Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.3% → 99.4%

Time: 8.6s

Alternatives: 5

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: 27.3% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{4 - {i}^{-2}} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ 0.25 (- 4.0 (pow i -2.0))))

C

double code(double i) {
	return 0.25 / (4.0 - pow(i, -2.0));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 0.25d0 / (4.0d0 - (i ** (-2.0d0)))
end function

Java

public static double code(double i) {
	return 0.25 / (4.0 - Math.pow(i, -2.0));
}

Python

def code(i):
	return 0.25 / (4.0 - math.pow(i, -2.0))

Julia

function code(i)
	return Float64(0.25 / Float64(4.0 - (i ^ -2.0)))
end

MATLAB

function tmp = code(i)
	tmp = 0.25 / (4.0 - (i ^ -2.0));
end

Wolfram

code[i_] := N[(0.25 / N[(4.0 - N[Power[i, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.3%

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

3. Simplified 76.8%

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

5. Taylor expanded in i around inf 57.7%

   \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{\color{blue}{{i}^{2} \cdot \left(4 - \frac{1}{{i}^{2}}\right)}}\right) \]
6. Step-by-step derivation

   1. *-un-lft-identity 57.7%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - \color{blue}{1 \cdot \frac{1}{{i}^{2}}}\right)}\right) \]

   2. pow-flip 57.6%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - 1 \cdot \color{blue}{{i}^{\left(-2\right)}}\right)}\right) \]

   3. metadata-eval 57.6%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - 1 \cdot {i}^{\color{blue}{-2}}\right)}\right) \]

7. Applied egg-rr 57.6%

   \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - \color{blue}{1 \cdot {i}^{-2}}\right)}\right) \]
8. Step-by-step derivation

   1. *-lft-identity 57.6%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - \color{blue}{{i}^{-2}}\right)}\right) \]

9. Simplified 57.6%

   \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - \color{blue}{{i}^{-2}}\right)}\right) \]
10. Step-by-step derivation

    1. add-log-exp 30.3%

       \[\leadsto \color{blue}{\log \left(e^{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)}\right)} \]

    2. *-un-lft-identity 30.3%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)}\right)} \]

    3. log-prod 30.3%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)}\right)} \]

    4. metadata-eval 30.3%

       \[\leadsto \color{blue}{0} + \log \left(e^{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)}\right) \]

    5. add-log-exp 57.6%

       \[\leadsto 0 + \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)} \]

    6. pow2 57.6%

       \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{i}{\color{blue}{\left(i \cdot i\right)} \cdot \left(4 - {i}^{-2}\right)}\right) \]

    7. associate-/r* 57.6%

       \[\leadsto 0 + 0.25 \cdot \left(i \cdot \color{blue}{\frac{\frac{i}{i \cdot i}}{4 - {i}^{-2}}}\right) \]

    8. pow1 57.6%

       \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{\frac{\color{blue}{{i}^{1}}}{i \cdot i}}{4 - {i}^{-2}}\right) \]

    9. pow2 57.6%

       \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{\frac{{i}^{1}}{\color{blue}{{i}^{2}}}}{4 - {i}^{-2}}\right) \]

    10. pow-div 98.7%

        \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{\color{blue}{{i}^{\left(1 - 2\right)}}}{4 - {i}^{-2}}\right) \]

    11. metadata-eval 98.7%

        \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{{i}^{\color{blue}{-1}}}{4 - {i}^{-2}}\right) \]

    12. unpow-1 98.7%

        \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{\color{blue}{\frac{1}{i}}}{4 - {i}^{-2}}\right) \]

11. Applied egg-rr 98.7%

    \[\leadsto \color{blue}{0 + 0.25 \cdot \left(i \cdot \frac{\frac{1}{i}}{4 - {i}^{-2}}\right)} \]
12. Step-by-step derivation

    1. +-lft-identity 98.7%

       \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{\frac{1}{i}}{4 - {i}^{-2}}\right)} \]

    2. associate-*r* 98.7%

       \[\leadsto \color{blue}{\left(0.25 \cdot i\right) \cdot \frac{\frac{1}{i}}{4 - {i}^{-2}}} \]

