Octave 3.8, jcobi/4, as called

Percentage Accurate: 28.0% → 100.0%

Time: 7.3s

Alternatives: 6

Speedup: 71.0×

Specification

\[i > 0\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 - 1} \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))

C

double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (2.0d0 * i) * (2.0d0 * i)
    code = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0d0)
end function

Java

public static double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 28.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 - 1} \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))

C

double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (2.0d0 * i) * (2.0d0 * i)
    code = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0d0)
end function

Java

public static double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5000000:\\ \;\;\;\;\frac{i \cdot \left(i \cdot 0.25\right)}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 5000000.0) (/ (* i (* i 0.25)) (fma i (* i 4.0) -1.0)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 5000000.0) {
		tmp = (i * (i * 0.25)) / fma(i, (i * 4.0), -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 5000000.0)
		tmp = Float64(Float64(i * Float64(i * 0.25)) / fma(i, Float64(i * 4.0), -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 5000000.0], N[(N[(i * N[(i * 0.25), $MachinePrecision]), $MachinePrecision] / N[(i * N[(i * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 5e6

   1. Initial program 32.2%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \frac{1}{4}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(i \cdot i\right)} \cdot \frac{1}{4}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{i \cdot \left(i \cdot \frac{1}{4}\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{i \cdot \left(i \cdot \frac{1}{4}\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{i \cdot \color{blue}{\left(\frac{1}{4} \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      6. lower-*.f64 100.0

         \[\leadsto \frac{i \cdot \color{blue}{\left(0.25 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{i \cdot \left(0.25 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   6. Taylor expanded in i around 0

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

      1. sub-neg N/A

         \[\leadsto \frac{i \cdot \left(\frac{1}{4} \cdot i\right)}{\color{blue}{4 \cdot {i}^{2} + \left(\mathsf{neg}\left(1\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{i \cdot \left(\frac{1}{4} \cdot i\right)}{4 \cdot \color{blue}{\left(i \cdot i\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{i \cdot \left(\frac{1}{4} \cdot i\right)}{\color{blue}{\left(4 \cdot i\right) \cdot i} + \left(\mathsf{neg}\left(1\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{i \cdot \left(\frac{1}{4} \cdot i\right)}{\color{blue}{i \cdot \left(4 \cdot i\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{i \cdot \left(\frac{1}{4} \cdot i\right)}{i \cdot \left(4 \cdot i\right) + \color{blue}{-1}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{i \cdot \left(\frac{1}{4} \cdot i\right)}{\color{blue}{\mathsf{fma}\left(i, 4 \cdot i, -1\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{i \cdot \left(\frac{1}{4} \cdot i\right)}{\mathsf{fma}\left(i, \color{blue}{i \cdot 4}, -1\right)} \]

      8. lower-*.f64 100.0

         \[\leadsto \frac{i \cdot \left(0.25 \cdot i\right)}{\mathsf{fma}\left(i, \color{blue}{i \cdot 4}, -1\right)} \]

   8. Simplified 100.0%

      \[\leadsto \frac{i \cdot \left(0.25 \cdot i\right)}{\color{blue}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}} \]

   if 5e6 < i

   1. Initial program 22.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5000000:\\ \;\;\;\;\frac{i \cdot \left(i \cdot 0.25\right)}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, i \cdot -4, -1\right), -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5)
   (* i (* i (fma (* i i) (fma i (* i -4.0) -1.0) -0.25)))
   (+ 0.0625 (/ 0.015625 (* i i)))))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * fma((i * i), fma(i, (i * -4.0), -1.0), -0.25));
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i * Float64(i * fma(Float64(i * i), fma(i, Float64(i * -4.0), -1.0), -0.25)));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(i * N[(i * N[(N[(i * i), $MachinePrecision] * N[(i * N[(i * -4.0), $MachinePrecision] + -1.0), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.4%

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

   3. Taylor expanded in i around 0

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

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(i \cdot i\right)} \cdot \left({i}^{2} \cdot \left(-4 \cdot {i}^{2} - 1\right) - \frac{1}{4}\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{i \cdot \left(i \cdot \left({i}^{2} \cdot \left(-4 \cdot {i}^{2} - 1\right) - \frac{1}{4}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(\left({i}^{2} \cdot \left(-4 \cdot {i}^{2} - 1\right) - \frac{1}{4}\right) \cdot i\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(\left({i}^{2} \cdot \left(-4 \cdot {i}^{2} - 1\right) - \frac{1}{4}\right) \cdot i\right)} \]

