Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.2% → 99.4%

Time: 5.8s

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: 27.2% 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, 2.8× 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 23.8%

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

   1. associate-/l/ N/A

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

   2. associate-*l* N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}\right)}\right) \]

   5. swap-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(i \cdot i\right)}\right)}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right) \cdot \color{blue}{\left(i \cdot i\right)}}\right)\right) \]

   7. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot \color{blue}{\frac{i \cdot i}{i \cdot i}}\right)\right) \]

   8. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot 1\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right)}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}\right)\right)\right) \]

3. Simplified 73.4%

   \[\leadsto \color{blue}{i \cdot \frac{i}{\left(i \cdot i\right) \cdot 16 + -4}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto i \cdot \frac{1}{\color{blue}{\frac{\left(i \cdot i\right) \cdot 16 + -4}{i}}} \]

   2. un-div-inv N/A

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

   3. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\left(i \cdot i\right) \cdot 16 + -4}{i}}{i}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{\left(i \cdot i\right) \cdot 16 + -4}{i}}{i}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\left(i \cdot i\right) \cdot 16 + -4}{i}\right), \color{blue}{i}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(i \cdot i\right) \cdot 16 + -4\right), i\right), i\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(i \cdot i\right) \cdot 16\right), -4\right), i\right), i\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(i \cdot \left(i \cdot 16\right)\right), -4\right), i\right), i\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \left(i \cdot 16\right)\right), -4\right), i\right), i\right)\right) \]

   10. *-lowering-*.f64 73.1%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, 16\right)\right), -4\right), i\right), i\right)\right) \]

6. Applied egg-rr 73.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i \cdot \left(i \cdot 16\right) + -4}{i}}{i}}} \]

7. Taylor expanded in i around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{16 \cdot {i}^{2} - 4}{{i}^{2}}\right)}\right) \]
8. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{16 \cdot {i}^{2}}{{i}^{2}} - \color{blue}{\frac{4}{{i}^{2}}}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(16 \cdot \frac{{i}^{2}}{{i}^{2}} - \frac{\color{blue}{4}}{{i}^{2}}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(16 \cdot \frac{{i}^{2}}{{i}^{2}} - \frac{4 \cdot 1}{{\color{blue}{i}}^{2}}\right)\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(16 \cdot \frac{{i}^{2}}{{i}^{2}} - 4 \cdot \color{blue}{\frac{1}{{i}^{2}}}\right)\right) \]

   5. fmsub-define N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fmsub}\left(16, \color{blue}{\left(\frac{{i}^{2}}{{i}^{2}}\right)}, \left(4 \cdot \frac{1}{{i}^{2}}\right)\right)\right) \]

   6. *-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fmsub}\left(16, 1, \left(4 \cdot \frac{1}{{i}^{2}}\right)\right)\right) \]

   7. fmsub-undefine N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(16 \cdot 1 - \color{blue}{4 \cdot \frac{1}{{i}^{2}}}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(16 - \color{blue}{4} \cdot \frac{1}{{i}^{2}}\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(16, \color{blue}{\left(4 \cdot \frac{1}{{i}^{2}}\right)}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(16, \left(\frac{4 \cdot 1}{\color{blue}{{i}^{2}}}\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(16, \left(\frac{4}{{\color{blue}{i}}^{2}}\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(16, \mathsf{/.f64}\left(4, \color{blue}{\left({i}^{2}\right)}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(16, \mathsf{/.f64}\left(4, \left(i \cdot \color{blue}{i}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(16, \mathsf{/.f64}\left(4, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right)\right) \]

9. Simplified 99.7%

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

Alternative 2: 99.5% accurate, 1.8× speedup

Math

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

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(i * Float64(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[(i * N[(i * N[(-0.25 - N[(i * i), $MachinePrecision]), $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 26.6%

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

      1. associate-/l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}\right)}\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(i \cdot i\right)}\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right) \cdot \color{blue}{\left(i \cdot i\right)}}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot \color{blue}{\frac{i \cdot i}{i \cdot i}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot 1\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{i \cdot \frac{i}{\left(i \cdot i\right) \cdot 16 + -4}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(i \cdot \left(-1 \cdot {i}^{2} - \frac{1}{4}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \color{blue}{\left(-1 \cdot {i}^{2} - \frac{1}{4}\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \left(-1 \cdot {i}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \left(-1 \cdot {i}^{2} + \frac{-1}{4}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \left(\frac{-1}{4} + \color{blue}{-1 \cdot {i}^{2}}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \left(\frac{-1}{4} + \left(\mathsf{neg}\left({i}^{2}\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \left(\frac{-1}{4} - \color{blue}{{i}^{2}}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\frac{-1}{4}, \color{blue}{\left({i}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\frac{-1}{4}, \left(i \cdot \color{blue}{i}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right)\right) \]

