Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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 - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y / (y - z)) - (x / (y - 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 / (y - z)) - (x / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ t_1 := \frac{y - x}{y - z}\\ t_2 := \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-252}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))) (t_1 (/ (- y x) (- y z))) (t_2 (/ x (- z y))))
   (if (<= t_1 -500.0)
     t_2
     (if (<= t_1 -1e-252)
       t_0
       (if (<= t_1 1e-25) (/ x z) (if (<= t_1 2.0) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = (y - x) / (y - z);
	double t_2 = x / (z - y);
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= -1e-252) {
		tmp = t_0;
	} else if (t_1 <= 1e-25) {
		tmp = x / z;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y / (y - z)
    t_1 = (y - x) / (y - z)
    t_2 = x / (z - y)
    if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= (-1d-252)) then
        tmp = t_0
    else if (t_1 <= 1d-25) then
        tmp = x / z
    else if (t_1 <= 2.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = (y - x) / (y - z);
	double t_2 = x / (z - y);
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= -1e-252) {
		tmp = t_0;
	} else if (t_1 <= 1e-25) {
		tmp = x / z;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	t_1 = (y - x) / (y - z)
	t_2 = x / (z - y)
	tmp = 0
	if t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= -1e-252:
		tmp = t_0
	elif t_1 <= 1e-25:
		tmp = x / z
	elif t_1 <= 2.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	t_1 = Float64(Float64(y - x) / Float64(y - z))
	t_2 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= -1e-252)
		tmp = t_0;
	elseif (t_1 <= 1e-25)
		tmp = Float64(x / z);
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	t_1 = (y - x) / (y - z);
	t_2 = x / (z - y);
	tmp = 0.0;
	if (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= -1e-252)
		tmp = t_0;
	elseif (t_1 <= 1e-25)
		tmp = x / z;
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, -1e-252], t$95$0, If[LessEqual[t$95$1, 1e-25], N[(x / z), $MachinePrecision], If[LessEqual[t$95$1, 2.0], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -500 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -500 < (/.f64 (-.f64 x y) (-.f64 z y)) < -9.99999999999999943e-253 or 1.00000000000000004e-25 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y}{z - y}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(\left(z - y\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(\left(z - y\right)\right)}} \]

      4. sub-neg N/A

         \[\leadsto \frac{y}{\mathsf{neg}\left(\color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{y}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(z\right)\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \frac{y}{\color{blue}{y} + \left(\mathsf{neg}\left(z\right)\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{y}{\color{blue}{y - z}} \]

      9. lower--.f64 82.4

         \[\leadsto \frac{y}{\color{blue}{y - z}} \]

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{y}{y - z}} \]

   if -9.99999999999999943e-253 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000004e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 62.8

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -500:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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 / (z - y)) - (y / (z - y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

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

  (/ (- x y) (- z y)))