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

Percentage Accurate: 99.9% → 100.0%

Time: 3.0s

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. remove-double-neg 99.9%

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

   4. distribute-frac-neg 99.9%

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

   5. sub-neg 99.9%

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

   6. neg-mul-1 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-/l* 99.9%

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

   9. *-rgt-identity 99.9%

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

   10. metadata-eval 99.9%

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

   11. distribute-lft-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. metadata-eval 100.0%

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

   14. metadata-eval 100.0%

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

   15. metadata-eval 100.0%

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

   16. distribute-frac-neg 100.0%

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

   17. neg-mul-1 100.0%

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

   18. *-commutative 100.0%

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

   19. associate-/l* 100.0%

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

   20. 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 74.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+80} \lor \neg \left(y \leq -7.2 \cdot 10^{-53}\right) \land \left(y \leq -2.8 \cdot 10^{-77} \lor \neg \left(y \leq -1 \cdot 10^{-125}\right) \land \left(y \leq -4.1 \cdot 10^{-144} \lor \neg \left(y \leq 85000000\right)\right)\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.15e+80)
         (and (not (<= y -7.2e-53))
              (or (<= y -2.8e-77)
                  (and (not (<= y -1e-125))
                       (or (<= y -4.1e-144) (not (<= y 85000000.0)))))))
   (* y -0.5)
   (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.15e+80) || (!(y <= -7.2e-53) && ((y <= -2.8e-77) || (!(y <= -1e-125) && ((y <= -4.1e-144) || !(y <= 85000000.0)))))) {
		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 ((y <= (-3.15d+80)) .or. (.not. (y <= (-7.2d-53))) .and. (y <= (-2.8d-77)) .or. (.not. (y <= (-1d-125))) .and. (y <= (-4.1d-144)) .or. (.not. (y <= 85000000.0d0))) 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 ((y <= -3.15e+80) || (!(y <= -7.2e-53) && ((y <= -2.8e-77) || (!(y <= -1e-125) && ((y <= -4.1e-144) || !(y <= 85000000.0)))))) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.15e+80) or (not (y <= -7.2e-53) and ((y <= -2.8e-77) or (not (y <= -1e-125) and ((y <= -4.1e-144) or not (y <= 85000000.0))))):
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.15e+80) || (!(y <= -7.2e-53) && ((y <= -2.8e-77) || (!(y <= -1e-125) && ((y <= -4.1e-144) || !(y <= 85000000.0))))))
		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 ((y <= -3.15e+80) || (~((y <= -7.2e-53)) && ((y <= -2.8e-77) || (~((y <= -1e-125)) && ((y <= -4.1e-144) || ~((y <= 85000000.0)))))))
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.15e+80], And[N[Not[LessEqual[y, -7.2e-53]], $MachinePrecision], Or[LessEqual[y, -2.8e-77], And[N[Not[LessEqual[y, -1e-125]], $MachinePrecision], Or[LessEqual[y, -4.1e-144], N[Not[LessEqual[y, 85000000.0]], $MachinePrecision]]]]]], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.14999999999999989e80 or -7.1999999999999998e-53 < y < -2.7999999999999999e-77 or -1.00000000000000001e-125 < y < -4.1e-144 or 8.5e7 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 83.1%

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

   if -3.14999999999999989e80 < y < -7.1999999999999998e-53 or -2.7999999999999999e-77 < y < -1.00000000000000001e-125 or -4.1e-144 < y < 8.5e7

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 81.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+80} \lor \neg \left(y \leq -7.2 \cdot 10^{-53}\right) \land \left(y \leq -2.8 \cdot 10^{-77} \lor \neg \left(y \leq -1 \cdot 10^{-125}\right) \land \left(y \leq -4.1 \cdot 10^{-144} \lor \neg \left(y \leq 85000000\right)\right)\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing
3. Add Preprocessing

Alternative 4: 51.8% 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. Add Preprocessing

3. Taylor expanded in x around 0 53.3%

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

4. Final simplification 53.3%

   \[\leadsto y \cdot -0.5 \]
5. Add Preprocessing

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 2024091 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"
  :precision binary64

  :alt
  (- (* 1.5 x) (* 0.5 y))

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