Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Average Error: 5.9 → 5.9

Time: 4.6s

Precision: binary64

Math

\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left(z \cdot z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, t \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+289)
   (fma x x (* y (* -4.0 (- (* z z) t))))
   (fma x x (* t (* y 4.0)))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+289) {
		tmp = fma(x, x, (y * (-4.0 * ((z * z) - t))));
	} else {
		tmp = fma(x, x, (t * (y * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+289)
		tmp = fma(x, x, Float64(y * Float64(-4.0 * Float64(Float64(z * z) - t))));
	else
		tmp = fma(x, x, Float64(t * Float64(y * 4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+289], N[(x * x + N[(y * N[(-4.0 * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.9
Target	5.8
Herbie	5.9

\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e289

   1. Initial program 0.1

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(z \cdot z - t\right) \cdot -4, x \cdot x\right)} \]

   3. Taylor expanded in y around 0 0.1

      \[\leadsto \color{blue}{\left(4 \cdot \left(y \cdot t\right) + {x}^{2}\right) - 4 \cdot \left(y \cdot {z}^{2}\right)} \]

   4. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left(z \cdot z - t\right)\right)\right)} \]

   if 2.0000000000000001e289 < (*.f64 z z)

   1. Initial program 56.0

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Simplified 56.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(z \cdot z - t\right) \cdot -4, x \cdot x\right)} \]

   3. Taylor expanded in z around 0 56.4

      \[\leadsto \color{blue}{4 \cdot \left(y \cdot t\right) + {x}^{2}} \]

   4. Simplified 56.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, t \cdot \left(4 \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left(z \cdot z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, t \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))