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

Percentage Accurate: 87.9% → 98.0%

Time: 6.7s

Alternatives: 6

Speedup: 0.8×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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: 87.9% 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: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x - x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (+ x (* y (- z x))) z) (- INFINITY))
   (fma (/ (- 1.0 y) z) x y)
   (+ y (/ (- x (* x y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * (z - x))) / z) <= -((double) INFINITY)) {
		tmp = fma(((1.0 - y) / z), x, y);
	} else {
		tmp = y + ((x - (x * y)) / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * Float64(z - x))) / z) <= Float64(-Inf))
		tmp = fma(Float64(Float64(1.0 - y) / z), x, y);
	else
		tmp = Float64(y + Float64(Float64(x - Float64(x * y)) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], (-Infinity)], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x + y), $MachinePrecision], N[(y + N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z) < -inf.0

   1. Initial program 62.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right) \cdot x + y} \]
   3. Step-by-step derivation

      1. fma-def 100.0%

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

      2. +-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. div-sub 100.0%

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

   4. Simplified 100.0%

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

   if -inf.0 < (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z)

   1. Initial program 96.1%

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

   2. Taylor expanded in x around inf 99.0%

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

   3. Taylor expanded in y around 0 99.0%

      \[\leadsto y + \frac{\color{blue}{-1 \cdot \left(y \cdot x\right) + x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. unsub-neg 99.0%

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

      4. *-commutative 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x - x \cdot y}{z}\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* y (- 1.0 (/ x z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	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) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 77.6%

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

   2. Taylor expanded in x around inf 91.7%

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

   3. Taylor expanded in y around inf 99.0%

      \[\leadsto \color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{z}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

   5. Simplified 99.0%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 99.1%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{x}{z} + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 3: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x - x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+49) (* y (- 1.0 (/ x z))) (+ y (/ (- x (* x y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+49) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + ((x - (x * y)) / z);
	}
	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) :: tmp
    if (y <= (-2.3d+49)) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + ((x - (x * y)) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+49) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + ((x - (x * y)) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.3e+49:
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + ((x - (x * y)) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+49)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(Float64(x - Float64(x * y)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.3e+49)
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + ((x - (x * y)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.3e+49], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.30000000000000002e49

   1. Initial program 62.0%

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

   2. Taylor expanded in x around inf 84.5%

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

   3. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{z}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -2.30000000000000002e49 < y

   1. Initial program 95.3%

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

   2. Taylor expanded in x around inf 98.6%

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

   3. Taylor expanded in y around 0 98.6%

      \[\leadsto y + \frac{\color{blue}{-1 \cdot \left(y \cdot x\right) + x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. *-commutative 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x - x \cdot y}{z}\\ \end{array} \]

Alternative 4: 61.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-63}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.04:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.6e-63) y (if (<= y 0.04) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.6e-63) {
		tmp = y;
	} else if (y <= 0.04) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	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) :: tmp
    if (y <= (-2.6d-63)) then
        tmp = y
    else if (y <= 0.04d0) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.6e-63) {
		tmp = y;
	} else if (y <= 0.04) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.6e-63:
		tmp = y
	elif y <= 0.04:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.6e-63)
		tmp = y;
	elseif (y <= 0.04)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.6e-63)
		tmp = y;
	elseif (y <= 0.04)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.6e-63], y, If[LessEqual[y, 0.04], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6000000000000001e-63 or 0.0400000000000000008 < y

   1. Initial program 79.7%

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

   2. Taylor expanded in x around 0 56.0%

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

   if -2.6000000000000001e-63 < y < 0.0400000000000000008

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-63}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 0.04:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5: 78.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

2. Taylor expanded in z around inf 69.6%

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

3. Taylor expanded in x around 0 78.8%

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

   1. +-commutative 78.8%

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

5. Simplified 78.8%

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

6. Final simplification 78.8%

   \[\leadsto y + \frac{x}{z} \]

Alternative 6: 40.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 88.9%

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

2. Taylor expanded in x around 0 41.3%

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

3. Final simplification 41.3%

   \[\leadsto y \]

Developer target: 94.0% accurate, 0.8× 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 2023173 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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