Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 75.8% → 76.1%

Time: 9.7s

Alternatives: 9

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := t\_2 \cdot t\_2 + t\_0 \cdot t\_0\\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := t\_1 \cdot t\_1 + t\_4 \cdot t\_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t\_3, t\_5\right)}}\\ \mathbf{if}\;t\_3 \geq t\_5:\\ \;\;\;\;t\_6 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (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_2) (* t_6 t_1))))

C

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_2;
	} else {
		tmp = t_6 * t_1;
	}
	return tmp;
}

Julia

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(fmax(t_3, t_5)))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

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_2;
	else
		tmp = t_6 * t_1;
	end
	tmp_2 = tmp;
end

Initial Program: 75.8% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := t\_2 \cdot t\_2 + t\_0 \cdot t\_0\\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := t\_1 \cdot t\_1 + t\_4 \cdot t\_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t\_3, t\_5\right)}}\\ \mathbf{if}\;t\_3 \geq t\_5:\\ \;\;\;\;t\_6 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (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_2) (* t_6 t_1))))

C

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_2;
	} else {
		tmp = t_6 * t_1;
	}
	return tmp;
}

Julia

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(fmax(t_3, t_5)))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

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_2;
	else
		tmp = t_6 * t_1;
	end
	tmp_2 = tmp;
end

Alternative 1: 76.1% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_1 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_2 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_3 := \mathsf{fma}\left(t\_1, t\_1, t\_2 \cdot t\_2\right)\\ t_4 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_5 := \mathsf{fma}\left(t\_4, t\_4, t\_0 \cdot t\_0\right)\\ t_6 := \sqrt{\mathsf{max}\left(t\_3, t\_5\right)}\\ 1 \cdot \begin{array}{l} \mathbf{if}\;t\_5 \geq t\_3:\\ \;\;\;\;\frac{t\_0}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_6}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* dX.u (floor w)))
       (t_1 (* dY.v (floor h)))
       (t_2 (* dY.u (floor w)))
       (t_3 (fma t_1 t_1 (* t_2 t_2)))
       (t_4 (* dX.v (floor h)))
       (t_5 (fma t_4 t_4 (* t_0 t_0)))
       (t_6 (sqrt (fmax t_3 t_5))))
  (* 1.0 (if (>= t_5 t_3) (/ t_0 t_6) (/ t_2 t_6)))))

C

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 = dX_46_u * floorf(w);
	float t_1 = dY_46_v * floorf(h);
	float t_2 = dY_46_u * floorf(w);
	float t_3 = fmaf(t_1, t_1, (t_2 * t_2));
	float t_4 = dX_46_v * floorf(h);
	float t_5 = fmaf(t_4, t_4, (t_0 * t_0));
	float t_6 = sqrtf(fmaxf(t_3, t_5));
	float tmp;
	if (t_5 >= t_3) {
		tmp = t_0 / t_6;
	} else {
		tmp = t_2 / t_6;
	}
	return 1.0f * tmp;
}

Julia

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

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 76.1%

   \[\leadsto 1 \cdot \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right) \geq \mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \end{array} \]
4. Add Preprocessing

Alternative 2: 75.8% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_4 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_5 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_6 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_7 := \sqrt{\mathsf{max}\left(t\_1 \cdot t\_1 - \left(\left(-dY.v\right) \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(t\_3, t\_3, t\_6 \cdot t\_6\right)\right)}\\ \mathbf{if}\;t\_4 \cdot t\_4 + t\_0 \cdot t\_0 \geq t\_2 \cdot t\_2 + t\_5 \cdot t\_5:\\ \;\;\;\;\frac{dX.u}{t\_7} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \frac{dY.u}{t\_7}\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* (floor h) dX.v))
       (t_1 (* dY.u (floor w)))
       (t_2 (* (floor w) dY.u))
       (t_3 (* dX.v (floor h)))
       (t_4 (* (floor w) dX.u))
       (t_5 (* (floor h) dY.v))
       (t_6 (* dX.u (floor w)))
       (t_7
        (sqrt
         (fmax
          (-
           (* t_1 t_1)
           (* (* (- dY.v) dY.v) (* (floor h) (floor h))))
          (fma t_3 t_3 (* t_6 t_6))))))
  (if (>= (+ (* t_4 t_4) (* t_0 t_0)) (+ (* t_2 t_2) (* t_5 t_5)))
    (* (/ dX.u t_7) (floor w))
    (* (floor w) (/ dY.u t_7)))))

