Compound Interest

Percentage Accurate: 28.5% → 95.9%

Time: 14.3s

Alternatives: 13

Speedup: 8.1×

• could not determine a ground truth

  1. i = 8.857830663597154e-15
  2. n = 8.599845265295517e+35

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: 28.5% 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: 95.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 99.9

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

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

   1. Initial program 25.3%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 99.6

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 95.7%

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

Alternative 2: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{-197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-267}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-91}:\\ \;\;\;\;n \cdot \frac{\left(n \cdot 100\right) \cdot \left(\log i - \log n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -1.25e-197)
     t_0
     (if (<= n 6.2e-267)
       0.0
       (if (<= n 4e-91)
         (* n (/ (* (* n 100.0) (- (log i) (log n))) i))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -1.25e-197) {
		tmp = t_0;
	} else if (n <= 6.2e-267) {
		tmp = 0.0;
	} else if (n <= 4e-91) {
		tmp = n * (((n * 100.0) * (log(i) - log(n))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -1.25e-197) {
		tmp = t_0;
	} else if (n <= 6.2e-267) {
		tmp = 0.0;
	} else if (n <= 4e-91) {
		tmp = n * (((n * 100.0) * (Math.log(i) - Math.log(n))) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -1.25e-197:
		tmp = t_0
	elif n <= 6.2e-267:
		tmp = 0.0
	elif n <= 4e-91:
		tmp = n * (((n * 100.0) * (math.log(i) - math.log(n))) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -1.25e-197)
		tmp = t_0;
	elseif (n <= 6.2e-267)
		tmp = 0.0;
	elseif (n <= 4e-91)
		tmp = Float64(n * Float64(Float64(Float64(n * 100.0) * Float64(log(i) - log(n))) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.25e-197], t$95$0, If[LessEqual[n, 6.2e-267], 0.0, If[LessEqual[n, 4e-91], N[(n * N[(N[(N[(n * 100.0), $MachinePrecision] * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2500000000000001e-197 or 4.00000000000000009e-91 < n

   1. Initial program 23.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 23.3

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in n around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft-out-- N/A

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

      6. div-sub N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-expm1.f64 84.9

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 85.0%

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

   if -1.2500000000000001e-197 < n < 6.2000000000000002e-267

   1. Initial program 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 33.4

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

   4. Applied rewrites 33.4%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 90.7

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 90.7%

       \[\leadsto 0 \]

   if 6.2000000000000002e-267 < n < 4.00000000000000009e-91

   1. Initial program 23.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 24.0

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 82.7

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

   7. Applied rewrites 82.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-197}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-267}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-91}:\\ \;\;\;\;n \cdot \frac{\left(n \cdot 100\right) \cdot \left(\log i - \log n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{-197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-170}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-91}:\\ \;\;\;\;\frac{\left(\log i - \log n\right) \cdot \left(100 \cdot \left(n \cdot n\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -1.25e-197)
     t_0
     (if (<= n 4.8e-170)
       (* 100.0 (/ (+ 1.0 -1.0) (/ i n)))
       (if (<= n 4e-91)
         (/ (* (- (log i) (log n)) (* 100.0 (* n n))) i)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -1.25e-197) {
		tmp = t_0;
	} else if (n <= 4.8e-170) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (n <= 4e-91) {
		tmp = ((log(i) - log(n)) * (100.0 * (n * n))) / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -1.25e-197) {
		tmp = t_0;
	} else if (n <= 4.8e-170) {
		tmp = 100.0 * ((1.0 + -1.0) / (i / n));
	} else if (n <= 4e-91) {
		tmp = ((Math.log(i) - Math.log(n)) * (100.0 * (n * n))) / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -1.25e-197:
		tmp = t_0
	elif n <= 4.8e-170:
		tmp = 100.0 * ((1.0 + -1.0) / (i / n))
	elif n <= 4e-91:
		tmp = ((math.log(i) - math.log(n)) * (100.0 * (n * n))) / i
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -1.25e-197)
		tmp = t_0;
	elseif (n <= 4.8e-170)
		tmp = Float64(100.0 * Float64(Float64(1.0 + -1.0) / Float64(i / n)));
	elseif (n <= 4e-91)
		tmp = Float64(Float64(Float64(log(i) - log(n)) * Float64(100.0 * Float64(n * n))) / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.25e-197], t$95$0, If[LessEqual[n, 4.8e-170], N[(100.0 * N[(N[(1.0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4e-91], N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * N[(100.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2500000000000001e-197 or 4.00000000000000009e-91 < n

   1. Initial program 23.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 23.3

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in n around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft-out-- N/A

