Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 76.1% → 76.1%
Time: 22.5s
Alternatives: 3
Speedup: 1.0×

Specification

?
\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]
\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_3 := t_2 \cdot t_2 + t_0 \cdot t_0\\ t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_5 := t_1 \cdot t_1 + t_4 \cdot t_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t_3, t_5\right)}}\\ \mathbf{if}\;t_3 \geq t_5:\\ \;\;\;\;t_6 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot t_4\\ \end{array} \end{array} \]
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (+ (* t_2 t_2) (* t_0 t_0)))
        (t_4 (* (floor h) dY.v))
        (t_5 (+ (* t_1 t_1) (* t_4 t_4)))
        (t_6 (/ 1.0 (sqrt (fmax t_3 t_5)))))
   (if (>= t_3 t_5) (* t_6 t_0) (* t_6 t_4))))
float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = (t_2 * t_2) + (t_0 * t_0);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = (t_1 * t_1) + (t_4 * t_4);
	float t_6 = 1.0f / sqrtf(fmaxf(t_3, t_5));
	float tmp;
	if (t_3 >= t_5) {
		tmp = t_6 * t_0;
	} else {
		tmp = t_6 * t_4;
	}
	return tmp;
}

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(t_3, t_5)))))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_0);
	else
		tmp = Float32(t_6 * t_4);
	end
	return tmp
end

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = (t_2 * t_2) + (t_0 * t_0);
	t_4 = floor(h) * dY_46_v;
	t_5 = (t_1 * t_1) + (t_4 * t_4);
	t_6 = single(1.0) / sqrt(max(t_3, t_5));
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_6 * t_0;
	else
		tmp = t_6 * t_4;
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\
t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\
t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\
t_3 := t_2 \cdot t_2 + t_0 \cdot t_0\\
t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\
t_5 := t_1 \cdot t_1 + t_4 \cdot t_4\\
t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t_3, t_5\right)}}\\
\mathbf{if}\;t_3 \geq t_5:\\
\;\;\;\;t_6 \cdot t_0\\

\mathbf{else}:\\
\;\;\;\;t_6 \cdot t_4\\


\end{array}
\end{array}

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 76.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_3 := t_2 \cdot t_2 + t_0 \cdot t_0\\ t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_5 := t_1 \cdot t_1 + t_4 \cdot t_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t_3, t_5\right)}}\\ \mathbf{if}\;t_3 \geq t_5:\\ \;\;\;\;t_6 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot t_4\\ \end{array} \end{array} \]
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (+ (* t_2 t_2) (* t_0 t_0)))
        (t_4 (* (floor h) dY.v))
        (t_5 (+ (* t_1 t_1) (* t_4 t_4)))
        (t_6 (/ 1.0 (sqrt (fmax t_3 t_5)))))
   (if (>= t_3 t_5) (* t_6 t_0) (* t_6 t_4))))
float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = (t_2 * t_2) + (t_0 * t_0);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = (t_1 * t_1) + (t_4 * t_4);
	float t_6 = 1.0f / sqrtf(fmaxf(t_3, t_5));
	float tmp;
	if (t_3 >= t_5) {
		tmp = t_6 * t_0;
	} else {
		tmp = t_6 * t_4;
	}
	return tmp;
}

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(t_3, t_5)))))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_0);
	else
		tmp = Float32(t_6 * t_4);
	end
	return tmp
end

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = (t_2 * t_2) + (t_0 * t_0);
	t_4 = floor(h) * dY_46_v;
	t_5 = (t_1 * t_1) + (t_4 * t_4);
	t_6 = single(1.0) / sqrt(max(t_3, t_5));
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_6 * t_0;
	else
		tmp = t_6 * t_4;
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\
t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\
t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\
t_3 := t_2 \cdot t_2 + t_0 \cdot t_0\\
t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\
t_5 := t_1 \cdot t_1 + t_4 \cdot t_4\\
t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t_3, t_5\right)}}\\
\mathbf{if}\;t_3 \geq t_5:\\
\;\;\;\;t_6 \cdot t_0\\

\mathbf{else}:\\
\;\;\;\;t_6 \cdot t_4\\


\end{array}
\end{array}

Alternative 1: 76.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_3 := t_0 \cdot t_0\\ t_4 := t_2 \cdot t_2 + t_3\\ t_5 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_6 := t_1 \cdot t_1 + t_5 \cdot t_5\\ \mathbf{if}\;t_4 \geq t_6:\\ \;\;\;\;t_0 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_3 + {dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, t_6\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_4, t_6\right)}}\\ \end{array} \end{array} \]
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (* t_0 t_0))
        (t_4 (+ (* t_2 t_2) t_3))
        (t_5 (* (floor h) dY.v))
        (t_6 (+ (* t_1 t_1) (* t_5 t_5))))
   (if (>= t_4 t_6)
     (*
      t_0
      (/ 1.0 (sqrt (fmax (+ t_3 (* (pow dX.u 2.0) (pow (floor w) 2.0))) t_6))))
     (* t_5 (/ 1.0 (sqrt (fmax t_4 t_6)))))))
float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = t_0 * t_0;
	float t_4 = (t_2 * t_2) + t_3;
	float t_5 = floorf(h) * dY_46_v;
	float t_6 = (t_1 * t_1) + (t_5 * t_5);
	float tmp;
	if (t_4 >= t_6) {
		tmp = t_0 * (1.0f / sqrtf(fmaxf((t_3 + (powf(dX_46_u, 2.0f) * powf(floorf(w), 2.0f))), t_6)));
	} else {
		tmp = t_5 * (1.0f / sqrtf(fmaxf(t_4, t_6)));
	}
	return tmp;
}