C

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 = dY_46_u * floorf(w);
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = dX_46_v * floorf(h);
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = floorf(h) * dY_46_v;
	float t_6 = dX_46_u * floorf(w);
	float t_7 = sqrtf(fmaxf(((t_1 * t_1) - ((-dY_46_v * dY_46_v) * (floorf(h) * floorf(h)))), fmaf(t_3, t_3, (t_6 * t_6))));
	float tmp;
	if (((t_4 * t_4) + (t_0 * t_0)) >= ((t_2 * t_2) + (t_5 * t_5))) {
		tmp = (dX_46_u / t_7) * floorf(w);
	} else {
		tmp = floorf(w) * (dY_46_u / t_7);
	}
	return tmp;
}

Julia

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(dY_46_u * floor(w))
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(dX_46_v * floor(h))
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = Float32(floor(h) * dY_46_v)
	t_6 = Float32(dX_46_u * floor(w))
	t_7 = sqrt(fmax(Float32(Float32(t_1 * t_1) - Float32(Float32(Float32(-dY_46_v) * dY_46_v) * Float32(floor(h) * floor(h)))), fma(t_3, t_3, Float32(t_6 * t_6))))
	tmp = Float32(0.0)
	if (Float32(Float32(t_4 * t_4) + Float32(t_0 * t_0)) >= Float32(Float32(t_2 * t_2) + Float32(t_5 * t_5)))
		tmp = Float32(Float32(dX_46_u / t_7) * floor(w));
	else
		tmp = Float32(floor(w) * Float32(dY_46_u / t_7));
	end
	return tmp
end

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
2. Step-by-step derivation

3. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \end{array} \]

7. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right) - \left(\left(-dY.v\right) \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \frac{dY.u}{\sqrt{\mathsf{max}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right) - \left(\left(-dY.v\right) \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \end{array} \]
8. Add Preprocessing

Alternative 3: 75.8% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_2 := \mathsf{fma}\left(dY.v \cdot dY.v, t\_1, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)\\ t_3 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_4 := \sqrt{\mathsf{max}\left(t\_2, \mathsf{fma}\left(t\_3, t\_3, t\_0 \cdot t\_0\right)\right)}\\ \mathbf{if}\;\mathsf{fma}\left(t\_0 \cdot dX.u, \left\lfloor w\right\rfloor , \left(t\_1 \cdot dX.v\right) \cdot dX.v\right) \geq t\_2:\\ \;\;\;\;\frac{dX.u}{t\_4} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \frac{dY.u}{t\_4}\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* dX.u (floor w)))
       (t_1 (* (floor h) (floor h)))
       (t_2
        (fma
         (* dY.v dY.v)
         t_1
         (* (* (floor w) (floor w)) (* dY.u dY.u))))
       (t_3 (* dX.v (floor h)))
       (t_4 (sqrt (fmax t_2 (fma t_3 t_3 (* t_0 t_0))))))
  (if (>= (fma (* t_0 dX.u) (floor w) (* (* t_1 dX.v) dX.v)) t_2)
    (* (/ dX.u t_4) (floor w))
    (* (floor w) (/ dY.u t_4)))))

