Compound Interest

Percentage Accurate: 28.9% → 94.6%

Time: 11.4s

Alternatives: 15

Speedup: 24.3×

• could not determine a ground truth

  1. i = 2.8989419017426785e-104
  2. n = -1.0968964818277218e+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: 28.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))) (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)
       (/ (* (expm1 (* (/ n 2.0) (log (pow t_0 2.0)))) 100.0) (/ i n))
       (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	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 = (expm1(((n / 2.0) * log(pow(t_0, 2.0)))) * 100.0) / (i / n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double t_1 = (Math.pow(t_0, n) - 1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.expm1(((n / 2.0) * Math.log(Math.pow(t_0, 2.0)))) * 100.0) / (i / n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	t_1 = (math.pow(t_0, n) - 1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.expm1((math.log1p((i / n)) * n)) * 100.0) / (i / n)
	elif t_1 <= math.inf:
		tmp = (math.expm1(((n / 2.0) * math.log(math.pow(t_0, 2.0)))) * 100.0) / (i / n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	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(Float64(expm1(Float64(Float64(n / 2.0) * log((t_0 ^ 2.0)))) * 100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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[(Exp[N[(N[(n / 2.0), $MachinePrecision] * N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $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 27.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.3

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 98.0

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

   4. Applied rewrites 98.0%

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

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

   1. Initial program 97.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.5

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 65.1

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log1p.f64 N/A

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

      4. log-pow-rev N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. log-pow N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   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 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 2: 94.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision] * n + N[(N[(-1.0 / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 27.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.3

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 98.0

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

   4. Applied rewrites 98.0%

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

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

   1. Initial program 97.5%

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

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

   if +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 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 3: 93.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision] * n + N[(N[(-1.0 / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 27.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.3

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 97.2

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

   4. Applied rewrites 97.2%

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

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

   1. Initial program 97.5%

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

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

   if +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 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 4: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \frac{\mathsf{fma}\left(\log n, -1, \log i\right)}{i}\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5e-311)
     (* (* t_0 100.0) n)
     (if (<= n 2.55e-142)
       (* 100.0 (* (* n (/ (fma (log n) -1.0 (log i)) i)) n))
       (* 100.0 (* t_0 n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5e-311) {
		tmp = (t_0 * 100.0) * n;
	} else if (n <= 2.55e-142) {
		tmp = 100.0 * ((n * (fma(log(n), -1.0, log(i)) / i)) * n);
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5e-311)
		tmp = Float64(Float64(t_0 * 100.0) * n);
	elseif (n <= 2.55e-142)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(fma(log(n), -1.0, log(i)) / i)) * n));
	else
		tmp = Float64(100.0 * Float64(t_0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5e-311], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.55e-142], N[(100.0 * N[(N[(n * N[(N[(N[Log[n], $MachinePrecision] * -1.0 + N[Log[i], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.00000000000023e-311

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   if -5.00000000000023e-311 < n < 2.55e-142

   1. Initial program 32.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 33.4

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 2.55e-142 < n

   1. Initial program 22.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-expm1.f64 85.5

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

   5. Applied rewrites 85.5%

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

Alternative 5: 80.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-134} \lor \neg \left(n \leq 4.4 \cdot 10^{-181}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.75e-134) (not (<= n 4.4e-181)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.75e-134) || !(n <= 4.4e-181)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.75e-134) || !(n <= 4.4e-181)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.75e-134) or not (n <= 4.4e-181):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.75e-134) || !(n <= 4.4e-181))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.75e-134], N[Not[LessEqual[n, 4.4e-181]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.7499999999999999e-134 or 4.39999999999999994e-181 < n

   1. Initial program 23.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

   if -1.7499999999999999e-134 < n < 4.39999999999999994e-181

   1. Initial program 48.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-134} \lor \neg \left(n \leq 4.4 \cdot 10^{-181}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-134} \lor \neg \left(n \leq 4.4 \cdot 10^{-181}\right):\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.8e-134) (not (<= n 4.4e-181)))
   (*
    100.0
    (*
     (/
      (*
       (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
       i)
      i)
     n))
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.8e-134) || !(n <= 4.4e-181)) {
		tmp = 100.0 * (((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * n);
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.8e-134) || !(n <= 4.4e-181))
		tmp = Float64(100.0 * Float64(Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * n));
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.8e-134], N[Not[LessEqual[n, 4.4e-181]], $MachinePrecision]], N[(100.0 * N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.7999999999999999e-134 or 4.39999999999999994e-181 < n

