Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.2% → 99.8%

Time: 6.8s

Alternatives: 4

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: 99.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{i}{\frac{-4}{i} - i \cdot -16} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ i (- (/ -4.0 i) (* i -16.0))))

C

double code(double i) {
	return i / ((-4.0 / i) - (i * -16.0));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = i / (((-4.0d0) / i) - (i * (-16.0d0)))
end function

Java

public static double code(double i) {
	return i / ((-4.0 / i) - (i * -16.0));
}

Python

def code(i):
	return i / ((-4.0 / i) - (i * -16.0))

Julia

function code(i)
	return Float64(i / Float64(Float64(-4.0 / i) - Float64(i * -16.0)))
end

MATLAB

function tmp = code(i)
	tmp = i / ((-4.0 / i) - (i * -16.0));
end

Wolfram

code[i_] := N[(i / N[(N[(-4.0 / i), $MachinePrecision] - N[(i * -16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.9%

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

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/l/ N/A

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

   10. associate-*r* N/A

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

3. Simplified 75.4%

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

   1. associate-*l/ N/A

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

   2. associate-/r/ N/A

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

   3. /-lowering-/.f64 N/A

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

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

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

   5. +-commutative N/A

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

   6. cancel-sign-sub N/A

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

   7. distribute-lft-neg-in N/A

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

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(i, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-4, \left(\mathsf{neg}\left(\left(\left(i \cdot i\right) \cdot 4\right) \cdot 4\right)\right)\right), i\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(i, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-4, \left(\mathsf{neg}\left(\left(i \cdot i\right) \cdot \left(4 \cdot 4\right)\right)\right)\right), i\right)\right) \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(i, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-4, \left(\left(i \cdot i\right) \cdot \left(\mathsf{neg}\left(4 \cdot 4\right)\right)\right)\right), i\right)\right) \]

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

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval 76.0%

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

6. Applied egg-rr 76.0%

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

   1. div-sub N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. pow2 N/A

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

   6. inv-pow N/A

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

   7. pow-prod-up N/A

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

   8. metadata-eval N/A

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

   9. unpow1 N/A

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

   10. *-commutative N/A

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

   11. --lowering--.f64 N/A

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

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

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

   13. *-lowering-*.f64 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 2: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = (i * i) * (-0.25 - (i * i))
	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 - Float64(i * i)));
	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 - (i * i));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 32.5%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/l/ N/A

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

      10. associate-*r* N/A

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 99.3%

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

   7. Simplified 99.3%

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

   if 0.5 < i

   1. Initial program 20.3%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/l/ N/A

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

      10. associate-*r* N/A

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

   3. Simplified 46.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 100.0%

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

Alternative 3: 98.9% accurate, 2.5× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 32.5%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/l/ N/A

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

      10. associate-*r* N/A

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

   if 0.5 < i

   1. Initial program 20.3%

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

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/l/ N/A

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

      10. associate-*r* N/A

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

   3. Simplified 46.6%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 100.0%

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

Alternative 4: 50.5% 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 26.9%

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

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/l/ N/A

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

   10. associate-*r* N/A

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

3. Simplified 75.4%

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

5. Taylor expanded in i around inf

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

7. Simplified 47.5%

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

Reproduce

herbie shell --seed 2024164 
(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)))