C

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 = dX_46_u * floorf(w);
	float t_1 = floorf(h) * floorf(h);
	float t_2 = fmaf((dY_46_v * dY_46_v), t_1, ((floorf(w) * floorf(w)) * (dY_46_u * dY_46_u)));
	float t_3 = dX_46_v * floorf(h);
	float t_4 = sqrtf(fmaxf(t_2, fmaf(t_3, t_3, (t_0 * t_0))));
	float tmp;
	if (fmaf((t_0 * dX_46_u), floorf(w), ((t_1 * dX_46_v) * dX_46_v)) >= t_2) {
		tmp = (dX_46_u / t_4) * floorf(w);
	} else {
		tmp = floorf(w) * (dY_46_u / t_4);
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_u * floor(w))
	t_1 = Float32(floor(h) * floor(h))
	t_2 = fma(Float32(dY_46_v * dY_46_v), t_1, Float32(Float32(floor(w) * floor(w)) * Float32(dY_46_u * dY_46_u)))
	t_3 = Float32(dX_46_v * floor(h))
	t_4 = sqrt(fmax(t_2, fma(t_3, t_3, Float32(t_0 * t_0))))
	tmp = Float32(0.0)
	if (fma(Float32(t_0 * dX_46_u), floor(w), Float32(Float32(t_1 * dX_46_v) * dX_46_v)) >= t_2)
		tmp = Float32(Float32(dX_46_u / t_4) * floor(w));
	else
		tmp = Float32(floor(w) * Float32(dY_46_u / t_4));
	end
	return tmp
end

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
2. Step-by-step derivation

3. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \end{array} \]

7. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right) \geq \mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right) \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \end{array} \]
10. Add Preprocessing

Alternative 4: 46.8% accurate, 1.2× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \frac{1}{\sqrt{\mathsf{max}\left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1, \frac{1}{{\left(\sqrt{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right|} \cdot \left(-1 \cdot \left(dY.u \cdot \sqrt{-1 \cdot \left|\frac{-1}{dY.u}\right|}\right)\right)\right)}^{-2}} + t\_2 \cdot t\_2\right)}}\\ \mathbf{if}\;0:\\ \;\;\;\;t\_3 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* (floor w) dX.u))
       (t_1 (* (floor h) dX.v))
       (t_2 (* (floor h) dY.v))
       (t_3
        (/
         1.0
         (sqrt
          (fmax
           (+ (* t_0 t_0) (* t_1 t_1))
           (+
            (/
             1.0
             (pow
              (*
               (sqrt (fabs (* (* dY.u (floor w)) (floor w))))
               (* -1.0 (* dY.u (sqrt (* -1.0 (fabs (/ -1.0 dY.u)))))))
              -2.0))
            (* t_2 t_2)))))))
  (if 0 (* t_3 t_0) (* t_3 (* (floor w) dY.u)))))

C

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) * dX_46_v;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = 1.0f / sqrtf(fmaxf(((t_0 * t_0) + (t_1 * t_1)), ((1.0f / powf((sqrtf(fabsf(((dY_46_u * floorf(w)) * floorf(w)))) * (-1.0f * (dY_46_u * sqrtf((-1.0f * fabsf((-1.0f / dY_46_u))))))), -2.0f)) + (t_2 * t_2))));
	float tmp;
	if (0.0f) {
		tmp = t_3 * t_0;
	} else {
		tmp = t_3 * (floorf(w) * dY_46_u);
	}
	return tmp;
}

Julia

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) * dX_46_v)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(Float32(1.0) / sqrt(fmax(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)), Float32(Float32(Float32(1.0) / (Float32(sqrt(abs(Float32(Float32(dY_46_u * floor(w)) * floor(w)))) * Float32(Float32(-1.0) * Float32(dY_46_u * sqrt(Float32(Float32(-1.0) * abs(Float32(Float32(-1.0) / dY_46_u))))))) ^ Float32(-2.0))) + Float32(t_2 * t_2)))))
	tmp = Float32(0.0)
	if (Float32(0.0))
		tmp = Float32(t_3 * t_0);
	else
		tmp = Float32(t_3 * Float32(floor(w) * dY_46_u));
	end
	return tmp
end