      5. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(i \cdot \left({i}^{2} \cdot \left(-4 \cdot {i}^{2} - 1\right) - \frac{1}{4}\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(i \cdot \left({i}^{2} \cdot \left(-4 \cdot {i}^{2} - 1\right) - \frac{1}{4}\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto i \cdot \left(i \cdot \color{blue}{\left({i}^{2} \cdot \left(-4 \cdot {i}^{2} - 1\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto i \cdot \left(i \cdot \left({i}^{2} \cdot \left(-4 \cdot {i}^{2} - 1\right) + \color{blue}{\frac{-1}{4}}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto i \cdot \left(i \cdot \color{blue}{\mathsf{fma}\left({i}^{2}, -4 \cdot {i}^{2} - 1, \frac{-1}{4}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(\color{blue}{i \cdot i}, -4 \cdot {i}^{2} - 1, \frac{-1}{4}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(\color{blue}{i \cdot i}, -4 \cdot {i}^{2} - 1, \frac{-1}{4}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, \color{blue}{-4 \cdot {i}^{2} + \left(\mathsf{neg}\left(1\right)\right)}, \frac{-1}{4}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, -4 \cdot \color{blue}{\left(i \cdot i\right)} + \left(\mathsf{neg}\left(1\right)\right), \frac{-1}{4}\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, \color{blue}{\left(-4 \cdot i\right) \cdot i} + \left(\mathsf{neg}\left(1\right)\right), \frac{-1}{4}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, \color{blue}{i \cdot \left(-4 \cdot i\right)} + \left(\mathsf{neg}\left(1\right)\right), \frac{-1}{4}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, i \cdot \left(-4 \cdot i\right) + \color{blue}{-1}, \frac{-1}{4}\right)\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, \color{blue}{\mathsf{fma}\left(i, -4 \cdot i, -1\right)}, \frac{-1}{4}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, \color{blue}{i \cdot -4}, -1\right), \frac{-1}{4}\right)\right) \]

      19. lower-*.f64 99.7

          \[\leadsto i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, \color{blue}{i \cdot -4}, -1\right), -0.25\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{i \cdot \left(i \cdot \mathsf{fma}\left(i \cdot i, \mathsf{fma}\left(i, i \cdot -4, -1\right), -0.25\right)\right)} \]

   if 0.5 < i

   1. Initial program 26.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

      6. lower-*.f64 98.4

         \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(\mathsf{fma}\left(i, i, 0.25\right) \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5)
   (* i (* (fma i i 0.25) (- i)))
   (+ 0.0625 (/ 0.015625 (* i i)))))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (fma(i, i, 0.25) * -i);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i * Float64(fma(i, i, 0.25) * Float64(-i)));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(i * N[(N[(i * i + 0.25), $MachinePrecision] * (-i)), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.4%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

         \[\leadsto {i}^{2} \cdot \color{blue}{\left(-1 \cdot {i}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto {i}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left({i}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \]

      3. distribute-neg-in N/A

         \[\leadsto {i}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\left({i}^{2} + \frac{1}{4}\right)\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto {i}^{2} \cdot \left(\mathsf{neg}\left(\left({i}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto {i}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left({i}^{2} - \frac{-1}{4}\right)}\right)\right) \]

      6. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({i}^{2} \cdot \left({i}^{2} - \frac{-1}{4}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({i}^{2} \cdot \left({i}^{2} - \frac{-1}{4}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{i}^{2} \cdot \left({i}^{2} - \frac{-1}{4}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot i\right)} \cdot \left({i}^{2} - \frac{-1}{4}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot i\right)} \cdot \left({i}^{2} - \frac{-1}{4}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{neg}\left(\left(i \cdot i\right) \cdot \color{blue}{\left({i}^{2} + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(i \cdot i\right) \cdot \left({i}^{2} + \color{blue}{\frac{1}{4}}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(i \cdot i\right) \cdot \left(\color{blue}{i \cdot i} + \frac{1}{4}\right)\right) \]

      14. lower-fma.f64 99.4

          \[\leadsto -\left(i \cdot i\right) \cdot \color{blue}{\mathsf{fma}\left(i, i, 0.25\right)} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{-\left(i \cdot i\right) \cdot \mathsf{fma}\left(i, i, 0.25\right)} \]

