Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.2% → 100.0%

Time: 6.4s

Alternatives: 6

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: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(i \cdot 2\right) \cdot \left(i \cdot 2\right)\\ \mathbf{if}\;i \leq 5 \cdot 10^{-9}:\\ \;\;\;\;i \cdot \left(0.25 \cdot \left(-i\right)\right)\\ \mathbf{elif}\;i \leq 5000000:\\ \;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{-1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* i 2.0) (* i 2.0))))
   (if (<= i 5e-9)
     (* i (* 0.25 (- i)))
     (if (<= i 5000000.0)
       (/ (/ (* (* i i) (* i i)) t_0) (+ -1.0 t_0))
       0.0625))))

C

double code(double i) {
	double t_0 = (i * 2.0) * (i * 2.0);
	double tmp;
	if (i <= 5e-9) {
		tmp = i * (0.25 * -i);
	} else if (i <= 5000000.0) {
		tmp = (((i * i) * (i * i)) / t_0) / (-1.0 + t_0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (i * 2.0d0) * (i * 2.0d0)
    if (i <= 5d-9) then
        tmp = i * (0.25d0 * -i)
    else if (i <= 5000000.0d0) then
        tmp = (((i * i) * (i * i)) / t_0) / ((-1.0d0) + t_0)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double t_0 = (i * 2.0) * (i * 2.0);
	double tmp;
	if (i <= 5e-9) {
		tmp = i * (0.25 * -i);
	} else if (i <= 5000000.0) {
		tmp = (((i * i) * (i * i)) / t_0) / (-1.0 + t_0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	t_0 = (i * 2.0) * (i * 2.0)
	tmp = 0
	if i <= 5e-9:
		tmp = i * (0.25 * -i)
	elif i <= 5000000.0:
		tmp = (((i * i) * (i * i)) / t_0) / (-1.0 + t_0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	t_0 = Float64(Float64(i * 2.0) * Float64(i * 2.0))
	tmp = 0.0
	if (i <= 5e-9)
		tmp = Float64(i * Float64(0.25 * Float64(-i)));
	elseif (i <= 5000000.0)
		tmp = Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(-1.0 + t_0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	t_0 = (i * 2.0) * (i * 2.0);
	tmp = 0.0;
	if (i <= 5e-9)
		tmp = i * (0.25 * -i);
	elseif (i <= 5000000.0)
		tmp = (((i * i) * (i * i)) / t_0) / (-1.0 + t_0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] * N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 5e-9], N[(i * N[(0.25 * (-i)), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5000000.0], N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 3 regimes

2. if i < 5.0000000000000001e-9

   1. Initial program 32.3%

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

   3. Simplified 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. fma-neg 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. *-commutative 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-neg-in 100.0%

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

      15. metadata-eval 100.0%

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

      16. swap-sqr 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in i around 0 100.0%

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

   if 5.0000000000000001e-9 < i < 5e6

   1. Initial program 99.4%

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

   if 5e6 < i

   1. Initial program 21.4%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in i around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{-9}:\\ \;\;\;\;i \cdot \left(0.25 \cdot \left(-i\right)\right)\\ \mathbf{elif}\;i \leq 5000000:\\ \;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right)}}{-1 + \left(i \cdot 2\right) \cdot \left(i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5000000:\\ \;\;\;\;0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 5000000.0) (* 0.25 (* i (/ i (fma i (* i 4.0) -1.0)))) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 5000000.0) {
		tmp = 0.25 * (i * (i / fma(i, (i * 4.0), -1.0)));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 5000000.0)
		tmp = Float64(0.25 * Float64(i * Float64(i / fma(i, Float64(i * 4.0), -1.0))));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 5000000.0], N[(0.25 * N[(i * N[(i / N[(i * N[(i * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 5e6

