Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.5% → 100.0%

Time: 8.1s

Alternatives: 7

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: 27.5% 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, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 20000000:\\ \;\;\;\;\frac{i \cdot i}{\mathsf{fma}\left(i \cdot i, 16, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 20000000.0) (/ (* i i) (fma (* i i) 16.0 -4.0)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 20000000.0) {
		tmp = (i * i) / fma((i * i), 16.0, -4.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 20000000.0)
		tmp = Float64(Float64(i * i) / fma(Float64(i * i), 16.0, -4.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 20000000.0], N[(N[(i * i), $MachinePrecision] / N[(N[(i * i), $MachinePrecision] * 16.0 + -4.0), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 2e7

   1. Initial program 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{i \cdot i}{4 \cdot \mathsf{fma}\left(4 \cdot i, i, -1\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. swap-sqr N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{i \cdot i}{\color{blue}{16} \cdot \left(i \cdot i\right) + \left(\mathsf{neg}\left(4\right)\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{i \cdot i}{16 \cdot \color{blue}{\left(i \cdot i\right)} + \left(\mathsf{neg}\left(4\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{i \cdot i}{\color{blue}{\left(i \cdot i\right) \cdot 16} + \left(\mathsf{neg}\left(4\right)\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{i \cdot i}{\color{blue}{\mathsf{fma}\left(i \cdot i, 16, \mathsf{neg}\left(4\right)\right)}} \]

      16. metadata-eval 100.0

          \[\leadsto \frac{i \cdot i}{\mathsf{fma}\left(i \cdot i, 16, \color{blue}{-4}\right)} \]

   6. Applied rewrites 100.0%

      \[\leadsto \frac{i \cdot i}{\color{blue}{\mathsf{fma}\left(i \cdot i, 16, -4\right)}} \]

   if 2e7 < i

   1. Initial program 24.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{16 - \frac{4}{i \cdot i}} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ 1.0 (- 16.0 (/ 4.0 (* i i)))))

C

double code(double i) {
	return 1.0 / (16.0 - (4.0 / (i * i)));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 1.0d0 / (16.0d0 - (4.0d0 / (i * i)))
end function

Java

public static double code(double i) {
	return 1.0 / (16.0 - (4.0 / (i * i)));
}

Python

def code(i):
	return 1.0 / (16.0 - (4.0 / (i * i)))

Julia

function code(i)
	return Float64(1.0 / Float64(16.0 - Float64(4.0 / Float64(i * i))))
end

MATLAB

function tmp = code(i)
	tmp = 1.0 / (16.0 - (4.0 / (i * i)));
end

Wolfram

code[i_] := N[(1.0 / N[(16.0 - N[(4.0 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. div-sub N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1}{16 - \frac{4}{i \cdot i}}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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)))