-eval N/A

      \[\leadsto n \cdot \left(\color{blue}{\left(100 \cdot \frac{1}{2}\right)} \cdot i + 100\right) \]

   4. associate-*r* N/A

      \[\leadsto n \cdot \left(\color{blue}{100 \cdot \left(\frac{1}{2} \cdot i\right)} + 100\right) \]

   5. metadata-eval N/A

      \[\leadsto n \cdot \left(100 \cdot \left(\frac{1}{2} \cdot i\right) + \color{blue}{100 \cdot 1}\right) \]

   6. distribute-lft-in N/A

      \[\leadsto n \cdot \color{blue}{\left(100 \cdot \left(\frac{1}{2} \cdot i + 1\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto n \cdot \left(100 \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot i\right)}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{n \cdot \left(100 \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto n \cdot \left(100 \cdot \color{blue}{\left(\frac{1}{2} \cdot i + 1\right)}\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto n \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot i\right) \cdot 100 + 1 \cdot 100\right)} \]

   11. *-commutative N/A

       \[\leadsto n \cdot \left(\color{blue}{\left(i \cdot \frac{1}{2}\right)} \cdot 100 + 1 \cdot 100\right) \]

   12. associate-*l* N/A

       \[\leadsto n \cdot \left(\color{blue}{i \cdot \left(\frac{1}{2} \cdot 100\right)} + 1 \cdot 100\right) \]

   13. metadata-eval N/A

       \[\leadsto n \cdot \left(i \cdot \color{blue}{50} + 1 \cdot 100\right) \]

   14. metadata-eval N/A

       \[\leadsto n \cdot \left(i \cdot 50 + \color{blue}{100}\right) \]

   15. accelerator-lowering-fma.f64 55.6

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, 50, 100\right)} \]

8. Simplified 55.6%

   \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(i, 50, 100\right)} \]
9. Add Preprocessing

Alternative 19: 49.3% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * 100.0d0
end function

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

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

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{n \cdot 100} \]

   2. *-lowering-*.f64 47.7

      \[\leadsto \color{blue}{n \cdot 100} \]

5. Simplified 47.7%

   \[\leadsto \color{blue}{n \cdot 100} \]
6. Add Preprocessing

Alternative 20: 2.8% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

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

   \[\leadsto \color{blue}{100 \cdot n + 100 \cdot \left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \color{blue}{100 \cdot \left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot 100} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot 100} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n\right)} \cdot 100 \]

   5. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(i \cdot n\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)} + n\right) \cdot 100 \]

   6. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) \cdot \left(i \cdot n\right)} + n\right) \cdot 100 \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}, i \cdot n, n\right)} \cdot 100 \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}}, i \cdot n, n\right) \cdot 100 \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} - \color{blue}{\frac{\frac{1}{2} \cdot 1}{n}}, i \cdot n, n\right) \cdot 100 \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2} - \frac{\color{blue}{\frac{1}{2}}}{n}, i \cdot n, n\right) \cdot 100 \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2} - \color{blue}{\frac{\frac{1}{2}}{n}}, i \cdot n, n\right) \cdot 100 \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2} - \frac{\frac{1}{2}}{n}, \color{blue}{n \cdot i}, n\right) \cdot 100 \]

   13. *-lowering-*.f64 55.6

       \[\leadsto \mathsf{fma}\left(0.5 - \frac{0.5}{n}, \color{blue}{n \cdot i}, n\right) \cdot 100 \]

5. Simplified 55.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - \frac{0.5}{n}, n \cdot i, n\right) \cdot 100} \]

6. Taylor expanded in n around 0

   \[\leadsto \color{blue}{-50 \cdot i} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{i \cdot -50} \]

   2. *-lowering-*.f64 2.6

      \[\leadsto \color{blue}{i \cdot -50} \]

8. Simplified 2.6%

   \[\leadsto \color{blue}{i \cdot -50} \]
9. Add Preprocessing

Developer Target 1: 34.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024199 
(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))))