   1. Initial program 35.6%

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

   3. Simplified 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

   if 5e6 < i

   1. Initial program 21.4%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in i around inf 100.0%

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

Alternative 3: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(i \cdot i\right)\\ \mathbf{if}\;i \leq 5 \cdot 10^{-9}:\\ \;\;\;\;i \cdot \left(0.25 \cdot \left(-i\right)\right)\\ \mathbf{elif}\;i \leq 5000000:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0 \cdot \left(-1 + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* 4.0 (* i i))))
   (if (<= i 5e-9)
     (* i (* 0.25 (- i)))
     (if (<= i 5000000.0)
       (/ (* (* i i) (* i i)) (* t_0 (+ -1.0 t_0)))
       0.0625))))

C

double code(double i) {
	double t_0 = 4.0 * (i * i);
	double tmp;
	if (i <= 5e-9) {
		tmp = i * (0.25 * -i);
	} else if (i <= 5000000.0) {
		tmp = ((i * i) * (i * i)) / (t_0 * (-1.0 + t_0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (i * i)
    if (i <= 5d-9) then
        tmp = i * (0.25d0 * -i)
    else if (i <= 5000000.0d0) then
        tmp = ((i * i) * (i * i)) / (t_0 * ((-1.0d0) + t_0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double t_0 = 4.0 * (i * i);
	double tmp;
	if (i <= 5e-9) {
		tmp = i * (0.25 * -i);
	} else if (i <= 5000000.0) {
		tmp = ((i * i) * (i * i)) / (t_0 * (-1.0 + t_0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	t_0 = 4.0 * (i * i)
	tmp = 0
	if i <= 5e-9:
		tmp = i * (0.25 * -i)
	elif i <= 5000000.0:
		tmp = ((i * i) * (i * i)) / (t_0 * (-1.0 + t_0))
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	t_0 = Float64(4.0 * Float64(i * i))
	tmp = 0.0
	if (i <= 5e-9)
		tmp = Float64(i * Float64(0.25 * Float64(-i)));
	elseif (i <= 5000000.0)
		tmp = Float64(Float64(Float64(i * i) * Float64(i * i)) / Float64(t_0 * Float64(-1.0 + t_0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	t_0 = 4.0 * (i * i);
	tmp = 0.0;
	if (i <= 5e-9)
		tmp = i * (0.25 * -i);
	elseif (i <= 5000000.0)
		tmp = ((i * i) * (i * i)) / (t_0 * (-1.0 + t_0));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := Block[{t$95$0 = N[(4.0 * N[(i * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 5e-9], N[(i * N[(0.25 * (-i)), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5000000.0], N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 3 regimes

2. if i < 5.0000000000000001e-9

   1. Initial program 32.3%

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

   3. Simplified 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. fma-neg 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. *-commutative 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-neg-in 100.0%

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

      15. metadata-eval 100.0%

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

      16. swap-sqr 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in i around 0 100.0%

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

   if 5.0000000000000001e-9 < i < 5e6

   1. Initial program 99.4%

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

      1. associate-/l/ 99.2%

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

      2. sub-neg 99.2%

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

      3. swap-sqr 99.2%

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

      4. metadata-eval 99.2%

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

      5. metadata-eval 99.2%

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

      6. swap-sqr 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   if 5e6 < i

   1. Initial program 21.4%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in i around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{-9}:\\ \;\;\;\;i \cdot \left(0.25 \cdot \left(-i\right)\right)\\ \mathbf{elif}\;i \leq 5000000:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(4 \cdot \left(i \cdot i\right)\right) \cdot \left(-1 + 4 \cdot \left(i \cdot i\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 34.7%

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

   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. fma-neg 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. *-commutative 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. +-commutative 100.0%

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

      14. distribute-neg-in 100.0%

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

      15. metadata-eval 100.0%

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

      16. swap-sqr 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in i around 0 97.7%

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

   if 0.5 < i

   1. Initial program 22.7%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in i around inf 99.1%

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

4. Final simplification 98.4%

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

Alternative 5: 99.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = 0.25d0 * (i * -i)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 34.7%

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

   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 97.7%

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

      1. neg-mul-1 97.7%

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

   7. Simplified 97.7%

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

   if 0.5 < i

   1. Initial program 22.7%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in i around inf 99.1%

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

Alternative 6: 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 29.1%

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

3. Simplified 76.8%

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

5. Taylor expanded in i around inf 47.5%

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

Reproduce

herbie shell --seed 2024085 
(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)))