Compound Interest

Percentage Accurate: 28.9% → 96.0%

Time: 13.1s

Alternatives: 15

Speedup: 24.3×

• could not determine a ground truth

  1. i = 1.0540360963436131e-83
  2. n = -4.4542706690541265e+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: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \mathsf{fma}\left(t\_0, 100, -100\right)\\ t_2 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-13}:\\ \;\;\;\;n \cdot \frac{t\_1}{i}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n))
        (t_1 (fma t_0 100.0 -100.0))
        (t_2 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_2 -1e-13)
     (* n (/ t_1 i))
     (if (<= t_2 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_2 INFINITY) (* t_1 (/ n i)) (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = fma(t_0, 100.0, -100.0);
	double t_2 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_2 <= -1e-13) {
		tmp = n * (t_1 / i);
	} else if (t_2 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1 * (n / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = fma(t_0, 100.0, -100.0)
	t_2 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_2 <= -1e-13)
		tmp = Float64(n * Float64(t_1 / i));
	elseif (t_2 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_2 <= Inf)
		tmp = Float64(t_1 * Float64(n / i));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 100.0 + -100.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-13], N[(n * N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$1 * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e-13

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   if -1e-13 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sqr-pow N/A

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

      4. sqr-pow N/A

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

      5. pow-to-exp N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-log1p.f64 99.8

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

   4. Applied rewrites 99.8%

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

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

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 98.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \left(100 \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-147}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 100\right)}{n}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \left(\left(\log i - \log n\right) \cdot \frac{n \cdot n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2.7e-219)
     (* n (* 100.0 t_0))
     (if (<= n 2.9e-147)
       (/ (* n (* n 100.0)) n)
       (if (<= n 1.05e-33)
         (* 100.0 (* (- (log i) (log n)) (/ (* n n) i)))
         (* (* n 100.0) t_0))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2.7e-219) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 2.9e-147) {
		tmp = (n * (n * 100.0)) / n;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((log(i) - log(n)) * ((n * n) / i));
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2.7e-219) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 2.9e-147) {
		tmp = (n * (n * 100.0)) / n;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((Math.log(i) - Math.log(n)) * ((n * n) / i));
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2.7e-219:
		tmp = n * (100.0 * t_0)
	elif n <= 2.9e-147:
		tmp = (n * (n * 100.0)) / n
	elif n <= 1.05e-33:
		tmp = 100.0 * ((math.log(i) - math.log(n)) * ((n * n) / i))
	else:
		tmp = (n * 100.0) * t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2.7e-219)
		tmp = Float64(n * Float64(100.0 * t_0));
	elseif (n <= 2.9e-147)
		tmp = Float64(Float64(n * Float64(n * 100.0)) / n);
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(log(i) - log(n)) * Float64(Float64(n * n) / i)));
	else
		tmp = Float64(Float64(n * 100.0) * t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.7e-219], N[(n * N[(100.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.9e-147], N[(N[(n * N[(n * 100.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.7e-219

   1. Initial program 26.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 60.7

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

   5. Applied rewrites 60.7%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 78.9

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

   7. Applied rewrites 78.9%

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

   if -2.7e-219 < n < 2.9000000000000001e-147

   1. Initial program 45.2%

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

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 12.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(n, 100 \cdot \mathsf{fma}\left(n, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), 1\right), i \cdot \mathsf{fma}\left(i, -0.5, -0.5\right)\right), \left(i \cdot i\right) \cdot 33.333333333333336\right)}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto \frac{\color{blue}{100 \cdot {n}^{2}}}{n} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 100}}{n} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 68.7

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

   11. Applied rewrites 68.7%

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

   if 2.9000000000000001e-147 < n < 1.05e-33

   1. Initial program 14.9%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

   if 1.05e-33 < n

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-147}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 100\right)}{n}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \left(\left(\log i - \log n\right) \cdot \frac{n \cdot n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;n \cdot \left(100 \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2e-310)
     (* n (* 100.0 t_0))
     (if (<= n 1.05e-33)
       (* 100.0 (/ (* n (- (log i) (log n))) (/ i n)))
       (* (* n 100.0) t_0)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2e-310) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((n * (log(i) - log(n))) / (i / n));
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2e-310) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((n * (Math.log(i) - Math.log(n))) / (i / n));
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2e-310:
		tmp = n * (100.0 * t_0)
	elif n <= 1.05e-33:
		tmp = 100.0 * ((n * (math.log(i) - math.log(n))) / (i / n))
	else:
		tmp = (n * 100.0) * t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2e-310)
		tmp = Float64(n * Float64(100.0 * t_0));
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(log(i) - log(n))) / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2e-310], N[(n * N[(100.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.999999999999994e-310

