Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.7% → 100.0%

Time: 7.4s

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: 27.7% 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, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 50000000:\\ \;\;\;\;\frac{i \cdot \left(i \cdot 0.25\right)}{-1 - \left(i \cdot i\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 50000000.0) (/ (* i (* i 0.25)) (- -1.0 (* (* i i) -4.0))) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 50000000.0) {
		tmp = (i * (i * 0.25)) / (-1.0 - ((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 <= 50000000.0d0) then
        tmp = (i * (i * 0.25d0)) / ((-1.0d0) - ((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 <= 50000000.0) {
		tmp = (i * (i * 0.25)) / (-1.0 - ((i * i) * -4.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 50000000.0:
		tmp = (i * (i * 0.25)) / (-1.0 - ((i * i) * -4.0))
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 50000000.0)
		tmp = Float64(Float64(i * Float64(i * 0.25)) / Float64(-1.0 - Float64(Float64(i * i) * -4.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 50000000.0)
		tmp = (i * (i * 0.25)) / (-1.0 - ((i * i) * -4.0));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 50000000.0], N[(N[(i * N[(i * 0.25), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(N[(i * i), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 5e7

   1. Initial program 33.7%

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

   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. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. fmm-def 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

      8. swap-sqr 100.0%

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

      9. associate-*r/ 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. swap-sqr 100.0%

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

      16. metadata-eval 100.0%

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

      17. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 5e7 < i

   1. Initial program 27.3%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in i around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 50000000:\\ \;\;\;\;\frac{i \cdot \left(i \cdot 0.25\right)}{-1 - \left(i \cdot i\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = (i * i) * -0.25;
	} 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
    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;
	} 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
	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));
	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;
	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] * (-0.25)), $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 31.7%

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

   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 98.9%

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

      1. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   if 0.5 < i

   1. Initial program 29.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in i around inf 99.1%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.1%

         \[\leadsto 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{{i}^{2}}} \]

      2. metadata-eval 99.1%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.015625}}{{i}^{2}} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{{i}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 49.2%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.0%

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

Alternative 3: 98.9% accurate, 2.3× speedup

Math

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

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(Float64(i * i) * Float64(-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[(N[(i * i), $MachinePrecision] * (-0.25)), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 31.7%

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

   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 98.9%

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

      1. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   if 0.5 < i

   1. Initial program 29.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in i around inf 98.5%

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

4. Final simplification 98.7%

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

Alternative 4: 75.0% 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 31.7%

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

   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 98.9%

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

      1. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

      1. neg-sub0 98.9%

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

      2. sub-neg 98.9%

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

      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 46.7%

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

      5. sqr-neg 46.7%

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

      6. sqrt-prod 46.7%

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

      7. add-sqr-sqrt 46.7%

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

   9. Applied egg-rr 46.7%

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

       1. +-lft-identity 46.7%

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

   11. Simplified 46.7%

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

   if 0.80000000000000004 < i

   1. Initial program 29.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in i around inf 98.5%

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

Alternative 5: 50.8% 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 30.7%

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

3. Simplified 76.1%

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

5. Taylor expanded in i around inf 49.1%

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

Reproduce

herbie shell --seed 2024151 
(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)))