Compound Interest

Percentage Accurate: 29.1% → 94.1%

Time: 12.9s

Alternatives: 16

Speedup: 8.1×

• could not determine a ground truth

  1. i = 4.936436874231483e-204
  2. n = -6.250538672199037e+36

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: 29.1% 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: 94.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 20.6

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 98.4

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

   4. Applied rewrites 98.4%

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

   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 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. remove-double-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. metadata-eval N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. times-frac N/A

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

      16. lower-fma.f64 N/A

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

   6. Applied rewrites 98.4%

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

   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. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 93.6%

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

Alternative 2: 93.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 20.6

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 96.5%

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

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

   1. Initial program 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. remove-double-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. metadata-eval N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. times-frac N/A

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

      16. lower-fma.f64 N/A

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

   6. Applied rewrites 98.4%

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

   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. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 92.3%

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

Alternative 3: 93.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

   if -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 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. remove-double-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. metadata-eval N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. times-frac N/A

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

      16. lower-fma.f64 N/A

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

   6. Applied rewrites 98.4%

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

   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. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 92.2%

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

Alternative 4: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{-204}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.00089:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -2.9e-204)
     t_0
     (if (<= n 9.5e-246)
       0.0
       (if (<= n 0.00089) (/ (* 100.0 i) (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -2.9e-204) {
		tmp = t_0;
	} else if (n <= 9.5e-246) {
		tmp = 0.0;
	} else if (n <= 0.00089) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -2.9e-204) {
		tmp = t_0;
	} else if (n <= 9.5e-246) {
		tmp = 0.0;
	} else if (n <= 0.00089) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -2.9e-204:
		tmp = t_0
	elif n <= 9.5e-246:
		tmp = 0.0
	elif n <= 0.00089:
		tmp = (100.0 * i) / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -2.9e-204)
		tmp = t_0;
	elseif (n <= 9.5e-246)
		tmp = 0.0;
	elseif (n <= 0.00089)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.9e-204], t$95$0, If[LessEqual[n, 9.5e-246], 0.0, If[LessEqual[n, 0.00089], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.90000000000000009e-204 or 8.8999999999999995e-4 < n

   1. Initial program 19.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if -2.90000000000000009e-204 < n < 9.5000000000000002e-246

   1. Initial program 69.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 13.4

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 87.8%

       \[\leadsto 0 \]

   if 9.5000000000000002e-246 < n < 8.8999999999999995e-4

   1. Initial program 15.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 15.9

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 66.0

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

   7. Applied rewrites 66.0%

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

Alternative 5: 66.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right) \cdot n, i, 0.5 \cdot n\right), i, n\right) \cdot i}{i} \cdot 100\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.00089:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (/
           (*
            (fma
             (fma
              (* (fma 0.041666666666666664 i 0.16666666666666666) n)
              i
              (* 0.5 n))
             i
             n)
            i)
           i)
          100.0)))
   (if (<= n -1.15e-89)
     t_0
     (if (<= n 9.5e-246)
       0.0
       (if (<= n 0.00089) (/ (* 100.0 i) (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = ((fma(fma((fma(0.041666666666666664, i, 0.16666666666666666) * n), i, (0.5 * n)), i, n) * i) / i) * 100.0;
	double tmp;
	if (n <= -1.15e-89) {
		tmp = t_0;
	} else if (n <= 9.5e-246) {
		tmp = 0.0;
	} else if (n <= 0.00089) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(fma(fma(Float64(fma(0.041666666666666664, i, 0.16666666666666666) * n), i, Float64(0.5 * n)), i, n) * i) / i) * 100.0)
	tmp = 0.0
	if (n <= -1.15e-89)
		tmp = t_0;
	elseif (n <= 9.5e-246)
		tmp = 0.0;
	elseif (n <= 0.00089)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * n), $MachinePrecision] * i + N[(0.5 * n), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -1.15e-89], t$95$0, If[LessEqual[n, 9.5e-246], 0.0, If[LessEqual[n, 0.00089], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.15e-89 or 8.8999999999999995e-4 < n