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(t_0 * t_0)
	t_4 = Float32(Float32(t_2 * t_2) + t_3)
	t_5 = Float32(floor(h) * dY_46_v)
	t_6 = Float32(Float32(t_1 * t_1) + Float32(t_5 * t_5))
	tmp = Float32(0.0)
	if (t_4 >= t_6)
		tmp = Float32(t_0 * Float32(Float32(1.0) / sqrt(((Float32(t_3 + Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))) != Float32(t_3 + Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))))) ? t_6 : ((t_6 != t_6) ? Float32(t_3 + Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))) : max(Float32(t_3 + Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))), t_6))))));
	else
		tmp = Float32(t_5 * Float32(Float32(1.0) / sqrt(((t_4 != t_4) ? t_6 : ((t_6 != t_6) ? t_4 : max(t_4, t_6))))));
	end
	return tmp
end

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = t_0 * t_0;
	t_4 = (t_2 * t_2) + t_3;
	t_5 = floor(h) * dY_46_v;
	t_6 = (t_1 * t_1) + (t_5 * t_5);
	tmp = single(0.0);
	if (t_4 >= t_6)
		tmp = t_0 * (single(1.0) / sqrt(max((t_3 + ((dX_46_u ^ single(2.0)) * (floor(w) ^ single(2.0)))), t_6)));
	else
		tmp = t_5 * (single(1.0) / sqrt(max(t_4, t_6)));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\
t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\
t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\
t_3 := t_0 \cdot t_0\\
t_4 := t_2 \cdot t_2 + t_3\\
t_5 := \left\lfloorh\right\rfloor \cdot dY.v\\
t_6 := t_1 \cdot t_1 + t_5 \cdot t_5\\
\mathbf{if}\;t_4 \geq t_6:\\
\;\;\;\;t_0 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_3 + {dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, t_6\right)}}\\

\mathbf{else}:\\
\;\;\;\;t_5 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_4, t_6\right)}}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 76.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := t_1 \cdot t_1\\ t_3 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_4 := t_3 \cdot t_3\\ t_5 := t_0 \cdot t_0 + t_4\\ t_6 := \left\lfloorw\right\rfloor \cdot dY.u\\ \mathbf{if}\;t_4 + {t_0}^{2} \geq t_2 + {t_6}^{2}:\\ \;\;\;\;t_3 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_5, t_6 \cdot t_6 + t_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_5, t_2 + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}}\\ \end{array} \end{array} \]
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* t_1 t_1))
        (t_3 (* (floor h) dX.v))
        (t_4 (* t_3 t_3))
        (t_5 (+ (* t_0 t_0) t_4))
        (t_6 (* (floor w) dY.u)))
   (if (>= (+ t_4 (pow t_0 2.0)) (+ t_2 (pow t_6 2.0)))
     (* t_3 (/ 1.0 (sqrt (fmax t_5 (+ (* t_6 t_6) t_2)))))
     (*
      t_1
      (/
       1.0
       (sqrt (fmax t_5 (+ t_2 (* (pow (floor w) 2.0) (pow dY.u 2.0))))))))))
float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = t_1 * t_1;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = t_3 * t_3;
	float t_5 = (t_0 * t_0) + t_4;
	float t_6 = floorf(w) * dY_46_u;
	float tmp;
	if ((t_4 + powf(t_0, 2.0f)) >= (t_2 + powf(t_6, 2.0f))) {
		tmp = t_3 * (1.0f / sqrtf(fmaxf(t_5, ((t_6 * t_6) + t_2))));
	} else {
		tmp = t_1 * (1.0f / sqrtf(fmaxf(t_5, (t_2 + (powf(floorf(w), 2.0f) * powf(dY_46_u, 2.0f))))));
	}
	return tmp;
}

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(t_1 * t_1)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(t_3 * t_3)
	t_5 = Float32(Float32(t_0 * t_0) + t_4)
	t_6 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (Float32(t_4 + (t_0 ^ Float32(2.0))) >= Float32(t_2 + (t_6 ^ Float32(2.0))))
		tmp = Float32(t_3 * Float32(Float32(1.0) / sqrt(((t_5 != t_5) ? Float32(Float32(t_6 * t_6) + t_2) : ((Float32(Float32(t_6 * t_6) + t_2) != Float32(Float32(t_6 * t_6) + t_2)) ? t_5 : max(t_5, Float32(Float32(t_6 * t_6) + t_2)))))));
	else
		tmp = Float32(t_1 * Float32(Float32(1.0) / sqrt(((t_5 != t_5) ? Float32(t_2 + Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) : ((Float32(t_2 + Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0)))) != Float32(t_2 + Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0))))) ? t_5 : max(t_5, Float32(t_2 + Float32((floor(w) ^ Float32(2.0)) * (dY_46_u ^ Float32(2.0))))))))));
	end
	return tmp
end