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

      6. div-sub N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-expm1.f64 84.9

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 85.0%

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

   if -1.2500000000000001e-197 < n < 4.7999999999999999e-170

   1. Initial program 58.1%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 79.7%

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

   if 4.7999999999999999e-170 < n < 4.00000000000000009e-91

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 75.1

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

   5. Applied rewrites 75.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-197}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-170}:\\ \;\;\;\;100 \cdot \frac{1 + -1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-91}:\\ \;\;\;\;\frac{\left(\log i - \log n\right) \cdot \left(100 \cdot \left(n \cdot n\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{-197}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -1.25e-197) t_0 (if (<= n 2.45e-91) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -1.25e-197) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -1.25e-197) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -1.25e-197:
		tmp = t_0
	elif n <= 2.45e-91:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -1.25e-197)
		tmp = t_0;
	elseif (n <= 2.45e-91)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.25e-197], t$95$0, If[LessEqual[n, 2.45e-91], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2500000000000001e-197 or 2.4499999999999999e-91 < n

   1. Initial program 23.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 23.3

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in n around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft-out-- N/A

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

      6. div-sub N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-expm1.f64 84.9

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 85.0%

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

   if -1.2500000000000001e-197 < n < 2.4499999999999999e-91

   1. Initial program 46.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 16.2

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

   4. Applied rewrites 16.2%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 69.3

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

   7. Applied rewrites 69.3%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 69.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-197}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -1.12e-92) t_0 (if (<= n 2.45e-91) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -1.12e-92) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} 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 <= -1.12e-92) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -1.12e-92:
		tmp = t_0
	elif n <= 2.45e-91:
		tmp = 0.0
	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 <= -1.12e-92)
		tmp = t_0;
	elseif (n <= 2.45e-91)
		tmp = 0.0;
	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, -1.12e-92], t$95$0, If[LessEqual[n, 2.45e-91], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.11999999999999999e-92 or 2.4499999999999999e-91 < n

   1. Initial program 20.8%

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

   3. Taylor expanded in n around inf

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

      1. 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 86.2

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

   5. Applied rewrites 86.2%

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

   if -1.11999999999999999e-92 < n < 2.4499999999999999e-91

   1. Initial program 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 63.4%

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

Alternative 6: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* 100.0 (* n (expm1 i))) i)))
   (if (<= n -1.12e-92) t_0 (if (<= n 2.45e-91) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = (100.0 * (n * expm1(i))) / i;
	double tmp;
	if (n <= -1.12e-92) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} 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 <= -1.12e-92) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * (n * math.expm1(i))) / i
	tmp = 0
	if n <= -1.12e-92:
		tmp = t_0
	elif n <= 2.45e-91:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * Float64(n * expm1(i))) / i)
	tmp = 0.0
	if (n <= -1.12e-92)
		tmp = t_0;
	elseif (n <= 2.45e-91)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.12e-92], t$95$0, If[LessEqual[n, 2.45e-91], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.11999999999999999e-92 or 2.4499999999999999e-91 < n

   1. Initial program 20.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   if -1.11999999999999999e-92 < n < 2.4499999999999999e-91

   1. Initial program 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 63.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \mathsf{expm1}\left(i\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100 \cdot \left(i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(n, 0.5, i \cdot \left(n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right)\right)\right), n\right)\right)}{i}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (/
          (*
           100.0
           (*
            i
            (fma
             i
             (fma
              n
              0.5
              (* i (* n (fma 0.041666666666666664 i 0.16666666666666666))))
             n)))
          i)))
   (if (<= n -1.15e-92) t_0 (if (<= n 2.45e-91) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = (100.0 * (i * fma(i, fma(n, 0.5, (i * (n * fma(0.041666666666666664, i, 0.16666666666666666)))), n))) / i;
	double tmp;
	if (n <= -1.15e-92) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * Float64(i * fma(i, fma(n, 0.5, Float64(i * Float64(n * fma(0.041666666666666664, i, 0.16666666666666666)))), n))) / i)
	tmp = 0.0
	if (n <= -1.15e-92)
		tmp = t_0;
	elseif (n <= 2.45e-91)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[(i * N[(i * N[(n * 0.5 + N[(i * N[(n * N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.15e-92], t$95$0, If[LessEqual[n, 2.45e-91], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.15000000000000008e-92 or 2.4499999999999999e-91 < n

   1. Initial program 20.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 68.4%

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

   if -1.15000000000000008e-92 < n < 2.4499999999999999e-91

   1. Initial program 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 63.4%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-92}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(n, 0.5, i \cdot \left(n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right)\right)\right), n\right)\right)}{i}\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(n, 0.5, i \cdot \left(n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right)\right)\right), n\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-92)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 2.45e-91)
     0.0
     (*
      n
      (fma
       i
       (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
       100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-92) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-92)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 2.45e-91)
		tmp = 0.0;
	else
		tmp = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-92], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.45e-91], 0.0, N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.11999999999999999e-92

