Octave 3.8, jcobi/4, as called

Percentage Accurate: 26.6% → 98.9%

Time: 3.4s

Alternatives: 3

Speedup: 25.0×

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: 26.6% 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: 98.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \frac{-i}{4}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.5) (* i (/ (- i) 4.0)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (-i / 4.0);
	} 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 / 4.0d0)
    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 / 4.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = i * (-i / 4.0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i * Float64(Float64(-i) / 4.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = i * (-i / 4.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 29.4%

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

      1. sqr-neg 29.4%

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

      2. sqr-neg 29.4%

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

      3. associate-/l* 57.2%

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

      4. sqr-neg 57.2%

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

      5. swap-sqr 57.2%

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

      6. metadata-eval 57.2%

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

      7. sqr-neg 57.2%

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

      8. sub-neg 57.2%

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

      9. swap-sqr 57.2%

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

      10. metadata-eval 57.2%

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

      11. metadata-eval 57.2%

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

   3. Simplified 57.2%

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

      1. associate-/l/ 57.2%

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

      2. times-frac 57.2%

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

      3. fma-def 57.2%

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

      4. pow2 57.2%

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

      5. associate-/l* 57.2%

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

      6. *-inverses 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in i around 0 99.6%

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

      1. neg-mul-1 99.6%

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

   8. Simplified 99.6%

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

   if 0.5 < i

   1. Initial program 25.4%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{1}{i \cdot i}}} \]

   4. Taylor expanded in i around inf 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \frac{-i}{4}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 2: 99.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{4 + \frac{-1}{i \cdot i}} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ 0.25 (+ 4.0 (/ -1.0 (* i i)))))

C

double code(double i) {
	return 0.25 / (4.0 + (-1.0 / (i * i)));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 0.25d0 / (4.0d0 + ((-1.0d0) / (i * i)))
end function

Java

public static double code(double i) {
	return 0.25 / (4.0 + (-1.0 / (i * i)));
}

Python

def code(i):
	return 0.25 / (4.0 + (-1.0 / (i * i)))

Julia

function code(i)
	return Float64(0.25 / Float64(4.0 + Float64(-1.0 / Float64(i * i))))
end

MATLAB

function tmp = code(i)
	tmp = 0.25 / (4.0 + (-1.0 / (i * i)));
end

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{1}{i \cdot i}}} \]

4. Final simplification 99.2%

   \[\leadsto \frac{0.25}{4 + \frac{-1}{i \cdot i}} \]

Alternative 3: 51.6% accurate, 25.0× 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 27.5%

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

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{1}{i \cdot i}}} \]

4. Taylor expanded in i around inf 49.6%

   \[\leadsto \color{blue}{0.0625} \]

5. Final simplification 49.6%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023309 
(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)))