Octave 3.8, jcobi/4, as called

Percentage Accurate: 32.1% → 99.9%
Time: 4.8s
Alternatives: 6
Speedup: 71.0×

Specification

?
\[i > 0\]
\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 - 1} \end{array} \end{array} \]
(FPCore (i)
 :precision binary32
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))
float code(float i) {
	float t_0 = (2.0f * i) * (2.0f * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0f);
}

real(4) function code(i)
    real(4), intent (in) :: i
    real(4) :: t_0
    t_0 = (2.0e0 * i) * (2.0e0 * i)
    code = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0e0)
end function

function code(i)
	t_0 = Float32(Float32(Float32(2.0) * i) * Float32(Float32(2.0) * i))
	return Float32(Float32(Float32(Float32(i * i) * Float32(i * i)) / t_0) / Float32(t_0 - Float32(1.0)))
end

function tmp = code(i)
	t_0 = (single(2.0) * i) * (single(2.0) * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - single(1.0));
end

\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}

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 6 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 32.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 - 1} \end{array} \end{array} \]
(FPCore (i)
 :precision binary32
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))
float code(float i) {
	float t_0 = (2.0f * i) * (2.0f * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0f);
}

real(4) function code(i)
    real(4), intent (in) :: i
    real(4) :: t_0
    t_0 = (2.0e0 * i) * (2.0e0 * i)
    code = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0e0)
end function

function code(i)
	t_0 = Float32(Float32(Float32(2.0) * i) * Float32(Float32(2.0) * i))
	return Float32(Float32(Float32(Float32(i * i) * Float32(i * i)) / t_0) / Float32(t_0 - Float32(1.0)))
end

function tmp = code(i)
	t_0 = (single(2.0) * i) * (single(2.0) * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - single(1.0));
end

\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}

Alternative 1: 99.9% accurate, 1.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2000:\\ \;\;\;\;\left(i \cdot i\right) \cdot \frac{0.25}{4 \cdot \left(i \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]
(FPCore (i)
 :precision binary32
 (if (<= i 2000.0) (* (* i i) (/ 0.25 (- (* 4.0 (* i i)) 1.0))) 0.0625))
float code(float i) {
	float tmp;
	if (i <= 2000.0f) {
		tmp = (i * i) * (0.25f / ((4.0f * (i * i)) - 1.0f));
	} else {
		tmp = 0.0625f;
	}
	return tmp;
}

real(4) function code(i)
    real(4), intent (in) :: i
    real(4) :: tmp
    if (i <= 2000.0e0) then
        tmp = (i * i) * (0.25e0 / ((4.0e0 * (i * i)) - 1.0e0))
    else
        tmp = 0.0625e0
    end if
    code = tmp
end function

function code(i)
	tmp = Float32(0.0)
	if (i <= Float32(2000.0))
		tmp = Float32(Float32(i * i) * Float32(Float32(0.25) / Float32(Float32(Float32(4.0) * Float32(i * i)) - Float32(1.0))));
	else
		tmp = Float32(0.0625);
	end
	return tmp
end

function tmp_2 = code(i)
	tmp = single(0.0);
	if (i <= single(2000.0))
		tmp = (i * i) * (single(0.25) / ((single(4.0) * (i * i)) - single(1.0)));
	else
		tmp = single(0.0625);
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;i \leq 2000:\\
\;\;\;\;\left(i \cdot i\right) \cdot \frac{0.25}{4 \cdot \left(i \cdot i\right) - 1}\\

\mathbf{else}:\\
\;\;\;\;0.0625\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if i < 2e3

    1. Initial program 46.2%

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

      1. lift-/.f32N/A

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

      2. lift-/.f32N/A

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

      3. associate-/l/N/A

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

      4. lift-*.f32N/A

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

      5. associate-/l*N/A

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

      6. lower-*.f32N/A

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

      7. lower-/.f32N/A

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

      8. lift-*.f32N/A

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

      9. lift-*.f32N/A

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

      10. associate-*r*N/A

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

      11. associate-*l*N/A

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

      12. lower-*.f32N/A

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

    4. Applied rewrites60.9%

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

    6. Applied rewrites99.4%

      \[\leadsto \color{blue}{\frac{i \cdot i}{1 - -2 \cdot i} \cdot \frac{0.25}{i \cdot 2 - 1}} \]
    7. Step-by-step derivation