   7. Simplified 98.9%

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

   if 0.5 < i

   1. Initial program 21.3%

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

      1. associate-/l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}\right)}\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(i \cdot i\right)}\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right) \cdot \color{blue}{\left(i \cdot i\right)}}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot \color{blue}{\frac{i \cdot i}{i \cdot i}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot 1\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}\right)\right)\right) \]

   3. Simplified 50.3%

      \[\leadsto \color{blue}{i \cdot \frac{i}{\left(i \cdot i\right) \cdot 16 + -4}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{1}{64} \cdot \frac{1}{{i}^{2}}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64} \cdot 1}{\color{blue}{{i}^{2}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64}}{{\color{blue}{i}}^{2}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \color{blue}{\left({i}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \left(i \cdot \color{blue}{i}\right)\right)\right) \]

      6. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right) \]

   7. Simplified 98.8%

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

Alternative 3: 99.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \frac{i}{-4}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5) (* i (/ i -4.0)) (+ 0.0625 (/ 0.015625 (* i i)))))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i / -4.0);
	} 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 / (-4.0d0))
    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 / -4.0);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = i * (i / -4.0)
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i * Float64(i / -4.0));
	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 / -4.0);
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(i * N[(i / -4.0), $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 26.6%

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

      1. associate-/l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}\right)}\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(i \cdot i\right)}\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right) \cdot \color{blue}{\left(i \cdot i\right)}}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot \color{blue}{\frac{i \cdot i}{i \cdot i}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot 1\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{i \cdot \frac{i}{\left(i \cdot i\right) \cdot 16 + -4}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{-4}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.0%

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

   if 0.5 < i

   1. Initial program 21.3%

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

      1. associate-/l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}\right)}\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(i \cdot i\right)}\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right) \cdot \color{blue}{\left(i \cdot i\right)}}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot \color{blue}{\frac{i \cdot i}{i \cdot i}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot 1\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}\right)\right)\right) \]

   3. Simplified 50.3%

      \[\leadsto \color{blue}{i \cdot \frac{i}{\left(i \cdot i\right) \cdot 16 + -4}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{1}{64} \cdot \frac{1}{{i}^{2}}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64} \cdot 1}{\color{blue}{{i}^{2}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{1}{64}}{{\color{blue}{i}}^{2}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \color{blue}{\left({i}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \left(i \cdot \color{blue}{i}\right)\right)\right) \]

      6. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(i, \color{blue}{i}\right)\right)\right) \]

   7. Simplified 98.8%

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

Alternative 4: 99.0% accurate, 2.5× 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(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 26.6%

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

      1. associate-/l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}\right)}\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(i \cdot i\right)}\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right) \cdot \color{blue}{\left(i \cdot i\right)}}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot \color{blue}{\frac{i \cdot i}{i \cdot i}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot 1\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{i \cdot \frac{i}{\left(i \cdot i\right) \cdot 16 + -4}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{-4}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 98.0%

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

   if 0.5 < i

   1. Initial program 21.3%

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

      1. associate-/l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}\right)}\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(i \cdot i\right)}\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right) \cdot \color{blue}{\left(i \cdot i\right)}}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot \color{blue}{\frac{i \cdot i}{i \cdot i}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot 1\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}\right)\right)\right) \]

   3. Simplified 50.3%

      \[\leadsto \color{blue}{i \cdot \frac{i}{\left(i \cdot i\right) \cdot 16 + -4}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf

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

   7. Simplified 98.5%

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

Alternative 5: 50.9% 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 23.8%

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

   1. associate-/l/ N/A

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

   2. associate-*l* N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right)}\right)}\right) \]

   5. swap-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(i \cdot i\right)}\right)}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i \cdot \left(i \cdot i\right)}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right) \cdot \color{blue}{\left(i \cdot i\right)}}\right)\right) \]

   7. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot \color{blue}{\frac{i \cdot i}{i \cdot i}}\right)\right) \]

   8. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)} \cdot 1\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(i, \left(\frac{i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \color{blue}{\left(\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)\right)}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(i, \left(\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}\right)\right)\right) \]

3. Simplified 73.4%

   \[\leadsto \color{blue}{i \cdot \frac{i}{\left(i \cdot i\right) \cdot 16 + -4}} \]
4. Add Preprocessing

5. Taylor expanded in i around inf

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

7. Simplified 54.0%

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

Reproduce

herbie shell --seed 2024138 
(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)))