   6. Taylor expanded in i around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{1}{4} \cdot {i}^{2} + {i}^{2} \cdot {i}^{2}\right)}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \color{blue}{\left(1 \cdot {i}^{2}\right)} + {i}^{2} \cdot {i}^{2}\right)\right) \]

      3. lft-mult-inverse N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(\color{blue}{\left(\frac{1}{{i}^{2}} \cdot {i}^{2}\right)} \cdot {i}^{2}\right) + {i}^{2} \cdot {i}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \color{blue}{\left(\frac{1}{{i}^{2}} \cdot \left({i}^{2} \cdot {i}^{2}\right)\right)} + {i}^{2} \cdot {i}^{2}\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(\frac{1}{{i}^{2}} \cdot \color{blue}{{i}^{\left(2 \cdot 2\right)}}\right) + {i}^{2} \cdot {i}^{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(\frac{1}{{i}^{2}} \cdot {i}^{\color{blue}{4}}\right) + {i}^{2} \cdot {i}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\frac{1}{4} \cdot \frac{1}{{i}^{2}}\right) \cdot {i}^{4}} + {i}^{2} \cdot {i}^{2}\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{{i}^{2}}\right) \cdot {i}^{4} + \color{blue}{{i}^{\left(2 \cdot 2\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{{i}^{2}}\right) \cdot {i}^{4} + {i}^{\color{blue}{4}}\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{1}{4} \cdot \frac{1}{{i}^{2}} + 1\right) \cdot {i}^{4}}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)} \cdot {i}^{4}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{i}^{4} \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left({i}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

      14. pow-sqr N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left({i}^{2} \cdot {i}^{2}\right)} \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(i \cdot i\right)} \cdot {i}^{2}\right) \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot \left(i \cdot {i}^{2}\right)\right)} \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{i \cdot \left(\left(i \cdot {i}^{2}\right) \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right)}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{i \cdot \left(\left(i \cdot {i}^{2}\right) \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right)}\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{neg}\left(i \cdot \color{blue}{\left(1 \cdot \left(i \cdot {i}^{2}\right) + \left(\frac{1}{4} \cdot \frac{1}{{i}^{2}}\right) \cdot \left(i \cdot {i}^{2}\right)\right)}\right) \]

   8. Simplified 99.4%

      \[\leadsto -\color{blue}{i \cdot \left(i \cdot \mathsf{fma}\left(i, i, 0.25\right)\right)} \]

   if 0.5 < i

   1. Initial program 26.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

      6. lower-*.f64 98.4

         \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(\mathsf{fma}\left(i, i, 0.25\right) \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(\mathsf{fma}\left(i, i, 0.25\right) \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5) (* i (* (fma i i 0.25) (- i))) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (fma(i, i, 0.25) * -i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i * Float64(fma(i, i, 0.25) * Float64(-i)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(i * N[(N[(i * i + 0.25), $MachinePrecision] * (-i)), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.4%

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

   3. Taylor expanded in i around 0

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

      1. sub-neg N/A

         \[\leadsto {i}^{2} \cdot \color{blue}{\left(-1 \cdot {i}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto {i}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left({i}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \]

      3. distribute-neg-in N/A

         \[\leadsto {i}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\left({i}^{2} + \frac{1}{4}\right)\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto {i}^{2} \cdot \left(\mathsf{neg}\left(\left({i}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto {i}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left({i}^{2} - \frac{-1}{4}\right)}\right)\right) \]

      6. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({i}^{2} \cdot \left({i}^{2} - \frac{-1}{4}\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({i}^{2} \cdot \left({i}^{2} - \frac{-1}{4}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{i}^{2} \cdot \left({i}^{2} - \frac{-1}{4}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot i\right)} \cdot \left({i}^{2} - \frac{-1}{4}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot i\right)} \cdot \left({i}^{2} - \frac{-1}{4}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{neg}\left(\left(i \cdot i\right) \cdot \color{blue}{\left({i}^{2} + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(i \cdot i\right) \cdot \left({i}^{2} + \color{blue}{\frac{1}{4}}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(i \cdot i\right) \cdot \left(\color{blue}{i \cdot i} + \frac{1}{4}\right)\right) \]