      1. lift-*.f32N/A

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

      2. lift-/.f32N/A

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

      3. lift-/.f32N/A

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

      4. frac-timesN/A

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

      5. associate-/l*N/A

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

      6. lift--.f32N/A

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

      7. flip3--N/A

        \[\leadsto \left(i \cdot i\right) \cdot \frac{\frac{1}{4}}{\left(1 - -2 \cdot i\right) \cdot \color{blue}{\frac{{\left(i \cdot 2\right)}^{3} - {1}^{3}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + \left(1 \cdot 1 + \left(i \cdot 2\right) \cdot 1\right)}}} \]

      8. associate-*r/N/A

        \[\leadsto \left(i \cdot i\right) \cdot \frac{\frac{1}{4}}{\color{blue}{\frac{\left(1 - -2 \cdot i\right) \cdot \left({\left(i \cdot 2\right)}^{3} - {1}^{3}\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + \left(1 \cdot 1 + \left(i \cdot 2\right) \cdot 1\right)}}} \]

      9. lift--.f32N/A

        \[\leadsto \left(i \cdot i\right) \cdot \frac{\frac{1}{4}}{\frac{\color{blue}{\left(1 - -2 \cdot i\right)} \cdot \left({\left(i \cdot 2\right)}^{3} - {1}^{3}\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + \left(1 \cdot 1 + \left(i \cdot 2\right) \cdot 1\right)}} \]

      10. lift-*.f32N/A

        \[\leadsto \left(i \cdot i\right) \cdot \frac{\frac{1}{4}}{\frac{\left(1 - \color{blue}{-2 \cdot i}\right) \cdot \left({\left(i \cdot 2\right)}^{3} - {1}^{3}\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + \left(1 \cdot 1 + \left(i \cdot 2\right) \cdot 1\right)}} \]

      11. fp-cancel-sub-sign-invN/A

        \[\leadsto \left(i \cdot i\right) \cdot \frac{\frac{1}{4}}{\frac{\color{blue}{\left(1 + \left(\mathsf{neg}\left(-2\right)\right) \cdot i\right)} \cdot \left({\left(i \cdot 2\right)}^{3} - {1}^{3}\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + \left(1 \cdot 1 + \left(i \cdot 2\right) \cdot 1\right)}} \]

      12. metadata-evalN/A

        \[\leadsto \left(i \cdot i\right) \cdot \frac{\frac{1}{4}}{\frac{\left(1 + \color{blue}{2} \cdot i\right) \cdot \left({\left(i \cdot 2\right)}^{3} - {1}^{3}\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + \left(1 \cdot 1 + \left(i \cdot 2\right) \cdot 1\right)}} \]

      13. +-commutativeN/A

        \[\leadsto \left(i \cdot i\right) \cdot \frac{\frac{1}{4}}{\frac{\color{blue}{\left(2 \cdot i + 1\right)} \cdot \left({\left(i \cdot 2\right)}^{3} - {1}^{3}\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + \left(1 \cdot 1 + \left(i \cdot 2\right) \cdot 1\right)}} \]

    8. Applied rewrites99.8%

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

    if 2e3 < i

    1. Initial program 15.2%

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

    3. Taylor expanded in i around inf

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

      1. Applied rewrites100.0%

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

    Alternative 2: 98.1% accurate, 2.4× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(i \cdot -0.25 - i \cdot \left(i \cdot i\right)\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \end{array} \end{array} \]
    (FPCore (i)
 :precision binary32
 (if (<= i 0.5)
   (* (- (* i -0.25) (* i (* i i))) i)
   (+ (/ 0.015625 (* i i)) 0.0625)))
    float code(float i) {
	float tmp;
	if (i <= 0.5f) {
		tmp = ((i * -0.25f) - (i * (i * i))) * i;
	} else {
		tmp = (0.015625f / (i * i)) + 0.0625f;
	}
	return tmp;
}

    real(4) function code(i)
    real(4), intent (in) :: i
    real(4) :: tmp
    if (i <= 0.5e0) then
        tmp = ((i * (-0.25e0)) - (i * (i * i))) * i
    else
        tmp = (0.015625e0 / (i * i)) + 0.0625e0
    end if
    code = tmp
end function

    function code(i)
	tmp = Float32(0.0)
	if (i <= Float32(0.5))
		tmp = Float32(Float32(Float32(i * Float32(-0.25)) - Float32(i * Float32(i * i))) * i);
	else
		tmp = Float32(Float32(Float32(0.015625) / Float32(i * i)) + Float32(0.0625));
	end
	return tmp
end