   1. Initial program 19.4%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 72.0%

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

   if -1.15e-89 < n < 9.5000000000000002e-246

   1. Initial program 54.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 15.4

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

   4. Applied rewrites 15.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 65.6

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

   7. Applied rewrites 65.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 65.6%

       \[\leadsto 0 \]

   if 9.5000000000000002e-246 < n < 8.8999999999999995e-4

   1. Initial program 15.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 15.9

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 66.0

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

   7. Applied rewrites 66.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right) \cdot n, i, 0.5 \cdot n\right), i, n\right) \cdot i}{i} \cdot 100\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.00089:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right) \cdot n, i, 0.5 \cdot n\right), i, n\right) \cdot i}{i} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.041666666666666664, i, n\right) \cdot i}{i} \cdot 100\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1750:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot i}{i} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.15e-89)
   (* (/ (* (fma (* (* (* i i) n) 0.041666666666666664) i n) i) i) 100.0)
   (if (<= n 9.5e-246)
     0.0
     (if (<= n 1750.0)
       (/ (* 100.0 i) (/ i n))
       (*
        (/
         (*
          (fma
           (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
           i
           100.0)
          i)
         i)
        n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.15e-89) {
		tmp = ((fma((((i * i) * n) * 0.041666666666666664), i, n) * i) / i) * 100.0;
	} else if (n <= 9.5e-246) {
		tmp = 0.0;
	} else if (n <= 1750.0) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = ((fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * i) / i) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.15e-89)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(i * i) * n) * 0.041666666666666664), i, n) * i) / i) * 100.0);
	elseif (n <= 9.5e-246)
		tmp = 0.0;
	elseif (n <= 1750.0)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * i) / i) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.15e-89], N[(N[(N[(N[(N[(N[(N[(i * i), $MachinePrecision] * n), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * i + n), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 9.5e-246], 0.0, If[LessEqual[n, 1750.0], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.15e-89

   1. Initial program 20.0%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 64.2%

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

   9. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \left({i}^{2} \cdot n\right), i, n\right) \cdot i}{i} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.2%

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

   if -1.15e-89 < n < 9.5000000000000002e-246

   1. Initial program 54.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 15.4

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

   4. Applied rewrites 15.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 65.6

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

   7. Applied rewrites 65.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 65.6%

       \[\leadsto 0 \]

   if 9.5000000000000002e-246 < n < 1750

   1. Initial program 15.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 15.0

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 77.8

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

   if 1750 < n

   1. Initial program 19.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 69.3

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 97.3

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

   7. Applied rewrites 97.3%

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

   9. Applied rewrites 97.2%

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

   10. Taylor expanded in i around 0

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

   12. Applied rewrites 80.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.041666666666666664, i, n\right) \cdot i}{i} \cdot 100\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1750:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot i}{i} \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right) \cdot n, i, n\right) \cdot 100\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1750:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot i}{i} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.5e-201)
   (*
    (fma
     (* (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) n)
     i
     n)
    100.0)
   (if (<= n 9.5e-246)
     0.0
     (if (<= n 1750.0)
       (/ (* 100.0 i) (/ i n))
       (*
        (/
         (*
          (fma
           (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
           i
           100.0)
          i)
         i)
        n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.5e-201) {
		tmp = fma((fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5) * n), i, n) * 100.0;
	} else if (n <= 9.5e-246) {
		tmp = 0.0;
	} else if (n <= 1750.0) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = ((fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * i) / i) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.5e-201)
		tmp = Float64(fma(Float64(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5) * n), i, n) * 100.0);
	elseif (n <= 9.5e-246)
		tmp = 0.0;
	elseif (n <= 1750.0)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * i) / i) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.5e-201], N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 9.5e-246], 0.0, If[LessEqual[n, 1750.0], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.49999999999999974e-201

   1. Initial program 20.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 57.1%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 59.0%

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

   if -6.49999999999999974e-201 < n < 9.5000000000000002e-246

   1. Initial program 69.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 13.4

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 87.8%

       \[\leadsto 0 \]

   if 9.5000000000000002e-246 < n < 1750

   1. Initial program 15.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 15.0

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 77.8

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

   if 1750 < n

   1. Initial program 19.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 69.3

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 97.3

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

   7. Applied rewrites 97.3%

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

   9. Applied rewrites 97.2%

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

   10. Taylor expanded in i around 0

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

   12. Applied rewrites 80.5%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right) \cdot n, i, n\right) \cdot 100\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 1750:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot i}{i} \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right) \cdot n, i, n\right) \cdot 100\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.00089:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (* (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) n)
           i
           n)
          100.0)))
   (if (<= n -6.5e-201)
     t_0
     (if (<= n 9.5e-246)
       0.0
       (if (<= n 0.00089) (/ (* 100.0 i) (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = fma((fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5) * n), i, n) * 100.0;
	double tmp;
	if (n <= -6.5e-201) {
		tmp = t_0;
	} else if (n <= 9.5e-246) {
		tmp = 0.0;
	} else if (n <= 0.00089) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(Float64(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5) * n), i, n) * 100.0)
	tmp = 0.0
	if (n <= -6.5e-201)
		tmp = t_0;
	elseif (n <= 9.5e-246)
		tmp = 0.0;
	elseif (n <= 0.00089)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -6.5e-201], t$95$0, If[LessEqual[n, 9.5e-246], 0.0, If[LessEqual[n, 0.00089], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.49999999999999974e-201 or 8.8999999999999995e-4 < n

