Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 88.1% → 99.8%

Time: 1.1s

Alternatives: 5

Speedup: 0.7×

• 20.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + (y * (z - x))) / z
end function

Java

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

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end

Wolfram

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

Initial Program: 88.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + (y * (z - x))) / z
end function

Java

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

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{z - x}{z}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 60000000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ (- z x) z))))
   (if (<= y -1.25e+19)
     t_0
     (if (<= y 60000000.0) (/ (+ x (* y (- z x))) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * ((z - x) / z);
	double tmp;
	if (y <= -1.25e+19) {
		tmp = t_0;
	} else if (y <= 60000000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * ((z - x) / z)
    if (y <= (-1.25d+19)) then
        tmp = t_0
    else if (y <= 60000000.0d0) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * ((z - x) / z);
	double tmp;
	if (y <= -1.25e+19) {
		tmp = t_0;
	} else if (y <= 60000000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * ((z - x) / z)
	tmp = 0
	if y <= -1.25e+19:
		tmp = t_0
	elif y <= 60000000.0:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(z - x) / z))
	tmp = 0.0
	if (y <= -1.25e+19)
		tmp = t_0;
	elseif (y <= 60000000.0)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * ((z - x) / z);
	tmp = 0.0;
	if (y <= -1.25e+19)
		tmp = t_0;
	elseif (y <= 60000000.0)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e+19], t$95$0, If[LessEqual[y, 60000000.0], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e19 or 6e7 < y

   1. Initial program 71.3%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 9.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{z} \]

   8. Applied rewrites 9.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.9%

       \[\leadsto y \cdot \frac{z - x}{z} \]

   if -1.25e19 < y < 6e7

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 11.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-298}:\\ \;\;\;\;y \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z x)))) (if (<= z -1.15e-298) (* y t_0) (+ x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (z - x);
	double tmp;
	if (z <= -1.15e-298) {
		tmp = y * t_0;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (z - x)
    if (z <= (-1.15d-298)) then
        tmp = y * t_0
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (z - x);
	double tmp;
	if (z <= -1.15e-298) {
		tmp = y * t_0;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (z - x)
	tmp = 0
	if z <= -1.15e-298:
		tmp = y * t_0
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(z - x))
	tmp = 0.0
	if (z <= -1.15e-298)
		tmp = Float64(y * t_0);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (z - x);
	tmp = 0.0;
	if (z <= -1.15e-298)
		tmp = y * t_0;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e-298], N[(y * t$95$0), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e-298

   1. Initial program 88.3%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{z} \]

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 6.8%

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

   if -1.15e-298 < z

   1. Initial program 89.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.6%

      \[\leadsto \color{blue}{x + y \cdot \left(z - x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{z - x}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* y (/ (- z x) z)))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return y * ((z - x) / z)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.8%

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

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{z} \]

8. Applied rewrites 6.0%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 64.0%

    \[\leadsto y \cdot \frac{z - x}{z} \]
12. Add Preprocessing

Alternative 4: 7.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.8%

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

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{z} \]

8. Applied rewrites 6.0%

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

Alternative 5: 3.2% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- z x))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return z - x

Julia

function code(x, y, z)
	return Float64(z - x)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z - x), $MachinePrecision]

Derivation

1. Initial program 88.7%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 3.1%

   \[\leadsto z - \color{blue}{x} \]
9. Add Preprocessing

Developer Target 1: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ y (/ x z)) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	return (y + (x / z)) - (y / (z / x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y + (x / z)) - (y / (z / x))
end function

Java

public static double code(double x, double y, double z) {
	return (y + (x / z)) - (y / (z / x));
}

Python

def code(x, y, z):
	return (y + (x / z)) - (y / (z / x))

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(x / z)) - Float64(y / Float64(z / x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + (x / z)) - (y / (z / x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024321 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (- (+ y (/ x z)) (/ y (/ z x))))

  (/ (+ x (* y (- z x))) z))