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) * dX_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = t_1 * t_1;
	t_3 = floor(h) * dX_46_v;
	t_4 = t_3 * t_3;
	t_5 = (t_0 * t_0) + t_4;
	t_6 = floor(w) * dY_46_u;
	tmp = single(0.0);
	if ((t_4 + (t_0 ^ single(2.0))) >= (t_2 + (t_6 ^ single(2.0))))
		tmp = t_3 * (single(1.0) / sqrt(max(t_5, ((t_6 * t_6) + t_2))));
	else
		tmp = t_1 * (single(1.0) / sqrt(max(t_5, (t_2 + ((floor(w) ^ single(2.0)) * (dY_46_u ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\
t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\
t_2 := t_1 \cdot t_1\\
t_3 := \left\lfloorh\right\rfloor \cdot dX.v\\
t_4 := t_3 \cdot t_3\\
t_5 := t_0 \cdot t_0 + t_4\\
t_6 := \left\lfloorw\right\rfloor \cdot dY.u\\
\mathbf{if}\;t_4 + {t_0}^{2} \geq t_2 + {t_6}^{2}:\\
\;\;\;\;t_3 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_5, t_6 \cdot t_6 + t_2\right)}}\\

\mathbf{else}:\\
\;\;\;\;t_1 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_5, t_2 + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}\right)}}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 76.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := {\left(\mathsf{hypot}\left(t_3, t_0\right)\right)}^{2}\\ \mathbf{if}\;t_3 \cdot t_3 + t_0 \cdot t_0 \geq t_1 \cdot t_1 + t_2 \cdot t_2:\\ \;\;\;\;t_0 \cdot \frac{1}{{1}^{0.5} \cdot \sqrt{\mathsf{max}\left(t_4, {t_1}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_4, {\left(\mathsf{hypot}\left(t_1, t_2\right)\right)}^{2}\right)}}\\ \end{array} \end{array} \]
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (pow (hypot t_3 t_0) 2.0)))
   (if (>= (+ (* t_3 t_3) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2)))
     (* t_0 (/ 1.0 (* (pow 1.0 0.5) (sqrt (fmax t_4 (pow t_1 2.0))))))
     (* t_2 (/ 1.0 (sqrt (fmax t_4 (pow (hypot t_1 t_2) 2.0))))))))
float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = powf(hypotf(t_3, t_0), 2.0f);
	float tmp;
	if (((t_3 * t_3) + (t_0 * t_0)) >= ((t_1 * t_1) + (t_2 * t_2))) {
		tmp = t_0 * (1.0f / (powf(1.0f, 0.5f) * sqrtf(fmaxf(t_4, powf(t_1, 2.0f)))));
	} else {
		tmp = t_2 * (1.0f / sqrtf(fmaxf(t_4, powf(hypotf(t_1, t_2), 2.0f))));
	}
	return tmp;
}

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = hypot(t_3, t_0) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) >= Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)))
		tmp = Float32(t_0 * Float32(Float32(1.0) / Float32((Float32(1.0) ^ Float32(0.5)) * sqrt(((t_4 != t_4) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? t_4 : max(t_4, (t_1 ^ Float32(2.0)))))))));
	else
		tmp = Float32(t_2 * Float32(Float32(1.0) / sqrt(((t_4 != t_4) ? (hypot(t_1, t_2) ^ Float32(2.0)) : (((hypot(t_1, t_2) ^ Float32(2.0)) != (hypot(t_1, t_2) ^ Float32(2.0))) ? t_4 : max(t_4, (hypot(t_1, t_2) ^ Float32(2.0))))))));
	end
	return tmp
end

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(w) * dX_46_u;
	t_4 = hypot(t_3, t_0) ^ single(2.0);
	tmp = single(0.0);
	if (((t_3 * t_3) + (t_0 * t_0)) >= ((t_1 * t_1) + (t_2 * t_2)))
		tmp = t_0 * (single(1.0) / ((single(1.0) ^ single(0.5)) * sqrt(max(t_4, (t_1 ^ single(2.0))))));
	else
		tmp = t_2 * (single(1.0) / sqrt(max(t_4, (hypot(t_1, t_2) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\
t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\
t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\
t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\
t_4 := {\left(\mathsf{hypot}\left(t_3, t_0\right)\right)}^{2}\\
\mathbf{if}\;t_3 \cdot t_3 + t_0 \cdot t_0 \geq t_1 \cdot t_1 + t_2 \cdot t_2:\\
\;\;\;\;t_0 \cdot \frac{1}{{1}^{0.5} \cdot \sqrt{\mathsf{max}\left(t_4, {t_1}^{2}\right)}}\\

\mathbf{else}:\\
\;\;\;\;t_2 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_4, {\left(\mathsf{hypot}\left(t_1, t_2\right)\right)}^{2}\right)}}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023343 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, v)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor h) dX.v)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor h) dY.v))))