   1. Initial program 19.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 67.4%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 68.1%

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

   if -6.49999999999999974e-201 < n < 9.5000000000000002e-246

   1. Initial program 69.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 13.4

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 87.8%

       \[\leadsto 0 \]

   if 9.5000000000000002e-246 < n < 8.8999999999999995e-4

   1. Initial program 15.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 15.9

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 66.0

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

   7. Applied rewrites 66.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right) \cdot n, i, n\right) \cdot 100\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.00089:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right) \cdot n, i, n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-246}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 0.00089:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
           i
           100.0)
          n)))
   (if (<= n -6.5e-201)
     t_0
     (if (<= n 9.5e-246)
       0.0
       (if (<= n 0.00089) (/ (* 100.0 i) (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -6.5e-201) {
		tmp = t_0;
	} else if (n <= 9.5e-246) {
		tmp = 0.0;
	} else if (n <= 0.00089) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -6.5e-201)
		tmp = t_0;
	elseif (n <= 9.5e-246)
		tmp = 0.0;
	elseif (n <= 0.00089)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -6.5e-201], t$95$0, If[LessEqual[n, 9.5e-246], 0.0, If[LessEqual[n, 0.00089], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.49999999999999974e-201 or 8.8999999999999995e-4 < n

   1. Initial program 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 71.6

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 87.9

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

   7. Applied rewrites 87.9%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 68.1%

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

   if -6.49999999999999974e-201 < n < 9.5000000000000002e-246

   1. Initial program 69.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 13.4

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 87.8%

       \[\leadsto 0 \]

   if 9.5000000000000002e-246 < n < 8.8999999999999995e-4

   1. Initial program 15.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 15.9

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 66.0

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

   7. Applied rewrites 66.0%

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

Alternative 10: 63.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
           i
           100.0)
          n)))
   (if (<= n -6.5e-201) t_0 (if (<= n 6.2e-90) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -6.5e-201) {
		tmp = t_0;
	} else if (n <= 6.2e-90) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -6.5e-201)
		tmp = t_0;
	elseif (n <= 6.2e-90)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -6.5e-201], t$95$0, If[LessEqual[n, 6.2e-90], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.49999999999999974e-201 or 6.2000000000000003e-90 < n

   1. Initial program 20.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 84.9

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 66.8%

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

   if -6.49999999999999974e-201 < n < 6.2000000000000003e-90

   1. Initial program 39.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 9.7

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

   4. Applied rewrites 9.7%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 67.5%

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

Alternative 11: 62.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.5e-201)
   (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
   (if (<= n 6.2e-90)
     0.0
     (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.5e-201) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= 6.2e-90) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.5e-201)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= 6.2e-90)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.5e-201], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 6.2e-90], 0.0, N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.49999999999999974e-201

   1. Initial program 20.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. Applied rewrites 55.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 58.7%

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

   if -6.49999999999999974e-201 < n < 6.2000000000000003e-90

   1. Initial program 39.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 9.7

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

   4. Applied rewrites 9.7%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 67.5%

       \[\leadsto 0 \]

   if 6.2000000000000003e-90 < n

   1. Initial program 19.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 89.2

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

   7. Applied rewrites 89.2%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 74.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -6.5e-201) t_0 (if (<= n 6.2e-90) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -6.5e-201) {
		tmp = t_0;
	} else if (n <= 6.2e-90) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -6.5e-201)
		tmp = t_0;
	elseif (n <= 6.2e-90)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -6.5e-201], t$95$0, If[LessEqual[n, 6.2e-90], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.49999999999999974e-201 or 6.2000000000000003e-90 < n

   1. Initial program 20.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 84.9

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 66.3%

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

   if -6.49999999999999974e-201 < n < 6.2000000000000003e-90

   1. Initial program 39.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 9.7

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

   4. Applied rewrites 9.7%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 67.5%

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

Alternative 13: 59.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.5e-201)
   (* (fma 50.0 i 100.0) n)
   (if (<= n 6.2e-90) 0.0 (* (* (fma 0.5 i 1.0) n) 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.5e-201) {
		tmp = fma(50.0, i, 100.0) * n;
	} else if (n <= 6.2e-90) {
		tmp = 0.0;
	} else {
		tmp = (fma(0.5, i, 1.0) * n) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.5e-201)
		tmp = Float64(fma(50.0, i, 100.0) * n);
	elseif (n <= 6.2e-90)
		tmp = 0.0;
	else
		tmp = Float64(Float64(fma(0.5, i, 1.0) * n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.5e-201], N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 6.2e-90], 0.0, N[(N[(N[(0.5 * i + 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.49999999999999974e-201