      14. lower-fma.f64 99.4

          \[\leadsto -\left(i \cdot i\right) \cdot \color{blue}{\mathsf{fma}\left(i, i, 0.25\right)} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{-\left(i \cdot i\right) \cdot \mathsf{fma}\left(i, i, 0.25\right)} \]

   6. Taylor expanded in i around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{1}{4} \cdot {i}^{2} + {i}^{2} \cdot {i}^{2}\right)}\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \color{blue}{\left(1 \cdot {i}^{2}\right)} + {i}^{2} \cdot {i}^{2}\right)\right) \]

      3. lft-mult-inverse N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(\color{blue}{\left(\frac{1}{{i}^{2}} \cdot {i}^{2}\right)} \cdot {i}^{2}\right) + {i}^{2} \cdot {i}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \color{blue}{\left(\frac{1}{{i}^{2}} \cdot \left({i}^{2} \cdot {i}^{2}\right)\right)} + {i}^{2} \cdot {i}^{2}\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(\frac{1}{{i}^{2}} \cdot \color{blue}{{i}^{\left(2 \cdot 2\right)}}\right) + {i}^{2} \cdot {i}^{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{4} \cdot \left(\frac{1}{{i}^{2}} \cdot {i}^{\color{blue}{4}}\right) + {i}^{2} \cdot {i}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\frac{1}{4} \cdot \frac{1}{{i}^{2}}\right) \cdot {i}^{4}} + {i}^{2} \cdot {i}^{2}\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{{i}^{2}}\right) \cdot {i}^{4} + \color{blue}{{i}^{\left(2 \cdot 2\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{{i}^{2}}\right) \cdot {i}^{4} + {i}^{\color{blue}{4}}\right)\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{1}{4} \cdot \frac{1}{{i}^{2}} + 1\right) \cdot {i}^{4}}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)} \cdot {i}^{4}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{i}^{4} \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left({i}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

      14. pow-sqr N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left({i}^{2} \cdot {i}^{2}\right)} \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(i \cdot i\right)} \cdot {i}^{2}\right) \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot \left(i \cdot {i}^{2}\right)\right)} \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{i \cdot \left(\left(i \cdot {i}^{2}\right) \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right)}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{i \cdot \left(\left(i \cdot {i}^{2}\right) \cdot \left(1 + \frac{1}{4} \cdot \frac{1}{{i}^{2}}\right)\right)}\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{neg}\left(i \cdot \color{blue}{\left(1 \cdot \left(i \cdot {i}^{2}\right) + \left(\frac{1}{4} \cdot \frac{1}{{i}^{2}}\right) \cdot \left(i \cdot {i}^{2}\right)\right)}\right) \]

   8. Simplified 99.4%

      \[\leadsto -\color{blue}{i \cdot \left(i \cdot \mathsf{fma}\left(i, i, 0.25\right)\right)} \]

   if 0.5 < i

   1. Initial program 26.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Simplified 97.6%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(\mathsf{fma}\left(i, i, 0.25\right) \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.5) (* i (* i -0.25)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = i * (i * (-0.25d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = i * (i * -0.25)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i * Float64(i * -0.25));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = i * (i * -0.25);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(i * N[(i * -0.25), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.4%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot {i}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{i}^{2} \cdot \frac{-1}{4}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(i \cdot i\right)} \cdot \frac{-1}{4} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{i \cdot \left(i \cdot \frac{-1}{4}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(i \cdot \frac{-1}{4}\right)} \]

      5. lower-*.f64 98.8

         \[\leadsto i \cdot \color{blue}{\left(i \cdot -0.25\right)} \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{i \cdot \left(i \cdot -0.25\right)} \]

   if 0.5 < i

   1. Initial program 26.1%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Simplified 97.6%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 50.6% accurate, 71.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (i) :precision binary64 0.0625)

C

double code(double i) {
	return 0.0625;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

public static double code(double i) {
	return 0.0625;
}

Python

def code(i):
	return 0.0625

Julia

function code(i)
	return 0.0625
end

MATLAB

function tmp = code(i)
	tmp = 0.0625;
end

Wolfram

code[i_] := 0.0625

Derivation

1. Initial program 27.2%

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

3. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
4. Step-by-step derivation

5. Simplified 52.4%

   \[\leadsto \color{blue}{0.0625} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (> i 0.0)
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))