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

Percentage Accurate: 99.9% → 100.0%

Time: 3.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot y + \frac{3}{2} \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{3}{2} \cdot x + \frac{-1}{2} \cdot y} \]

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot y + x\\ \mathbf{if}\;y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* -0.5 y) x)))
   (if (<= y -5e+28) t_0 (if (<= y 8.6e-13) (* 1.5 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (-0.5 * y) + x;
	double tmp;
	if (y <= -5e+28) {
		tmp = t_0;
	} else if (y <= 8.6e-13) {
		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 = ((-0.5d0) * y) + x
    if (y <= (-5d+28)) then
        tmp = t_0
    else if (y <= 8.6d-13) 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 = (-0.5 * y) + x;
	double tmp;
	if (y <= -5e+28) {
		tmp = t_0;
	} else if (y <= 8.6e-13) {
		tmp = 1.5 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-0.5 * y) + x
	tmp = 0
	if y <= -5e+28:
		tmp = t_0
	elif y <= 8.6e-13:
		tmp = 1.5 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-0.5 * y) + x)
	tmp = 0.0
	if (y <= -5e+28)
		tmp = t_0;
	elseif (y <= 8.6e-13)
		tmp = Float64(1.5 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-0.5 * y) + x;
	tmp = 0.0;
	if (y <= -5e+28)
		tmp = t_0;
	elseif (y <= 8.6e-13)
		tmp = 1.5 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.5 * y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5e+28], t$95$0, If[LessEqual[y, 8.6e-13], N[(1.5 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999957e28 or 8.5999999999999997e-13 < y

   1. Initial program 100.0%

      \[x + \frac{x - y}{2} \]
   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. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -4.99999999999999957e28 < y < 8.5999999999999997e-13

   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 81.1

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

   5. Applied rewrites 81.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;-0.5 \cdot y + x\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+44}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e+44) (* -0.5 y) (if (<= y 1.1e-12) (* 1.5 x) (* -0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e+44) {
		tmp = -0.5 * y;
	} else if (y <= 1.1e-12) {
		tmp = 1.5 * x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9d+44)) then
        tmp = (-0.5d0) * y
    else if (y <= 1.1d-12) then
        tmp = 1.5d0 * x
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9e+44) {
		tmp = -0.5 * y;
	} else if (y <= 1.1e-12) {
		tmp = 1.5 * x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e+44:
		tmp = -0.5 * y
	elif y <= 1.1e-12:
		tmp = 1.5 * x
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e+44)
		tmp = Float64(-0.5 * y);
	elseif (y <= 1.1e-12)
		tmp = Float64(1.5 * x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e+44)
		tmp = -0.5 * y;
	elseif (y <= 1.1e-12)
		tmp = 1.5 * x;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e+44], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, 1.1e-12], N[(1.5 * x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -9e44 or 1.09999999999999996e-12 < y

   1. Initial program 100.0%

      \[x + \frac{x - y}{2} \]
   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 18.8

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

   5. Applied rewrites 18.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 82.0

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

   8. Applied rewrites 82.0%

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

   if -9e44 < y < 1.09999999999999996e-12

   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 80.6

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

   5. Applied rewrites 80.6%

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

Alternative 4: 99.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, -0.5, x\right)} \]
5. Add Preprocessing

Alternative 5: 49.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]
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 51.4

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

5. Applied rewrites 51.4%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 49.6

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

8. Applied rewrites 49.6%

   \[\leadsto \color{blue}{-0.5 \cdot y} \]
9. 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 2024298 
(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)))