Octave 3.8, jcobi/4, as called

Percentage Accurate: 28.2% → 100.0%

Time: 4.0s

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: 28.2% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 20000000:\\ \;\;\;\;-0.25 \cdot \frac{i \cdot i}{\mathsf{fma}\left(i \cdot i, -4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 20000000.0) (* -0.25 (/ (* i i) (fma (* i i) -4.0 1.0))) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 20000000.0) {
		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 <= 20000000.0)
		tmp = Float64(-0.25 * Float64(Float64(i * i) / fma(Float64(i * i), -4.0, 1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 20000000.0], N[(-0.25 * N[(N[(i * i), $MachinePrecision] / N[(N[(i * i), $MachinePrecision] * -4.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 2e7

   1. Initial program 30.1%

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

   3. Simplified 100.0%

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

   if 2e7 < i

   1. Initial program 28.2%

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

   3. Simplified 54.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{i \cdot i}{\mathsf{fma}\left(i \cdot i, -4, 1\right)}} \]
   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 2: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(i \cdot 2\right) \cdot \left(i \cdot 2\right)\\ \mathbf{if}\;i \leq 10^{-41}:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{elif}\;i \leq 20000000:\\ \;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* i 2.0) (* i 2.0))))
   (if (<= i 1e-41)
     (* i (* i -0.25))
     (if (<= i 20000000.0)
       (/ (/ (* (* i i) (* i i)) t_0) (+ t_0 -1.0))
       0.0625))))

C

double code(double i) {
	double t_0 = (i * 2.0) * (i * 2.0);
	double tmp;
	if (i <= 1e-41) {
		tmp = i * (i * -0.25);
	} else if (i <= 20000000.0) {
		tmp = (((i * i) * (i * i)) / t_0) / (t_0 + -1.0);
	} 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 <= 1d-41) then
        tmp = i * (i * (-0.25d0))
    else if (i <= 20000000.0d0) then
        tmp = (((i * i) * (i * i)) / t_0) / (t_0 + (-1.0d0))
    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 <= 1e-41) {
		tmp = i * (i * -0.25);
	} else if (i <= 20000000.0) {
		tmp = (((i * i) * (i * i)) / t_0) / (t_0 + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	t_0 = (i * 2.0) * (i * 2.0)
	tmp = 0
	if i <= 1e-41:
		tmp = i * (i * -0.25)
	elif i <= 20000000.0:
		tmp = (((i * i) * (i * i)) / t_0) / (t_0 + -1.0)
	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 <= 1e-41)
		tmp = Float64(i * Float64(i * -0.25));
	elseif (i <= 20000000.0)
		tmp = Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 + -1.0));
	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 <= 1e-41)
		tmp = i * (i * -0.25);
	elseif (i <= 20000000.0)
		tmp = (((i * i) * (i * i)) / t_0) / (t_0 + -1.0);
	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, 1e-41], N[(i * N[(i * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 20000000.0], N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 3 regimes

2. if i < 1.00000000000000001e-41

   1. Initial program 17.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{{i}^{2} \cdot -0.25} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{i}^{2} \cdot -0.25} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 1.00000000000000001e-41 < i < 2e7

   1. Initial program 99.9%

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

   if 2e7 < i

   1. Initial program 28.2%

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

   3. Simplified 54.0%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 10^{-41}:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{elif}\;i \leq 20000000:\\ \;\;\;\;\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right)}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. 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 27.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 98.9%

      \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \color{blue}{{i}^{2} \cdot -0.25} \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{{i}^{2} \cdot -0.25} \]
   8. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]

      2. unpow2 98.9%

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

      3. associate-*r* 98.9%

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

      4. *-commutative 98.9%

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

   9. Applied egg-rr 98.9%

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

   if 0.5 < i

   1. Initial program 31.0%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in i around inf 97.6%

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

4. Final simplification 98.2%

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

Alternative 4: 50.0% 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 29.2%

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

3. Simplified 77.7%

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

5. Taylor expanded in i around inf 50.6%

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

Reproduce

herbie shell --seed 2024096 
(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)))