   1. Initial program 30.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 59.0

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

   5. Applied rewrites 59.0%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.4

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

   7. Applied rewrites 75.4%

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

   if -1.999999999999994e-310 < n < 1.05e-33

   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 n around 0

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if 1.05e-33 < n

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -4.25 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -4.25e-35)
     t_0
     (if (<= n -2.7e-219)
       (* (expm1 i) (* 100.0 (/ n i)))
       (if (<= n 1.05e-33) (* 100.0 (/ (+ 1.0 -1.0) (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -4.25e-35) {
		tmp = t_0;
	} else if (n <= -2.7e-219) {
		tmp = expm1(i) * (100.0 * (n / i));
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -4.25e-35) {
		tmp = t_0;
	} else if (n <= -2.7e-219) {
		tmp = Math.expm1(i) * (100.0 * (n / i));
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -4.25e-35:
		tmp = t_0
	elif n <= -2.7e-219:
		tmp = math.expm1(i) * (100.0 * (n / i))
	elif n <= 1.05e-33:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -4.25e-35)
		tmp = t_0;
	elseif (n <= -2.7e-219)
		tmp = Float64(expm1(i) * Float64(100.0 * Float64(n / i)));
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.25e-35], t$95$0, If[LessEqual[n, -2.7e-219], N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.2500000000000001e-35 or 1.05e-33 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if -4.2500000000000001e-35 < n < -2.7e-219

   1. Initial program 32.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 63.2

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

   5. Applied rewrites 63.2%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 59.8

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

   7. Applied rewrites 59.8%

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

   if -2.7e-219 < n < 1.05e-33

   1. Initial program 36.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 62.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.25 \cdot 10^{-35}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;\mathsf{expm1}\left(i\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \left(100 \cdot t\_0\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2.7e-219)
     (* n (* 100.0 t_0))
     (if (<= n 1.05e-33)
       (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
       (* (* n 100.0) t_0)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2.7e-219) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2.7e-219) {
		tmp = n * (100.0 * t_0);
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = (n * 100.0) * t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2.7e-219:
		tmp = n * (100.0 * t_0)
	elif n <= 1.05e-33:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	else:
		tmp = (n * 100.0) * t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2.7e-219)
		tmp = Float64(n * Float64(100.0 * t_0));
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.7e-219], N[(n * N[(100.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.7e-219

   1. Initial program 26.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 60.7

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

   5. Applied rewrites 60.7%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 78.9

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

   7. Applied rewrites 78.9%

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

   if -2.7e-219 < n < 1.05e-33

   1. Initial program 36.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 62.7%

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

   if 1.05e-33 < n

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -2.7e-219)
     t_0
     (if (<= n 1.05e-33) (* 100.0 (/ (+ 1.0 -1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -2.7e-219) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -2.7e-219) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -2.7e-219:
		tmp = t_0
	elif n <= 1.05e-33:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -2.7e-219)
		tmp = t_0;
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.7e-219], t$95$0, If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.7e-219 or 1.05e-33 < n

   1. Initial program 24.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 64.8

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

   5. Applied rewrites 64.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   if -2.7e-219 < n < 1.05e-33

   1. Initial program 36.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 62.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-219}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -2.9e-131)
     t_0
     (if (<= n 1.05e-33) (* 100.0 (/ (+ 1.0 -1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -2.9e-131) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -2.9e-131) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -2.9e-131:
		tmp = t_0
	elif n <= 1.05e-33:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -2.9e-131)
		tmp = t_0;
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.9e-131], t$95$0, If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.9000000000000002e-131 or 1.05e-33 < n

   1. Initial program 22.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -2.9000000000000002e-131 < n < 1.05e-33

   1. Initial program 40.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 61.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-131}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-157}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.2e-157)
   (*
    n
    (fma i (fma i (fma i 4.166666666666667 16.666666666666668) 50.0) 100.0))
   (if (<= n 1.05e-33)
     (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
     (*
      (* n 100.0)
      (/
       (*
        i
        (fma
         i
         (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5)
         1.0))
       i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.2e-157) {
		tmp = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = (n * 100.0) * ((i * fma(i, fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0)) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.2e-157)
		tmp = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0));
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(Float64(i * fma(i, fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.2e-157], N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(i * N[(i * N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.1999999999999996e-157

   1. Initial program 23.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 60.0

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

   8. Applied rewrites 60.0%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-in N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. distribute-rgt-in N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 N/A

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

   11. Applied rewrites 60.8%

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

   if -6.1999999999999996e-157 < n < 1.05e-33

   1. Initial program 40.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 62.9%

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

   if 1.05e-33 < n

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 85.6

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

   10. Applied rewrites 85.6%

       \[\leadsto \frac{\color{blue}{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-157}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (fma
           i
           (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
           100.0))))
   (if (<= n -6.2e-157)
     t_0
     (if (<= n 1.05e-33) (* 100.0 (/ (+ 1.0 -1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	double tmp;
	if (n <= -6.2e-157) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0))
	tmp = 0.0
	if (n <= -6.2e-157)
		tmp = t_0;
	elseif (n <= 1.05e-33)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-157], t$95$0, If[LessEqual[n, 1.05e-33], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.1999999999999996e-157 or 1.05e-33 < n

