Compound Interest

Percentage Accurate: 28.4% → 95.7%

Time: 13.9s

Alternatives: 17

Speedup: 24.3×

• could not determine a ground truth

  1. i = 5.817224152043745e-249
  2. n = -2.8018565224256934e+39

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.4% 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: 95.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \mathsf{fma}\left(t\_0, 100, -100\right)\\ t_2 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-75}:\\ \;\;\;\;n \cdot \frac{t\_1}{i}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{n \cdot t\_1}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

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-75)
     (* n (/ t_1 i))
     (if (<= t_2 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_2 INFINITY) (/ (* n t_1) 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-75) {
		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 = (n * t_1) / 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-75)
		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(Float64(n * t_1) / 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-75], 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[(N[(n * t$95$1), $MachinePrecision] / i), $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)) < -9.9999999999999996e-76

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      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 99.9%

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

   if -9.9999999999999996e-76 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 21.0%

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

      1. lift-/.f64 N/A

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

      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 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. 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. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   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 72.0

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

   5. Applied rewrites 72.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-75}:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{n \cdot \mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;n \cdot \left(i \cdot \left(\left(i \cdot i\right) \cdot 4.166666666666667\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) (- INFINITY))
   (* n (* i (* (* i i) 4.166666666666667)))
   (fma i (* n (fma 16.666666666666668 i 50.0)) (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= -((double) INFINITY)) {
		tmp = n * (i * ((i * i) * 4.166666666666667));
	} else {
		tmp = fma(i, (n * fma(16.666666666666668, i, 50.0)), (n * 100.0));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n)) <= Float64(-Inf))
		tmp = Float64(n * Float64(i * Float64(Float64(i * i) * 4.166666666666667)));
	else
		tmp = fma(i, Float64(n * fma(16.666666666666668, i, 50.0)), Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(n * N[(i * N[(N[(i * i), $MachinePrecision] * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 20.8

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

   5. Applied rewrites 20.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 20.8

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

   7. Applied rewrites 20.8%

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

   8. Taylor expanded in i around 0

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

      1. +-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. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in i around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

          \[\leadsto n \cdot \left(\frac{25}{6} \cdot \left(\color{blue}{{i}^{2}} \cdot i\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto n \cdot \color{blue}{\left(\left(\frac{25}{6} \cdot {i}^{2}\right) \cdot i\right)} \]

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. unpow2 N/A

           \[\leadsto n \cdot \left(i \cdot \left(\frac{25}{6} \cdot \color{blue}{{i}^{2}}\right)\right) \]

       13. lower-*.f64 N/A

           \[\leadsto n \cdot \left(i \cdot \color{blue}{\left(\frac{25}{6} \cdot {i}^{2}\right)}\right) \]

       14. unpow2 N/A

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

       15. lower-*.f64 100.0

           \[\leadsto n \cdot \left(i \cdot \left(4.166666666666667 \cdot \color{blue}{\left(i \cdot i\right)}\right)\right) \]

   13. Applied rewrites 100.0%

       \[\leadsto \color{blue}{n \cdot \left(i \cdot \left(4.166666666666667 \cdot \left(i \cdot i\right)\right)\right)} \]

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

   1. Initial program 26.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.4

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

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), n \cdot 100\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

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

Alternative 3: 58.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;n \cdot \left(i \cdot \left(\left(i \cdot i\right) \cdot 4.166666666666667\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) (- INFINITY))
   (* n (* i (* (* i i) 4.166666666666667)))
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= -((double) INFINITY)) {
		tmp = n * (i * ((i * i) * 4.166666666666667));
	} else {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n)) <= Float64(-Inf))
		tmp = Float64(n * Float64(i * Float64(Float64(i * i) * 4.166666666666667)));
	else
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(n * N[(i * N[(N[(i * i), $MachinePrecision] * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 20.8

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

   5. Applied rewrites 20.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 20.8

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

   7. Applied rewrites 20.8%

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

   8. Taylor expanded in i around 0

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

      1. +-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. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in i around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

          \[\leadsto n \cdot \left(\frac{25}{6} \cdot \left(\color{blue}{{i}^{2}} \cdot i\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto n \cdot \color{blue}{\left(\left(\frac{25}{6} \cdot {i}^{2}\right) \cdot i\right)} \]

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. unpow2 N/A

           \[\leadsto n \cdot \left(i \cdot \left(\frac{25}{6} \cdot \color{blue}{{i}^{2}}\right)\right) \]

