Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.4% → 99.4%

Time: 7.5s

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: 27.4% 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: 99.4% 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 32.4%

   \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
2. Add Preprocessing
3. 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 99.9%

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

Alternative 2: 99.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = (i * i) * (-0.25 - (i * i));
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	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) - (i * i))
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

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

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = (i * i) * (-0.25 - (i * i))
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = (i * i) * (-0.25 - (i * i));
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(N[(i * i), $MachinePrecision] * N[(-0.25 - N[(i * i), $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 29.5%

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

   3. 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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{i \cdot \left(i \cdot \left(-0.25 - i \cdot i\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.5%

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

   if 0.5 < i

   1. Initial program 25.2%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.5%

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

Reproduce

herbie shell --seed 2024223 
(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)))