   1. Initial program 24.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 24.4

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

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in n around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft-out-- N/A

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

      6. div-sub N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-expm1.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 58.8%

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

   if -1.11999999999999999e-92 < n < 2.4499999999999999e-91

   1. Initial program 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 63.4%

       \[\leadsto 0 \]

   if 2.4499999999999999e-91 < n

   1. Initial program 17.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 18.0

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in n around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft-out-- N/A

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

      6. div-sub N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-expm1.f64 93.4

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

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 76.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-92)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 2.45e-91)
     0.0
     (* 100.0 (* n (fma i (fma i 0.16666666666666666 0.5) 1.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-92) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n * fma(i, fma(i, 0.16666666666666666, 0.5), 1.0));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-92)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 2.45e-91)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n * fma(i, fma(i, 0.16666666666666666, 0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-92], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.45e-91], 0.0, N[(100.0 * N[(n * N[(i * N[(i * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.11999999999999999e-92

   1. Initial program 24.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 24.4

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

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in n around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft-out-- N/A

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

      6. div-sub N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-expm1.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 58.8%

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

   if -1.11999999999999999e-92 < n < 2.4499999999999999e-91

   1. Initial program 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 63.4%

       \[\leadsto 0 \]

   if 2.4499999999999999e-91 < n

   1. Initial program 17.5%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 0.16666666666666666, 0.5\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i (fma i 16.666666666666668 50.0) 100.0))))
   (if (<= n -1.12e-92) t_0 (if (<= n 2.45e-91) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	double tmp;
	if (n <= -1.12e-92) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0))
	tmp = 0.0
	if (n <= -1.12e-92)
		tmp = t_0;
	elseif (n <= 2.45e-91)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.12e-92], t$95$0, If[LessEqual[n, 2.45e-91], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.11999999999999999e-92 or 2.4499999999999999e-91 < n

   1. Initial program 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 21.2

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

   4. Applied rewrites 21.2%

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

   5. Taylor expanded in n around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft-out-- N/A

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

      6. div-sub N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-expm1.f64 86.6

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 64.8%

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

   if -1.11999999999999999e-92 < n < 2.4499999999999999e-91

   1. Initial program 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 63.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i 50.0 100.0))))
   (if (<= n -1.12e-92) t_0 (if (<= n 2.45e-91) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, 50.0, 100.0);
	double tmp;
	if (n <= -1.12e-92) {
		tmp = t_0;
	} else if (n <= 2.45e-91) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, 50.0, 100.0))
	tmp = 0.0
	if (n <= -1.12e-92)
		tmp = t_0;
	elseif (n <= 2.45e-91)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.12e-92], t$95$0, If[LessEqual[n, 2.45e-91], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.11999999999999999e-92 or 2.4499999999999999e-91 < n

   1. Initial program 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 21.2

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

   4. Applied rewrites 21.2%

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

   5. Taylor expanded in n around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. distribute-lft-out-- N/A

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

      6. div-sub N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-expm1.f64 86.6

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 60.9%

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

   if -1.11999999999999999e-92 < n < 2.4499999999999999e-91

   1. Initial program 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 20.9

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

   4. Applied rewrites 20.9%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 63.4%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-92}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{-91}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -215:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.02 \cdot 10^{+61}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -215.0) 0.0 (if (<= i 1.02e+61) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -215.0) {
		tmp = 0.0;
	} else if (i <= 1.02e+61) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-215.0d0)) then
        tmp = 0.0d0
    else if (i <= 1.02d+61) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -215.0) {
		tmp = 0.0;
	} else if (i <= 1.02e+61) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -215.0:
		tmp = 0.0
	elif i <= 1.02e+61:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -215.0)
		tmp = 0.0;
	elseif (i <= 1.02e+61)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -215.0)
		tmp = 0.0;
	elseif (i <= 1.02e+61)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -215.0], 0.0, If[LessEqual[i, 1.02e+61], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -215 or 1.01999999999999999e61 < i

   1. Initial program 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 32.8

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

   7. Applied rewrites 32.8%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 32.8%

       \[\leadsto 0 \]

   if -215 < i < 1.01999999999999999e61

   1. Initial program 10.3%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.5

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

   5. Applied rewrites 75.5%

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

Alternative 13: 17.7% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 27.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. sub-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/r/ N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. distribute-neg-frac N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 20.6

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

4. Applied rewrites 20.6%

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

5. Taylor expanded in i around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt N/A

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

   4. lower-/.f64 19.6

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

7. Applied rewrites 19.6%

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

8. Taylor expanded in i around 0

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

10. Applied rewrites 19.6%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 34.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024237 
(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))))