   1. Initial program 20.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 56.5%

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

   if -6.49999999999999974e-201 < n < 6.2000000000000003e-90

   1. Initial program 39.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      16. lower-neg.f64 9.7

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 9.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 67.5

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 67.5%

       \[\leadsto 0 \]

   if 6.2000000000000003e-90 < n

   1. Initial program 19.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. lower-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(e^{i} - 1\right) \cdot n}}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{\left(e^{i} - 1\right) \cdot n}}{i} \]

      4. lower-expm1.f64 88.2

         \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)} \cdot n}{i} \]

   5. Applied rewrites 88.2%

      \[\leadsto 100 \cdot \color{blue}{\frac{\mathsf{expm1}\left(i\right) \cdot n}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \left(n + \color{blue}{\frac{1}{2} \cdot \left(i \cdot n\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 50.0 i 100.0) n)))
   (if (<= n -6.5e-201) t_0 (if (<= n 6.2e-90) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = fma(50.0, i, 100.0) * n;
	double tmp;
	if (n <= -6.5e-201) {
		tmp = t_0;
	} else if (n <= 6.2e-90) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(50.0, i, 100.0) * n)
	tmp = 0.0
	if (n <= -6.5e-201)
		tmp = t_0;
	elseif (n <= 6.2e-90)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -6.5e-201], t$95$0, If[LessEqual[n, 6.2e-90], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.49999999999999974e-201 or 6.2000000000000003e-90 < n

   1. Initial program 20.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{i} \cdot n \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      10. lift--.f64 N/A

          \[\leadsto \left(\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      12. pow-to-exp N/A

          \[\leadsto \left(\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      13. lower-expm1.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot \frac{100}{i}\right) \cdot n \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{n}\right) \cdot n}\right) \cdot \frac{100}{i}\right) \cdot n \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + \frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      16. lower-log1p.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      17. lower-/.f64 73.6

          \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \color{blue}{\frac{100}{i}}\right) \cdot n \]

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n} \]

   5. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right)} \cdot n \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      4. lower-expm1.f64 84.9

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   7. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right)} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \left(100 + \color{blue}{50 \cdot i}\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 62.9%

       \[\leadsto \mathsf{fma}\left(50, \color{blue}{i}, 100\right) \cdot n \]

   if -6.49999999999999974e-201 < n < 6.2000000000000003e-90

   1. Initial program 39.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      11. lift-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      16. lower-neg.f64 9.7

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 9.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 67.5

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 67.5%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.16 \cdot 10^{-16}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 700:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.16e-16) 0.0 (if (<= i 700.0) (* 100.0 n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.16e-16) {
		tmp = 0.0;
	} else if (i <= 700.0) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.16d-16)) then
        tmp = 0.0d0
    else if (i <= 700.0d0) then
        tmp = 100.0d0 * n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.16e-16) {
		tmp = 0.0;
	} else if (i <= 700.0) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.16e-16:
		tmp = 0.0
	elif i <= 700.0:
		tmp = 100.0 * n
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.16e-16)
		tmp = 0.0;
	elseif (i <= 700.0)
		tmp = Float64(100.0 * n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.16e-16)
		tmp = 0.0;
	elseif (i <= 700.0)
		tmp = 100.0 * n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.16e-16], 0.0, If[LessEqual[i, 700.0], N[(100.0 * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.1600000000000001e-16 or 700 < i

   1. Initial program 42.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      11. lift-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      16. lower-neg.f64 38.0

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 38.0%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 25.4

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.4%

       \[\leadsto 0 \]

   if -1.1600000000000001e-16 < i < 700

   1. Initial program 8.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. lower-*.f64 84.9

         \[\leadsto \color{blue}{100 \cdot n} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 18.3% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 23.5%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

   2. lift--.f64 N/A

      \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

   3. div-sub N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

   5. clear-num N/A

      \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

   6. sub-neg N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

   8. associate-/r/ N/A

      \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

   10. lower-/.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

   11. lift-+.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

   16. lower-neg.f64 19.1

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

4. Applied rewrites 19.1%

   \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

5. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

   2. distribute-rgt1-in N/A

      \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

   3. metadata-eval N/A

      \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

   4. mul0-lft N/A

      \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{i} \]

   6. lower-/.f64 16.4

      \[\leadsto \color{blue}{\frac{0}{i}} \]

7. Applied rewrites 16.4%

   \[\leadsto \color{blue}{\frac{0}{i}} \]

8. Taylor expanded in i around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 16.4%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 34.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024327 
(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))))