   1. Initial program 23.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 68.0

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

   5. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 83.5

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

   7. Applied rewrites 83.5%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 65.3%

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

   if -2.7999999999999999e-134 < n < 4.39999999999999994e-181

   1. Initial program 48.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.8%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-134} \lor \neg \left(n \leq 4.4 \cdot 10^{-181}\right):\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-134}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-182}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(100 \cdot n\right) \cdot \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e-134)
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (if (<= n 1.9e-182)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (if (<= n 5.6e-24)
       (/ (* 100.0 i) (/ i n))
       (/ (* (* 100.0 n) (* (fma 0.5 i 1.0) i)) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e-134) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else if (n <= 1.9e-182) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 5.6e-24) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = ((100.0 * n) * (fma(0.5, i, 1.0) * i)) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e-134)
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	elseif (n <= 1.9e-182)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	elseif (n <= 5.6e-24)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = Float64(Float64(Float64(100.0 * n) * Float64(fma(0.5, i, 1.0) * i)) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e-134], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.9e-182], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.6e-24], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(100.0 * n), $MachinePrecision] * N[(N[(0.5 * i + 1.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.7999999999999999e-134

   1. Initial program 24.7%

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

   3. Taylor expanded in 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 55.4%

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

   if -2.7999999999999999e-134 < n < 1.9000000000000002e-182

   1. Initial program 48.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.8%

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

   if 1.9000000000000002e-182 < n < 5.6000000000000003e-24

   1. Initial program 17.1%

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

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

      6. lower-*.f64 17.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 92.7

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 60.5

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

   7. Applied rewrites 60.5%

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

   if 5.6000000000000003e-24 < n

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 74.2

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

   5. Applied rewrites 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 94.3

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

   7. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 93.1

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

   9. Applied rewrites 93.1%

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

   10. Taylor expanded in i around 0

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

   12. Applied rewrites 80.5%

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

Alternative 8: 64.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-134}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-181}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e-134)
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (if (<= n 4.4e-181)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e-134) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else if (n <= 4.4e-181) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e-134)
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	elseif (n <= 4.4e-181)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e-134], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.4e-181], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.7999999999999999e-134

   1. Initial program 24.7%

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

   3. Taylor expanded in 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 55.4%

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

   if -2.7999999999999999e-134 < n < 4.39999999999999994e-181

   1. Initial program 48.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.8%

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

   if 4.39999999999999994e-181 < n

   1. Initial program 23.0%

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

   3. Taylor expanded in 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.3%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{+46}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-31}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.5e+46)
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (if (<= n 1.75e-31)
     (/ (* 100.0 i) (/ i n))
     (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.5e+46) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else if (n <= 1.75e-31) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.5e+46)
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	elseif (n <= 1.75e-31)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.5e+46], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-31], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.50000000000000008e46

   1. Initial program 26.6%

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

   3. Taylor expanded in 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.1%

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

   if -6.50000000000000008e46 < n < 1.74999999999999993e-31

   1. Initial program 32.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if 1.74999999999999993e-31 < n

   1. Initial program 25.5%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 77.3%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n)))

C

double code(double i, double n) {
	return 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
}

Julia

function code(i, n)
	return Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 48.8%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 54.4%

   \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]
9. Add Preprocessing

Alternative 11: 56.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n)))

C

double code(double i, double n) {
	return 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
}

Julia

function code(i, n)
	return Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 48.8%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 54.4%

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

Alternative 12: 54.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot i\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.0) (* 100.0 n) (* 100.0 (* (* n i) 0.5))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.0) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((n * i) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 2.0d0) then
        tmp = 100.0d0 * n
    else
        tmp = 100.0d0 * ((n * i) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.0) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((n * i) * 0.5);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.0:
		tmp = 100.0 * n
	else:
		tmp = 100.0 * ((n * i) * 0.5)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.0)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(Float64(n * i) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 2.0)
		tmp = 100.0 * n;
	else
		tmp = 100.0 * ((n * i) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.0], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(N[(n * i), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2

   1. Initial program 23.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 2 < i

   1. Initial program 46.2%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 32.0%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 32.0%

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

   12. Taylor expanded in i around 0

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

   14. Applied rewrites 26.5%

       \[\leadsto 100 \cdot \left(\left(n \cdot i\right) \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (* (fma (* 0.16666666666666666 i) i 1.0) n)))

C

double code(double i, double n) {
	return 100.0 * (fma((0.16666666666666666 * i), i, 1.0) * n);
}

Julia

function code(i, n)
	return Float64(100.0 * Float64(fma(Float64(0.16666666666666666 * i), i, 1.0) * n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[(0.16666666666666666 * i), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 48.8%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 54.4%

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

9. Taylor expanded in i around inf

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

11. Applied rewrites 53.8%

    \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \]
12. Add Preprocessing

Alternative 14: 54.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 (fma (* n i) 0.5 n)))

C

double code(double i, double n) {
	return 100.0 * fma((n * i), 0.5, n);
}

Julia

function code(i, n)
	return Float64(100.0 * fma(Float64(n * i), 0.5, n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(n * i), $MachinePrecision] * 0.5 + n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.9%

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 48.8%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 54.4%

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

9. Taylor expanded in i around 0

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

11. Applied rewrites 53.0%

    \[\leadsto 100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right) \]
12. Add Preprocessing

Alternative 15: 48.6% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.9%

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

3. Taylor expanded in i around 0

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

   1. lower-*.f64 47.5

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

5. Applied rewrites 47.5%

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

Developer Target 1: 35.5% 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 2024326 
(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))))