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.6

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-in N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. distribute-rgt-in N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 N/A

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

   11. Applied rewrites 70.1%

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

   if -6.1999999999999996e-157 < n < 1.05e-33

   1. Initial program 40.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 62.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-157}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (fma
           i
           (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
           100.0))))
   (if (<= n -6.2e-157) t_0 (if (<= n 1.05e-33) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	double tmp;
	if (n <= -6.2e-157) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0))
	tmp = 0.0
	if (n <= -6.2e-157)
		tmp = t_0;
	elseif (n <= 1.05e-33)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-157], t$95$0, If[LessEqual[n, 1.05e-33], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.1999999999999996e-157 or 1.05e-33 < n

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.6

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-in N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. distribute-rgt-in N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 N/A

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

   11. Applied rewrites 70.1%

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

   if -6.1999999999999996e-157 < n < 1.05e-33

   1. Initial program 40.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-frac-neg2 N/A

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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 62.9

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

   7. Applied rewrites 62.9%

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

Alternative 11: 60.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-157}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.2e-157)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 1.05e-33) (/ 0.0 i) (* n (fma 50.0 i 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.2e-157) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 1.05e-33) {
		tmp = 0.0 / i;
	} else {
		tmp = n * fma(50.0, i, 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.2e-157)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 1.05e-33)
		tmp = Float64(0.0 / i);
	else
		tmp = Float64(n * fma(50.0, i, 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.2e-157], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-33], N[(0.0 / i), $MachinePrecision], N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.1999999999999996e-157

   1. Initial program 23.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 58.8

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

   5. Applied rewrites 58.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 78.6

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

   7. Applied rewrites 78.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.6

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

   9. Applied rewrites 78.6%

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

   10. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 60.4

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

   12. Applied rewrites 60.4%

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

   if -6.1999999999999996e-157 < n < 1.05e-33

   1. Initial program 40.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-frac-neg2 N/A

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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if 1.05e-33 < n

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in i around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 80.2

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

   8. Applied rewrites 80.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-157}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma 50.0 i 100.0))))
   (if (<= n -6.2e-157) t_0 (if (<= n 1.05e-33) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(50.0, i, 100.0);
	double tmp;
	if (n <= -6.2e-157) {
		tmp = t_0;
	} else if (n <= 1.05e-33) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(50.0, i, 100.0))
	tmp = 0.0
	if (n <= -6.2e-157)
		tmp = t_0;
	elseif (n <= 1.05e-33)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-157], t$95$0, If[LessEqual[n, 1.05e-33], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.1999999999999996e-157 or 1.05e-33 < n

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in i around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 65.4

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

   8. Applied rewrites 65.4%

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

   if -6.1999999999999996e-157 < n < 1.05e-33

   1. Initial program 40.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift--.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. div-sub N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. distribute-lft-neg-in N/A

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

      18. distribute-frac-neg2 N/A

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

   4. Applied rewrites 13.0%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 62.9

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

   7. Applied rewrites 62.9%

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

Alternative 13: 54.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 9.5 \cdot 10^{+25}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 9.5e+25) (* n 100.0) (* n (* i 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 9.5e+25) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 9.5d+25) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 9.5e+25) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 9.5e+25:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 9.5e+25)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 9.5e+25)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 9.5e+25], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 9.5000000000000005e25

   1. Initial program 23.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if 9.5000000000000005e25 < i

   1. Initial program 42.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 44.9

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in i around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 21.5

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in i around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot 50} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(n \cdot i\right)} \cdot 50 \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{n \cdot \left(i \cdot 50\right)} \]

       4. *-commutative N/A

          \[\leadsto n \cdot \color{blue}{\left(50 \cdot i\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{n \cdot \left(50 \cdot i\right)} \]

       6. *-commutative N/A

          \[\leadsto n \cdot \color{blue}{\left(i \cdot 50\right)} \]

       7. lower-*.f64 21.5

          \[\leadsto n \cdot \color{blue}{\left(i \cdot 50\right)} \]

   11. Applied rewrites 21.5%

       \[\leadsto \color{blue}{n \cdot \left(i \cdot 50\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 54.3% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.2%

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

3. Taylor expanded in n around inf

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

   1. lower-expm1.f64 57.4

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

5. Applied rewrites 57.4%

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

6. Taylor expanded in i around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 56.1

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

8. Applied rewrites 56.1%

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

Alternative 15: 48.7% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.2%

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

3. Taylor expanded in i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 51.3

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

5. Applied rewrites 51.3%

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

Developer Target 1: 35.2% 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 2024214 
(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))))