    function tmp_2 = code(i)
	tmp = single(0.0);
	if (i <= single(0.5))
		tmp = ((i * single(-0.25)) - (i * (i * i))) * i;
	else
		tmp = (single(0.015625) / (i * i)) + single(0.0625);
	end
	tmp_2 = tmp;
end

    \begin{array}{l}

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

\mathbf{else}:\\
\;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\


\end{array}
\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if i < 0.5

      1. Initial program 38.7%

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

      3. Taylor expanded in i around 0

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

        1. *-commutativeN/A

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

        2. lower-*.f32N/A

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

      5. Applied rewrites98.1%

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

      7. Applied rewrites98.1%

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

      8. Taylor expanded in i around 0

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

        1. Applied rewrites97.6%

          \[\leadsto \left(\left(-0.25 - i \cdot i\right) \cdot i\right) \cdot i \]
        2. Step-by-step derivation

          1. Applied rewrites97.6%

            \[\leadsto \left(i \cdot -0.25 + i \cdot \left(\left(-i\right) \cdot i\right)\right) \cdot i \]

          if 0.5 < i

          1. Initial program 24.6%

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

          3. Taylor expanded in i around inf

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

            1. +-commutativeN/A

              \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

            2. lower-+.f32N/A

              \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

            3. associate-*r/N/A

              \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

            4. metadata-evalN/A

              \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

            5. lower-/.f32N/A

              \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

            6. unpow2N/A

              \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

            7. lower-*.f3297.3

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

          5. Applied rewrites97.3%

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

        4. Final simplification97.4%

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

        Alternative 3: 98.1% accurate, 2.7× speedup?

        \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(\left(-0.25 - i \cdot i\right) \cdot i\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \end{array} \end{array} \]
        (FPCore (i)
 :precision binary32
 (if (<= i 0.5) (* (* (- -0.25 (* i i)) i) i) (+ (/ 0.015625 (* i i)) 0.0625)))
        float code(float i) {
	float tmp;
	if (i <= 0.5f) {
		tmp = ((-0.25f - (i * i)) * i) * i;
	} else {
		tmp = (0.015625f / (i * i)) + 0.0625f;
	}
	return tmp;
}

        real(4) function code(i)
    real(4), intent (in) :: i
    real(4) :: tmp
    if (i <= 0.5e0) then
        tmp = (((-0.25e0) - (i * i)) * i) * i
    else
        tmp = (0.015625e0 / (i * i)) + 0.0625e0
    end if
    code = tmp
end function

        function code(i)
	tmp = Float32(0.0)
	if (i <= Float32(0.5))
		tmp = Float32(Float32(Float32(Float32(-0.25) - Float32(i * i)) * i) * i);
	else
		tmp = Float32(Float32(Float32(0.015625) / Float32(i * i)) + Float32(0.0625));
	end
	return tmp
end

        function tmp_2 = code(i)
	tmp = single(0.0);
	if (i <= single(0.5))
		tmp = ((single(-0.25) - (i * i)) * i) * i;
	else
		tmp = (single(0.015625) / (i * i)) + single(0.0625);
	end
	tmp_2 = tmp;
end

        \begin{array}{l}

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

\mathbf{else}:\\
\;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\


\end{array}
\end{array}
        Derivation
        1. Split input into 2 regimes

        2. if i < 0.5

          1. Initial program 38.7%

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

          3. Taylor expanded in i around 0

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

            1. *-commutativeN/A

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

            2. lower-*.f32N/A

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

          5. Applied rewrites98.1%

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

          7. Applied rewrites98.1%

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

          8. Taylor expanded in i around 0

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

            1. Applied rewrites97.6%

              \[\leadsto \left(\left(-0.25 - i \cdot i\right) \cdot i\right) \cdot i \]

            if 0.5 < i

            1. Initial program 24.6%

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

            3. Taylor expanded in i around inf

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

              1. +-commutativeN/A

                \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

              2. lower-+.f32N/A

                \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

              3. associate-*r/N/A

                \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

              4. metadata-evalN/A

                \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

              5. lower-/.f32N/A

                \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

              6. unpow2N/A

                \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

              7. lower-*.f3297.3

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

            5. Applied rewrites97.3%

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

          Alternative 4: 97.2% accurate, 2.8× speedup?