       13. lower-*.f64 N/A

           \[\leadsto n \cdot \left(i \cdot \color{blue}{\left(\frac{25}{6} \cdot {i}^{2}\right)}\right) \]

       14. unpow2 N/A

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

       15. lower-*.f64 100.0

           \[\leadsto n \cdot \left(i \cdot \left(4.166666666666667 \cdot \color{blue}{\left(i \cdot i\right)}\right)\right) \]

   13. Applied rewrites 100.0%

       \[\leadsto \color{blue}{n \cdot \left(i \cdot \left(4.166666666666667 \cdot \left(i \cdot i\right)\right)\right)} \]

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

   1. Initial program 26.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 75.0

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

   5. Applied rewrites 75.0%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 80.3

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 58.4

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

   10. Applied rewrites 58.4%

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

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

Alternative 4: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -4.6 \cdot 10^{-209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-197}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{1 + -1}{i}\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -4.6e-209)
     t_0
     (if (<= n 6e-197)
       (* (* n 100.0) (/ (+ 1.0 -1.0) i))
       (if (<= n 8.6e-47) (/ (* i 100.0) (/ i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -4.6e-209) {
		tmp = t_0;
	} else if (n <= 6e-197) {
		tmp = (n * 100.0) * ((1.0 + -1.0) / i);
	} else if (n <= 8.6e-47) {
		tmp = (i * 100.0) / (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 <= -4.6e-209) {
		tmp = t_0;
	} else if (n <= 6e-197) {
		tmp = (n * 100.0) * ((1.0 + -1.0) / i);
	} else if (n <= 8.6e-47) {
		tmp = (i * 100.0) / (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 <= -4.6e-209:
		tmp = t_0
	elif n <= 6e-197:
		tmp = (n * 100.0) * ((1.0 + -1.0) / i)
	elif n <= 8.6e-47:
		tmp = (i * 100.0) / (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 <= -4.6e-209)
		tmp = t_0;
	elseif (n <= 6e-197)
		tmp = Float64(Float64(n * 100.0) * Float64(Float64(1.0 + -1.0) / i));
	elseif (n <= 8.6e-47)
		tmp = Float64(Float64(i * 100.0) / 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, -4.6e-209], t$95$0, If[LessEqual[n, 6e-197], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(1.0 + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.6e-47], N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.5999999999999999e-209 or 8.5999999999999995e-47 < n

   1. Initial program 26.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 83.2

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

   5. Applied rewrites 83.2%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 87.3

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

   7. Applied rewrites 87.3%

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

   if -4.5999999999999999e-209 < n < 6.00000000000000051e-197

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      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 69.2

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

   4. Applied rewrites 69.2%

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 92.8%

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

   if 6.00000000000000051e-197 < n < 8.5999999999999995e-47

   1. Initial program 3.8%

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

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.0

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

   7. Applied rewrites 74.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{-209}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-197}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{1 + -1}{i}\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.4 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -1.4e-105) t_0 (if (<= n 1.65e-37) (/ (* i 100.0) (/ i n)) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -1.4e-105) {
		tmp = t_0;
	} else if (n <= 1.65e-37) {
		tmp = (i * 100.0) / (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 <= -1.4e-105) {
		tmp = t_0;
	} else if (n <= 1.65e-37) {
		tmp = (i * 100.0) / (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 <= -1.4e-105:
		tmp = t_0
	elif n <= 1.65e-37:
		tmp = (i * 100.0) / (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 <= -1.4e-105)
		tmp = t_0;
	elseif (n <= 1.65e-37)
		tmp = Float64(Float64(i * 100.0) / 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, -1.4e-105], t$95$0, If[LessEqual[n, 1.65e-37], N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.4e-105 or 1.64999999999999991e-37 < n

   1. Initial program 26.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   if -1.4e-105 < n < 1.64999999999999991e-37

   1. Initial program 34.1%

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

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.2

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

   7. Applied rewrites 72.2%

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

Alternative 6: 67.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100 \cdot \left(n \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i\right)\right)}{i}\\ \mathbf{if}\;n \leq -2.4 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (/
          (*
           100.0
           (*
            n
            (fma
             (* i i)
             (fma i (fma i 0.041666666666666664 0.16666666666666666) 0.5)
             i)))
          i)))
   (if (<= n -2.4e+85) t_0 (if (<= n 1.65e-37) (/ (* i 100.0) (/ i n)) t_0))))

