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

Percentage Accurate: 99.9% → 100.0%

Time: 3.7s

Alternatives: 5

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. sub-neg 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-neg-frac 99.9%

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

   6. neg-mul-1 99.9%

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

   7. associate-/l* 99.8%

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

   8. associate-/r/ 99.9%

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

   9. *-commutative 99.9%

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

   10. fma-def 99.9%

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

   11. metadata-eval 99.9%

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

   12. remove-double-neg 99.9%

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

   13. neg-mul-1 99.9%

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

   14. remove-double-neg 99.9%

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

   15. neg-mul-1 99.9%

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

   16. associate-/l* 99.9%

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

   17. associate-/r/ 99.9%

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

   18. distribute-rgt-out 99.9%

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

   19. distribute-lft-neg-in 99.9%

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

   20. distribute-rgt-neg-in 99.9%

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

   21. metadata-eval 99.9%

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

   22. metadata-eval 99.9%

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

   23. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, x \cdot 1.5\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \color{blue}{y \cdot -0.5 + x \cdot 1.5} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{y \cdot -0.5 + x \cdot 1.5} \]
6. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \color{blue}{x \cdot 1.5 + y \cdot -0.5} \]

   2. fma-def 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 69.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+19} \lor \neg \left(y \leq -7.1 \cdot 10^{-47}\right) \land \left(y \leq -2.8 \cdot 10^{-141} \lor \neg \left(y \leq 5.2 \cdot 10^{-111} \lor \neg \left(y \leq 4.8 \cdot 10^{-36}\right) \land y \leq 9.4 \cdot 10^{+58}\right)\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.4e+19)
         (and (not (<= y -7.1e-47))
              (or (<= y -2.8e-141)
                  (not
                   (or (<= y 5.2e-111)
                       (and (not (<= y 4.8e-36)) (<= y 9.4e+58)))))))
   (* y -0.5)
   (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.4e+19) || (!(y <= -7.1e-47) && ((y <= -2.8e-141) || !((y <= 5.2e-111) || (!(y <= 4.8e-36) && (y <= 9.4e+58)))))) {
		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 <= (-4.4d+19)) .or. (.not. (y <= (-7.1d-47))) .and. (y <= (-2.8d-141)) .or. (.not. (y <= 5.2d-111) .or. (.not. (y <= 4.8d-36)) .and. (y <= 9.4d+58))) 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 <= -4.4e+19) || (!(y <= -7.1e-47) && ((y <= -2.8e-141) || !((y <= 5.2e-111) || (!(y <= 4.8e-36) && (y <= 9.4e+58)))))) {
		tmp = y * -0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.4e+19) or (not (y <= -7.1e-47) and ((y <= -2.8e-141) or not ((y <= 5.2e-111) or (not (y <= 4.8e-36) and (y <= 9.4e+58))))):
		tmp = y * -0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.4e+19) || (!(y <= -7.1e-47) && ((y <= -2.8e-141) || !((y <= 5.2e-111) || (!(y <= 4.8e-36) && (y <= 9.4e+58))))))
		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 <= -4.4e+19) || (~((y <= -7.1e-47)) && ((y <= -2.8e-141) || ~(((y <= 5.2e-111) || (~((y <= 4.8e-36)) && (y <= 9.4e+58)))))))
		tmp = y * -0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.4e+19], And[N[Not[LessEqual[y, -7.1e-47]], $MachinePrecision], Or[LessEqual[y, -2.8e-141], N[Not[Or[LessEqual[y, 5.2e-111], And[N[Not[LessEqual[y, 4.8e-36]], $MachinePrecision], LessEqual[y, 9.4e+58]]]], $MachinePrecision]]]], N[(y * -0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4e19 or -7.1000000000000002e-47 < y < -2.80000000000000012e-141 or 5.19999999999999965e-111 < y < 4.8e-36 or 9.39999999999999944e58 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.8%

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

   if -4.4e19 < y < -7.1000000000000002e-47 or -2.80000000000000012e-141 < y < 5.19999999999999965e-111 or 4.8e-36 < y < 9.39999999999999944e58

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 86.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+19} \lor \neg \left(y \leq -7.1 \cdot 10^{-47}\right) \land \left(y \leq -2.8 \cdot 10^{-141} \lor \neg \left(y \leq 5.2 \cdot 10^{-111} \lor \neg \left(y \leq 4.8 \cdot 10^{-36}\right) \land y \leq 9.4 \cdot 10^{+58}\right)\right):\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * -0.5), $MachinePrecision] + N[(x * 1.5), $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. sub-neg 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-neg-frac 99.9%

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

   6. neg-mul-1 99.9%

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

   7. associate-/l* 99.8%

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

   8. associate-/r/ 99.9%

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

   9. *-commutative 99.9%

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

   10. fma-def 99.9%

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

   11. metadata-eval 99.9%

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

   12. remove-double-neg 99.9%

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

   13. neg-mul-1 99.9%

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

   14. remove-double-neg 99.9%

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

   15. neg-mul-1 99.9%

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

   16. associate-/l* 99.9%

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

   17. associate-/r/ 99.9%

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

   18. distribute-rgt-out 99.9%

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

   19. distribute-lft-neg-in 99.9%

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

   20. distribute-rgt-neg-in 99.9%

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

   21. metadata-eval 99.9%

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

   22. metadata-eval 99.9%

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

   23. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, x \cdot 1.5\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \color{blue}{y \cdot -0.5 + x \cdot 1.5} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{y \cdot -0.5 + x \cdot 1.5} \]

6. Final simplification 99.9%

   \[\leadsto y \cdot -0.5 + x \cdot 1.5 \]

Alternative 5: 50.1% 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 49.5%

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

3. Final simplification 49.5%

   \[\leadsto y \cdot -0.5 \]

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 2023173 
(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)))