          \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(\left(-0.25 - i \cdot i\right) \cdot i\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]
          (FPCore (i)
 :precision binary32
 (if (<= i 0.5) (* (* (- -0.25 (* i i)) i) i) 0.0625))
          float code(float i) {
	float tmp;
	if (i <= 0.5f) {
		tmp = ((-0.25f - (i * i)) * i) * i;
	} else {
		tmp = 0.0625f;
	}
	return tmp;
}

          real(4) function code(i)
    real(4), intent (in) :: i
    real(4) :: tmp
    if (i <= 0.5e0) then
        tmp = (((-0.25e0) - (i * i)) * i) * i
    else
        tmp = 0.0625e0
    end if
    code = tmp
end function

          function code(i)
	tmp = Float32(0.0)
	if (i <= Float32(0.5))
		tmp = Float32(Float32(Float32(Float32(-0.25) - Float32(i * i)) * i) * i);
	else
		tmp = Float32(0.0625);
	end
	return tmp
end

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

          \begin{array}{l}

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

\mathbf{else}:\\
\;\;\;\;0.0625\\


\end{array}
\end{array}
          Derivation
          1. Split input into 2 regimes

          2. if i < 0.5

            1. Initial program 38.7%

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

            3. Taylor expanded in i around 0

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

              1. *-commutativeN/A

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

              2. lower-*.f32N/A

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

            5. Applied rewrites98.1%

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

            7. Applied rewrites98.1%

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

            8. Taylor expanded in i around 0

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

              1. Applied rewrites97.6%

                \[\leadsto \left(\left(-0.25 - i \cdot i\right) \cdot i\right) \cdot i \]

              if 0.5 < i

              1. Initial program 24.6%

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

              3. Taylor expanded in i around inf

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

                1. Applied rewrites95.1%

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

              Alternative 5: 96.4% accurate, 4.2× speedup?

              \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(-0.25 \cdot i\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]
              (FPCore (i) :precision binary32 (if (<= i 0.5) (* (* -0.25 i) i) 0.0625))
              float code(float i) {
	float tmp;
	if (i <= 0.5f) {
		tmp = (-0.25f * i) * i;
	} else {
		tmp = 0.0625f;
	}
	return tmp;
}

              real(4) function code(i)
    real(4), intent (in) :: i
    real(4) :: tmp
    if (i <= 0.5e0) then
        tmp = ((-0.25e0) * i) * i
    else
        tmp = 0.0625e0
    end if
    code = tmp
end function

              function code(i)
	tmp = Float32(0.0)
	if (i <= Float32(0.5))
		tmp = Float32(Float32(Float32(-0.25) * i) * i);
	else
		tmp = Float32(0.0625);
	end
	return tmp
end

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

              \begin{array}{l}

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

\mathbf{else}:\\
\;\;\;\;0.0625\\


\end{array}
\end{array}
              Derivation
              1. Split input into 2 regimes

              2. if i < 0.5

                1. Initial program 38.7%

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

                3. Taylor expanded in i around 0

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

                  1. metadata-evalN/A

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

                  2. associate-*r*N/A

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

                  3. mul-1-negN/A

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

                  4. unpow2N/A

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

                  5. distribute-lft-neg-inN/A

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

                  6. associate-*r*N/A

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

                  7. lower-*.f32N/A

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

                  8. mul-1-negN/A

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

                  9. associate-*r*N/A

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

                  10. metadata-evalN/A

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

                  11. lower-*.f3296.2

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

                5. Applied rewrites96.2%

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

                if 0.5 < i

                1. Initial program 24.6%

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

                3. Taylor expanded in i around inf

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

                  1. Applied rewrites95.1%

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

                Alternative 6: 51.7% accurate, 71.0× speedup?

                \[\begin{array}{l} \\ 0.0625 \end{array} \]
                (FPCore (i) :precision binary32 0.0625)
                float code(float i) {
	return 0.0625f;
}

                real(4) function code(i)
    real(4), intent (in) :: i
    code = 0.0625e0
end function

                function code(i)
	return Float32(0.0625)
end

                function tmp = code(i)
	tmp = single(0.0625);
end

                \begin{array}{l}

\\
0.0625
\end{array}
                Derivation

                1. Initial program 30.8%

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

                3. Taylor expanded in i around inf

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

                  1. Applied rewrites55.5%

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

                  Reproduce

                  ?
                  herbie shell --seed 2024341 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary32
  :pre (> i 0.0)
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))