    3. associate-/l* 98.7%

       \[\leadsto \color{blue}{\frac{\left(0.25 \cdot i\right) \cdot \frac{1}{i}}{4 - {i}^{-2}}} \]

    4. associate-*l* 98.7%

       \[\leadsto \frac{\color{blue}{0.25 \cdot \left(i \cdot \frac{1}{i}\right)}}{4 - {i}^{-2}} \]

    5. rgt-mult-inverse 98.8%

       \[\leadsto \frac{0.25 \cdot \color{blue}{1}}{4 - {i}^{-2}} \]

    6. metadata-eval 98.8%

       \[\leadsto \frac{\color{blue}{0.25}}{4 - {i}^{-2}} \]

13. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{0.25}{4 - {i}^{-2}}} \]
14. Add Preprocessing

Alternative 2: 99.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 40.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 97.9%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 97.9%

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

   7. Simplified 97.9%

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

   if 0.5 < i

   1. Initial program 35.9%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in i around inf 98.1%

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

Alternative 3: 99.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.3%

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

3. Simplified 76.8%

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

5. Taylor expanded in i around inf 57.7%

   \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{\color{blue}{{i}^{2} \cdot \left(4 - \frac{1}{{i}^{2}}\right)}}\right) \]
6. Step-by-step derivation

   1. *-un-lft-identity 57.7%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - \color{blue}{1 \cdot \frac{1}{{i}^{2}}}\right)}\right) \]

   2. pow-flip 57.6%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - 1 \cdot \color{blue}{{i}^{\left(-2\right)}}\right)}\right) \]

   3. metadata-eval 57.6%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - 1 \cdot {i}^{\color{blue}{-2}}\right)}\right) \]

7. Applied egg-rr 57.6%

   \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - \color{blue}{1 \cdot {i}^{-2}}\right)}\right) \]
8. Step-by-step derivation

   1. *-lft-identity 57.6%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - \color{blue}{{i}^{-2}}\right)}\right) \]

9. Simplified 57.6%

   \[\leadsto 0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - \color{blue}{{i}^{-2}}\right)}\right) \]
10. Step-by-step derivation

    1. add-log-exp 30.3%

       \[\leadsto \color{blue}{\log \left(e^{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)}\right)} \]

    2. *-un-lft-identity 30.3%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)}\right)} \]

    3. log-prod 30.3%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)}\right)} \]

    4. metadata-eval 30.3%

       \[\leadsto \color{blue}{0} + \log \left(e^{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)}\right) \]

    5. add-log-exp 57.6%

       \[\leadsto 0 + \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{{i}^{2} \cdot \left(4 - {i}^{-2}\right)}\right)} \]

    6. pow2 57.6%

       \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{i}{\color{blue}{\left(i \cdot i\right)} \cdot \left(4 - {i}^{-2}\right)}\right) \]

    7. associate-/r* 57.6%

       \[\leadsto 0 + 0.25 \cdot \left(i \cdot \color{blue}{\frac{\frac{i}{i \cdot i}}{4 - {i}^{-2}}}\right) \]

    8. pow1 57.6%

       \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{\frac{\color{blue}{{i}^{1}}}{i \cdot i}}{4 - {i}^{-2}}\right) \]

    9. pow2 57.6%

       \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{\frac{{i}^{1}}{\color{blue}{{i}^{2}}}}{4 - {i}^{-2}}\right) \]

    10. pow-div 98.7%

        \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{\color{blue}{{i}^{\left(1 - 2\right)}}}{4 - {i}^{-2}}\right) \]

    11. metadata-eval 98.7%

        \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{{i}^{\color{blue}{-1}}}{4 - {i}^{-2}}\right) \]

    12. unpow-1 98.7%

        \[\leadsto 0 + 0.25 \cdot \left(i \cdot \frac{\color{blue}{\frac{1}{i}}}{4 - {i}^{-2}}\right) \]

11. Applied egg-rr 98.7%

    \[\leadsto \color{blue}{0 + 0.25 \cdot \left(i \cdot \frac{\frac{1}{i}}{4 - {i}^{-2}}\right)} \]
12. Step-by-step derivation