MATLAB

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) * dX_46_v;
	t_2 = floor(h) * dY_46_v;
	t_3 = single(1.0) / sqrt(max(((t_0 * t_0) + (t_1 * t_1)), ((single(1.0) / ((sqrt(abs(((dY_46_u * floor(w)) * floor(w)))) * (single(-1.0) * (dY_46_u * sqrt((single(-1.0) * abs((single(-1.0) / dY_46_u))))))) ^ single(-2.0))) + (t_2 * t_2))));
	tmp = single(0.0);
	if (single(0.0))
		tmp = t_3 * t_0;
	else
		tmp = t_3 * (floor(w) * dY_46_u);
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

2. Taylor expanded in undef-var around zero

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Step-by-step derivation

4. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

6. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \frac{1}{{\left(\left|dY.u \cdot \left\lfloor w\right\rfloor \right|\right)}^{-2}} + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \frac{1}{{\left(\left|dY.u \cdot \left\lfloor w\right\rfloor \right|\right)}^{-2}} + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
7. Step-by-step derivation

8. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \frac{1}{{\left(\sqrt{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right|} \cdot \sqrt{\left|dY.u\right|}\right)}^{-2}} + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \frac{1}{{\left(\sqrt{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right|} \cdot \sqrt{\left|dY.u\right|}\right)}^{-2}} + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

9. Taylor expanded in dY.u around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \frac{1}{{\left(\sqrt{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right|} \cdot \left(-1 \cdot \left(dY.u \cdot \sqrt{-1 \cdot \left|\frac{-1}{dY.u}\right|}\right)\right)\right)}^{-2}} + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \frac{1}{{\left(\sqrt{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right|} \cdot \left(-1 \cdot \left(dY.u \cdot \sqrt{-1 \cdot \left|\frac{-1}{dY.u}\right|}\right)\right)\right)}^{-2}} + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
10. Step-by-step derivation

11. Applied rewrites 46.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \frac{1}{{\left(\sqrt{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right|} \cdot \left(-1 \cdot \left(dY.u \cdot \sqrt{-1 \cdot \left|\frac{-1}{dY.u}\right|}\right)\right)\right)}^{-2}} + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \frac{1}{{\left(\sqrt{\left|\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right|} \cdot \left(-1 \cdot \left(dY.u \cdot \sqrt{-1 \cdot \left|\frac{-1}{dY.u}\right|}\right)\right)\right)}^{-2}} + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
12. Add Preprocessing

Alternative 5: 42.7% accurate, 1.8× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := \mathsf{fma}\left(t\_0, t\_0, t\_1 \cdot t\_1\right)\\ t_3 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_4 := dX.u \cdot \left\lfloor w\right\rfloor \\ \mathbf{if}\;0:\\ \;\;\;\;\frac{\left(-\left\lfloor w\right\rfloor \right) \cdot dX.u}{-\sqrt{\mathsf{max}\left(t\_2, \mathsf{fma}\left(t\_3, t\_3, t\_4 \cdot t\_4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(t\_2, \mathsf{fma}\left(t\_4 \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* dY.v (floor h)))
       (t_1 (* dY.u (floor w)))
       (t_2 (fma t_0 t_0 (* t_1 t_1)))
       (t_3 (* dX.v (floor h)))
       (t_4 (* dX.u (floor w))))
  (if 0
    (/
     (* (- (floor w)) dX.u)
     (- (sqrt (fmax t_2 (fma t_3 t_3 (* t_4 t_4))))))
    (*
     (/
      (floor w)
      (sqrt
       (fmax
        t_2
        (fma
         (* t_4 dX.u)
         (floor w)
         (* (* (* (floor h) (floor h)) dX.v) dX.v)))))
     dY.u))))

