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

Percentage Accurate: 99.9% → 100.0%

Time: 4.8s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * 1.5 + 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 99.9%

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

   1. +-commutative 99.9%

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

   2. metadata-eval 99.9%

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

   3. cancel-sign-sub-inv 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. fmm-def 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

   9. *-commutative 100.0%

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

5. Simplified 100.0%

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

Alternative 2: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+68} \lor \neg \left(x \leq 2.15 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;x + -0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.2e+68) (not (<= x 2.15e-91))) (* x 1.5) (+ x (* -0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.2e+68) || !(x <= 2.15e-91)) {
		tmp = x * 1.5;
	} else {
		tmp = x + (-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 ((x <= (-2.2d+68)) .or. (.not. (x <= 2.15d-91))) then
        tmp = x * 1.5d0
    else
        tmp = x + ((-0.5d0) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.2e+68) || !(x <= 2.15e-91)) {
		tmp = x * 1.5;
	} else {
		tmp = x + (-0.5 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.2e+68) or not (x <= 2.15e-91):
		tmp = x * 1.5
	else:
		tmp = x + (-0.5 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.2e+68) || !(x <= 2.15e-91))
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(x + Float64(-0.5 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.2e+68) || ~((x <= 2.15e-91)))
		tmp = x * 1.5;
	else
		tmp = x + (-0.5 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.2e+68], N[Not[LessEqual[x, 2.15e-91]], $MachinePrecision]], N[(x * 1.5), $MachinePrecision], N[(x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.19999999999999987e68 or 2.15e-91 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 84.6%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative 84.6%

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

   5. Simplified 84.6%

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

   if -2.19999999999999987e68 < x < 2.15e-91

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+68} \lor \neg \left(x \leq 2.15 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;x + -0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{-50} \lor \neg \left(x \leq 1.12 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.06e-50) (not (<= x 1.12e-91))) (* x 1.5) (* -0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.06e-50) || !(x <= 1.12e-91)) {
		tmp = x * 1.5;
	} 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 ((x <= (-1.06d-50)) .or. (.not. (x <= 1.12d-91))) then
        tmp = x * 1.5d0
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.06e-50) || !(x <= 1.12e-91)) {
		tmp = x * 1.5;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.06e-50) or not (x <= 1.12e-91):
		tmp = x * 1.5
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.06e-50) || !(x <= 1.12e-91))
		tmp = Float64(x * 1.5);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.06e-50) || ~((x <= 1.12e-91)))
		tmp = x * 1.5;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.06e-50], N[Not[LessEqual[x, 1.12e-91]], $MachinePrecision]], N[(x * 1.5), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05999999999999995e-50 or 1.12e-91 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 80.1%

      \[\leadsto \color{blue}{1.5 \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative 80.1%

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

   5. Simplified 80.1%

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

   if -1.05999999999999995e-50 < x < 1.12e-91

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{y \cdot \left(\left(0.5 \cdot \frac{x}{y} + \frac{x}{y}\right) - 0.5\right)} \]
   4. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto y \cdot \color{blue}{\left(\left(0.5 \cdot \frac{x}{y} + \frac{x}{y}\right) + \left(-0.5\right)\right)} \]

      2. distribute-lft1-in 99.9%

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

      3. metadata-eval 99.9%

         \[\leadsto y \cdot \left(\color{blue}{1.5} \cdot \frac{x}{y} + \left(-0.5\right)\right) \]

      4. metadata-eval 99.9%

         \[\leadsto y \cdot \left(1.5 \cdot \frac{x}{y} + \color{blue}{-0.5}\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(1.5 \cdot \frac{x}{y} + -0.5\right)} \]

   6. Taylor expanded in x around 0 84.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{-50} \lor \neg \left(x \leq 1.12 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. div-sub 99.9%

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

   3. associate-+l- 99.9%

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

   4. div-inv 99.9%

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

   5. metadata-eval 99.9%

      \[\leadsto x \cdot \color{blue}{0.5} - \left(\frac{y}{2} - x\right) \]

   6. div-inv 99.9%

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

   7. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 5: 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 6: 50.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 55.2%

   \[\leadsto \color{blue}{1.5 \cdot x} \]
4. Step-by-step derivation

   1. *-commutative 55.2%

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

5. Simplified 55.2%

   \[\leadsto \color{blue}{x \cdot 1.5} \]
6. Add Preprocessing

Alternative 7: 11.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 55.5%

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

4. Taylor expanded in x around inf 12.5%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Developer Target 1: 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 2024185 
(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)))