    1. +-lft-identity 98.7%

       \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{\frac{1}{i}}{4 - {i}^{-2}}\right)} \]

    2. associate-*r* 98.7%

       \[\leadsto \color{blue}{\left(0.25 \cdot i\right) \cdot \frac{\frac{1}{i}}{4 - {i}^{-2}}} \]

    3. associate-/l* 98.7%

       \[\leadsto \color{blue}{\frac{\left(0.25 \cdot i\right) \cdot \frac{1}{i}}{4 - {i}^{-2}}} \]

    4. associate-*l* 98.7%

       \[\leadsto \frac{\color{blue}{0.25 \cdot \left(i \cdot \frac{1}{i}\right)}}{4 - {i}^{-2}} \]

    5. rgt-mult-inverse 98.8%

       \[\leadsto \frac{0.25 \cdot \color{blue}{1}}{4 - {i}^{-2}} \]

    6. metadata-eval 98.8%

       \[\leadsto \frac{\color{blue}{0.25}}{4 - {i}^{-2}} \]

13. Simplified 98.8%

    \[\leadsto \color{blue}{\frac{0.25}{4 - {i}^{-2}}} \]
14. Step-by-step derivation

    1. sqr-pow 98.7%

       \[\leadsto \frac{0.25}{4 - \color{blue}{{i}^{\left(\frac{-2}{2}\right)} \cdot {i}^{\left(\frac{-2}{2}\right)}}} \]

    2. metadata-eval 98.7%

       \[\leadsto \frac{0.25}{4 - {i}^{\color{blue}{-1}} \cdot {i}^{\left(\frac{-2}{2}\right)}} \]

    3. unpow-1 98.7%

       \[\leadsto \frac{0.25}{4 - \color{blue}{\frac{1}{i}} \cdot {i}^{\left(\frac{-2}{2}\right)}} \]

    4. metadata-eval 98.7%

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

    5. unpow-1 98.7%

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

15. Applied egg-rr 98.7%

    \[\leadsto \frac{0.25}{4 - \color{blue}{\frac{1}{i} \cdot \frac{1}{i}}} \]
16. Add Preprocessing

Alternative 4: 74.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.8:\\ \;\;\;\;0.25 \cdot \left(i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.8) (* 0.25 (* i i)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.8) {
		tmp = 0.25 * (i * i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.8d0) then
        tmp = 0.25d0 * (i * i)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.8) {
		tmp = 0.25 * (i * i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.8:
		tmp = 0.25 * (i * i)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.8)
		tmp = Float64(0.25 * Float64(i * i));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.8)
		tmp = 0.25 * (i * i);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.8], N[(0.25 * N[(i * i), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.80000000000000004

   1. Initial program 40.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 97.9%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 97.9%

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

   7. Simplified 97.9%

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

      1. neg-sub0 97.9%

         \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(0 - i\right)}\right) \]

      2. sub-neg 97.9%

         \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(0 + \left(-i\right)\right)}\right) \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \color{blue}{\sqrt{-i} \cdot \sqrt{-i}}\right)\right) \]

      4. sqrt-unprod 41.3%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \color{blue}{\sqrt{\left(-i\right) \cdot \left(-i\right)}}\right)\right) \]

      5. sqr-neg 41.3%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \sqrt{\color{blue}{i \cdot i}}\right)\right) \]

      6. sqrt-prod 41.3%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \color{blue}{\sqrt{i} \cdot \sqrt{i}}\right)\right) \]

      7. add-sqr-sqrt 41.3%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \color{blue}{i}\right)\right) \]

   9. Applied egg-rr 41.3%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(0 + i\right)}\right) \]
   10. Step-by-step derivation

       1. +-lft-identity 41.3%

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

   11. Simplified 41.3%

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

   if 0.80000000000000004 < i

   1. Initial program 35.9%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in i around inf 98.1%

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

Alternative 5: 51.0% 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 38.3%

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

3. Simplified 76.8%

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

5. Taylor expanded in i around inf 51.1%

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

Reproduce

herbie shell --seed 2024152 
(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)))