C

double code(double i, double n) {
	double t_0 = (100.0 * (n * fma((i * i), fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), i))) / i;
	double tmp;
	if (n <= -2.4e+85) {
		tmp = t_0;
	} else if (n <= 1.65e-37) {
		tmp = (i * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * Float64(n * fma(Float64(i * i), fma(i, fma(i, 0.041666666666666664, 0.16666666666666666), 0.5), i))) / i)
	tmp = 0.0
	if (n <= -2.4e+85)
		tmp = t_0;
	elseif (n <= 1.65e-37)
		tmp = Float64(Float64(i * 100.0) / Float64(i / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[(n * N[(N[(i * i), $MachinePrecision] * N[(i * N[(i * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.4e+85], t$95$0, If[LessEqual[n, 1.65e-37], N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.39999999999999997e85 or 1.64999999999999991e-37 < n

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 75.8

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

   8. Applied rewrites 75.8%

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

   if -2.39999999999999997e85 < n < 1.64999999999999991e-37

   1. Initial program 33.6%

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

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

   4. Applied rewrites 33.6%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.2

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

   7. Applied rewrites 65.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{+85}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i\right)\right)}{i}\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), i\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right)\\ \mathbf{if}\;n \leq -2.4 \cdot 10^{+85}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, t\_0, 100\right)\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, n \cdot t\_0, n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)))
   (if (<= n -2.4e+85)
     (* n (fma i t_0 100.0))
     (if (<= n 8.6e-47)
       (/ (* i 100.0) (/ i n))
       (fma i (* n t_0) (* n 100.0))))))

C

double code(double i, double n) {
	double t_0 = fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0);
	double tmp;
	if (n <= -2.4e+85) {
		tmp = n * fma(i, t_0, 100.0);
	} else if (n <= 8.6e-47) {
		tmp = (i * 100.0) / (i / n);
	} else {
		tmp = fma(i, (n * t_0), (n * 100.0));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0)
	tmp = 0.0
	if (n <= -2.4e+85)
		tmp = Float64(n * fma(i, t_0, 100.0));
	elseif (n <= 8.6e-47)
		tmp = Float64(Float64(i * 100.0) / Float64(i / n));
	else
		tmp = fma(i, Float64(n * t_0), Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision]}, If[LessEqual[n, -2.4e+85], N[(n * N[(i * t$95$0 + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.6e-47], N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(i * N[(n * t$95$0), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.39999999999999997e85

   1. Initial program 29.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 89.5

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

   5. Applied rewrites 89.5%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 61.7

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

   10. Applied rewrites 61.7%

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

   if -2.39999999999999997e85 < n < 8.5999999999999995e-47

   1. Initial program 34.1%

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

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

   4. Applied rewrites 34.2%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.5

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

   7. Applied rewrites 65.5%

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

   if 8.5999999999999995e-47 < n

   1. Initial program 20.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 92.1

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

   5. Applied rewrites 92.1%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 94.4

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in i around 0

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

      1. +-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. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 80.0

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

   10. Applied rewrites 80.0%

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

   11. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-in N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 80.0

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

   13. Applied rewrites 80.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.4 \cdot 10^{+85}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{if}\;n \leq -2.4 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2.4e+85) t_0 (if (<= n 8.6e-47) (/ (* i 100.0) (/ 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 <= -2.4e+85) {
		tmp = t_0;
	} else if (n <= 8.6e-47) {
		tmp = (i * 100.0) / (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 <= -2.4e+85)
		tmp = t_0;
	elseif (n <= 8.6e-47)
		tmp = Float64(Float64(i * 100.0) / 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, -2.4e+85], t$95$0, If[LessEqual[n, 8.6e-47], N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.39999999999999997e85 or 8.5999999999999995e-47 < n

   1. Initial program 24.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 91.1

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

   5. Applied rewrites 91.1%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 72.7

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

   10. Applied rewrites 72.7%

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

   if -2.39999999999999997e85 < n < 8.5999999999999995e-47

   1. Initial program 34.1%

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

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

   4. Applied rewrites 34.2%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.5

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

   7. Applied rewrites 65.5%

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

Alternative 9: 67.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{if}\;n \leq -8 \cdot 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-197}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{1 + -1}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -8e-169)
     t_0
     (if (<= n 8.5e-197) (* (* n 100.0) (/ (+ 1.0 -1.0) 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 <= -8e-169) {
		tmp = t_0;
	} else if (n <= 8.5e-197) {
		tmp = (n * 100.0) * ((1.0 + -1.0) / 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 <= -8e-169)
		tmp = t_0;
	elseif (n <= 8.5e-197)
		tmp = Float64(Float64(n * 100.0) * Float64(Float64(1.0 + -1.0) / 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, -8e-169], t$95$0, If[LessEqual[n, 8.5e-197], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(1.0 + -1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.00000000000000016e-169 or 8.5e-197 < n

