Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3

Percentage Accurate: 99.9% → 100.0%

Time: 5.0s

Alternatives: 5

Speedup: 1.8×

Specification

Math

\[\begin{array}{l} \\ x + \frac{x - y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

C

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((x - y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Python

def code(x, y):
	return x + ((x - y) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x - y) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{x - y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

C

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((x - y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

Python

def code(x, y):
	return x + ((x - y) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x - y) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 1.5, y \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x 1.5 (* y -0.5)))

C

double code(double x, double y) {
	return fma(x, 1.5, (y * -0.5));
}

Julia

function code(x, y)
	return fma(x, 1.5, Float64(y * -0.5))
end

Wolfram

code[x_, y_] := N[(x * 1.5 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \frac{x - y}{2}} \]

   2. lift-/.f64 N/A

      \[\leadsto x + \color{blue}{\frac{x - y}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto x + \frac{\color{blue}{x - y}}{2} \]

   4. div-sub N/A

      \[\leadsto x + \color{blue}{\left(\frac{x}{2} - \frac{y}{2}\right)} \]

   5. associate-+r- N/A

      \[\leadsto \color{blue}{\left(x + \frac{x}{2}\right) - \frac{y}{2}} \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(x + \frac{x}{2}\right) - \frac{y}{2}} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{x}{2} + x\right)} - \frac{y}{2} \]

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, x\right)} - \frac{y}{2} \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2}}, x, x\right) - \frac{y}{2} \]

   12. clear-num N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, x\right) - \color{blue}{\frac{1}{\frac{2}{y}}} \]

   13. associate-/r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, x\right) - \color{blue}{\frac{1}{2} \cdot y} \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, x\right) - \color{blue}{\frac{1}{2} \cdot y} \]

   15. metadata-eval 99.9

       \[\leadsto \mathsf{fma}\left(0.5, x, x\right) - \color{blue}{0.5} \cdot y \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, x\right) - 0.5 \cdot y} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, x\right) - \frac{1}{2} \cdot y} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, x\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

   3. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + x\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) \]

   4. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + 1\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{3}{2}} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{3}{2}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{3}{2}, \mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{3}{2}, \mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot y}\right)\right) \]

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

      \[\leadsto \mathsf{fma}\left(x, \frac{3}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y}\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{3}{2}, \color{blue}{\frac{-1}{2}} \cdot y\right) \]

   11. lower-*.f64 100.0

       \[\leadsto \mathsf{fma}\left(x, 1.5, \color{blue}{-0.5 \cdot y}\right) \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.5, -0.5 \cdot y\right)} \]

7. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, 1.5, y \cdot -0.5\right) \]
8. Add Preprocessing

Alternative 2: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot -0.5 + x\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* y -0.5) x)))
   (if (<= y -2.8e-19) t_0 (if (<= y 2.9e+41) (* 1.5 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * -0.5) + x;
	double tmp;
	if (y <= -2.8e-19) {
		tmp = t_0;
	} else if (y <= 2.9e+41) {
		tmp = 1.5 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * (-0.5d0)) + x
    if (y <= (-2.8d-19)) then
        tmp = t_0
    else if (y <= 2.9d+41) then
        tmp = 1.5d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * -0.5) + x;
	double tmp;
	if (y <= -2.8e-19) {
		tmp = t_0;
	} else if (y <= 2.9e+41) {
		tmp = 1.5 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * -0.5) + x
	tmp = 0
	if y <= -2.8e-19:
		tmp = t_0
	elif y <= 2.9e+41:
		tmp = 1.5 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * -0.5) + x)
	tmp = 0.0
	if (y <= -2.8e-19)
		tmp = t_0;
	elseif (y <= 2.9e+41)
		tmp = Float64(1.5 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * -0.5) + x;
	tmp = 0.0;
	if (y <= -2.8e-19)
		tmp = t_0;
	elseif (y <= 2.9e+41)
		tmp = 1.5 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * -0.5), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -2.8e-19], t$95$0, If[LessEqual[y, 2.9e+41], N[(1.5 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000003e-19 or 2.89999999999999988e41 < y

   1. Initial program 99.9%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto x + \color{blue}{\frac{-1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \color{blue}{y \cdot \frac{-1}{2}} \]

      2. lower-*.f64 78.9

         \[\leadsto x + \color{blue}{y \cdot -0.5} \]

   5. Applied rewrites 78.9%

      \[\leadsto x + \color{blue}{y \cdot -0.5} \]

   if -2.80000000000000003e-19 < y < 2.89999999999999988e41

   1. Initial program 99.8%

      \[x + \frac{x - y}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 73.7

         \[\leadsto \color{blue}{1.5 \cdot x} \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-19}:\\ \;\;\;\;y \cdot -0.5 + x\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5 + x\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1.5 \cdot x - 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* 1.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	return (1.5 * x) - (0.5 * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.5d0 * x) - (0.5d0 * y)
end function

Java

public static double code(double x, double y) {
	return (1.5 * x) - (0.5 * y);
}

Python

def code(x, y):
	return (1.5 * x) - (0.5 * y)

Julia

function code(x, y)
	return Float64(Float64(1.5 * x) - Float64(0.5 * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.5 * x) - (0.5 * y);
end

Wolfram

code[x_, y_] := N[(N[(1.5 * x), $MachinePrecision] - N[(0.5 * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 3/2 x) (* 1/2 y)))

  (+ x (/ (- x y) 2.0)))