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

Percentage Accurate: 99.9% → 100.0%

Time: 2.3s

Alternatives: 4

Speedup: 1.0×

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, 0.1× 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. Step-by-step derivation

   1. div-sub 99.9%

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

   2. associate-+r- 99.9%

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

   3. *-lft-identity 99.9%

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

   4. metadata-eval 99.9%

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

   5. cancel-sign-sub-inv 99.9%

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

   6. *-lft-identity 99.9%

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

   7. metadata-eval 99.9%

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

   8. associate-*r/ 99.9%

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

   9. associate-/l* 99.8%

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

   10. associate-/r/ 99.9%

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

   11. distribute-rgt-out-- 99.9%

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

   12. fma-neg 100.0%

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

   13. distribute-frac-neg 100.0%

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

   14. metadata-eval 100.0%

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

   15. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(x, 1 - \color{blue}{-0.5}, \frac{-y}{2}\right) \]

   16. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{1.5}, \frac{-y}{2}\right) \]

   17. neg-mul-1 100.0%

       \[\leadsto \mathsf{fma}\left(x, 1.5, \frac{\color{blue}{-1 \cdot y}}{2}\right) \]

   18. associate-/l* 99.9%

       \[\leadsto \mathsf{fma}\left(x, 1.5, \color{blue}{\frac{-1}{\frac{2}{y}}}\right) \]

   19. associate-/r/ 100.0%

       \[\leadsto \mathsf{fma}\left(x, 1.5, \color{blue}{\frac{-1}{2} \cdot y}\right) \]

   20. *-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(x, 1.5, \color{blue}{y \cdot \frac{-1}{2}}\right) \]

   21. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{-85}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+68}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.66e-85) (* x 1.5) (if (<= x 1.15e+68) (* y -0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.66e-85) {
		tmp = x * 1.5;
	} else if (x <= 1.15e+68) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.66d-85)) then
        tmp = x * 1.5d0
    else if (x <= 1.15d+68) then
        tmp = y * (-0.5d0)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.66e-85) {
		tmp = x * 1.5;
	} else if (x <= 1.15e+68) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.66e-85:
		tmp = x * 1.5
	elif x <= 1.15e+68:
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.66e-85)
		tmp = Float64(x * 1.5);
	elseif (x <= 1.15e+68)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.66e-85)
		tmp = x * 1.5;
	elseif (x <= 1.15e+68)
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.66e-85], N[(x * 1.5), $MachinePrecision], If[LessEqual[x, 1.15e+68], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.66e-85 or 1.15e68 < x

   1. Initial program 99.8%

      \[x + \frac{x - y}{2} \]

   2. Taylor expanded in x around inf 74.5%

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

   if -1.66e-85 < x < 1.15e68

   1. Initial program 100.0%

      \[x + \frac{x - y}{2} \]

   2. Taylor expanded in x around 0 81.1%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{-85}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+68}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 3: 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]

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]

2. Final simplification 99.9%

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

Alternative 4: 50.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]

2. Taylor expanded in x around 0 51.5%

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

3. Final simplification 51.5%

   \[\leadsto y \cdot -0.5 \]

Developer target: 99.9% accurate, 1.0× 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 2023279 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* 1.5 x) (* 0.5 y))

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