   1. Initial program 23.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 78.7

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

   5. Applied rewrites 78.7%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 83.3

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 65.8

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

   10. Applied rewrites 65.8%

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

   if -8.00000000000000016e-169 < n < 8.5e-197

   1. Initial program 61.5%

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

      1. lift-/.f64 N/A

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

      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 75.7

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 61.8%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 85.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-169}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-197}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{1 + -1}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* n (fma i (fma i (fma i 4.166666666666667 16.666666666666668) 50.0) 100.0)))

C

double code(double i, double n) {
	return n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
}

Julia

function code(i, n)
	return Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0))
end

Wolfram

code[i_, n_] := N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in n around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-expm1.f64 73.9

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

5. Applied rewrites 73.9%

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

   1. lift-expm1.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. times-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 79.1

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

7. Applied rewrites 79.1%

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

8. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 59.2

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

10. Applied rewrites 59.2%

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

Alternative 11: 57.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* n (fma i (fma i 16.666666666666668 50.0) 100.0)))

C

double code(double i, double n) {
	return n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
}

Julia

function code(i, n)
	return Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0))
end

Wolfram

code[i_, n_] := N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in n around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-expm1.f64 73.9

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

5. Applied rewrites 73.9%

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

   1. lift-expm1.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. times-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 79.1

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

7. Applied rewrites 79.1%

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

8. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 57.3

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

10. Applied rewrites 57.3%

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

Alternative 12: 55.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.55 \cdot 10^{+20}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.55e+20) (* n 100.0) (* i (* n 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.55e+20) {
		tmp = n * 100.0;
	} else {
		tmp = i * (n * 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 <= 1.55d+20) then
        tmp = n * 100.0d0
    else
        tmp = i * (n * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.55e+20) {
		tmp = n * 100.0;
	} else {
		tmp = i * (n * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.55e+20:
		tmp = n * 100.0
	else:
		tmp = i * (n * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.55e+20)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(i * Float64(n * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.55e+20)
		tmp = n * 100.0;
	else
		tmp = i * (n * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.55e+20], N[(n * 100.0), $MachinePrecision], N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.55e20

   1. Initial program 24.6%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if 1.55e20 < i

   1. Initial program 42.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in i around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{i \cdot \left(n \cdot \left(\color{blue}{50} \cdot i + 100 \cdot 1\right)\right)}{i} \]

      14. metadata-eval N/A

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

      15. lower-fma.f64 39.4

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

   8. Applied rewrites 39.4%

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

   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. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 32.5

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

   11. Applied rewrites 32.5%

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

Alternative 13: 55.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in n around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-expm1.f64 73.9

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

5. Applied rewrites 73.9%

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

   1. lift-expm1.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. times-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 79.1

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

7. Applied rewrites 79.1%

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

8. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 56.1

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

10. Applied rewrites 56.1%

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

Alternative 14: 55.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 10.9

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in n around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 56.1

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

8. Applied rewrites 56.1%

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

Alternative 15: 55.1% 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 28.2%

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

3. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 10.9

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in n around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

      \[\leadsto n \cdot \left(\color{blue}{50} \cdot i + 100 \cdot 1\right) \]

   9. metadata-eval N/A

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

   10. 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 16: 49.8% 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 28.2%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
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 49.7

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

5. Applied rewrites 49.7%

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

Alternative 17: 2.8% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

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

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

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

Derivation

1. Initial program 28.2%

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

3. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 10.9

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

5. Applied rewrites 10.9%

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

6. Taylor expanded in n around 0

   \[\leadsto \color{blue}{-50 \cdot i} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{i \cdot -50} \]

   2. lower-*.f64 2.7

      \[\leadsto \color{blue}{i \cdot -50} \]

8. Applied rewrites 2.7%

   \[\leadsto \color{blue}{i \cdot -50} \]
9. Add Preprocessing

Developer Target 1: 34.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(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))))