C

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 = dY_46_v * floorf(h);
	float t_1 = dY_46_u * floorf(w);
	float t_2 = fmaf(t_0, t_0, (t_1 * t_1));
	float t_3 = dX_46_v * floorf(h);
	float t_4 = dX_46_u * floorf(w);
	float tmp;
	if (0.0f) {
		tmp = (-floorf(w) * dX_46_u) / -sqrtf(fmaxf(t_2, fmaf(t_3, t_3, (t_4 * t_4))));
	} else {
		tmp = (floorf(w) / sqrtf(fmaxf(t_2, fmaf((t_4 * dX_46_u), floorf(w), (((floorf(h) * floorf(h)) * dX_46_v) * dX_46_v))))) * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_v * floor(h))
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = fma(t_0, t_0, Float32(t_1 * t_1))
	t_3 = Float32(dX_46_v * floor(h))
	t_4 = Float32(dX_46_u * floor(w))
	tmp = Float32(0.0)
	if (Float32(0.0))
		tmp = Float32(Float32(Float32(-floor(w)) * dX_46_u) / Float32(-sqrt(fmax(t_2, fma(t_3, t_3, Float32(t_4 * t_4))))));
	else
		tmp = Float32(Float32(floor(w) / sqrt(fmax(t_2, fma(Float32(t_4 * dX_46_u), floor(w), Float32(Float32(Float32(floor(h) * floor(h)) * dX_46_v) * dX_46_v))))) * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

2. Taylor expanded in undef-var around zero

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Step-by-step derivation

4. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

6. Applied rewrites 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Applied rewrites 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]
9. Step-by-step derivation

10. Applied rewrites 42.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{\left(-\left\lfloor w\right\rfloor \right) \cdot dX.u}{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]
11. Add Preprocessing

Alternative 6: 42.7% accurate, 1.9× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_2 := \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\\ t_3 := dY.v \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_3, t\_3, t\_0 \cdot t\_0\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , t\_2 \cdot dX.v\right)\right)}}{\left\lfloor w\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_3, t\_3, t\_1 \cdot \left(dY.u \cdot dY.u\right)\right), \mathsf{fma}\left(t\_2, dX.v, t\_1 \cdot \left(dX.u \cdot dX.u\right)\right)\right)}} \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* dY.u (floor w)))
       (t_1 (* (floor w) (floor w)))
       (t_2 (* (* (floor h) (floor h)) dX.v))
       (t_3 (* dY.v (floor h))))
  (if 0
    (/
     dX.u
     (/
      (sqrt
       (fmax
        (fma t_3 t_3 (* t_0 t_0))
        (fma (* (* dX.u (floor w)) dX.u) (floor w) (* t_2 dX.v))))
      (floor w)))
    (*
     (/
      (floor w)
      (sqrt
       (fmax
        (fma t_3 t_3 (* t_1 (* dY.u dY.u)))
        (fma t_2 dX.v (* t_1 (* dX.u dX.u))))))
     dY.u))))

C

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 = dY_46_u * floorf(w);
	float t_1 = floorf(w) * floorf(w);
	float t_2 = (floorf(h) * floorf(h)) * dX_46_v;
	float t_3 = dY_46_v * floorf(h);
	float tmp;
	if (0.0f) {
		tmp = dX_46_u / (sqrtf(fmaxf(fmaf(t_3, t_3, (t_0 * t_0)), fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (t_2 * dX_46_v)))) / floorf(w));
	} else {
		tmp = (floorf(w) / sqrtf(fmaxf(fmaf(t_3, t_3, (t_1 * (dY_46_u * dY_46_u))), fmaf(t_2, dX_46_v, (t_1 * (dX_46_u * dX_46_u)))))) * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_u * floor(w))
	t_1 = Float32(floor(w) * floor(w))
	t_2 = Float32(Float32(floor(h) * floor(h)) * dX_46_v)
	t_3 = Float32(dY_46_v * floor(h))
	tmp = Float32(0.0)
	if (Float32(0.0))
		tmp = Float32(dX_46_u / Float32(sqrt(fmax(fma(t_3, t_3, Float32(t_0 * t_0)), fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(t_2 * dX_46_v)))) / floor(w)));
	else
		tmp = Float32(Float32(floor(w) / sqrt(fmax(fma(t_3, t_3, Float32(t_1 * Float32(dY_46_u * dY_46_u))), fma(t_2, dX_46_v, Float32(t_1 * Float32(dX_46_u * dX_46_u)))))) * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

2. Taylor expanded in undef-var around zero

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Step-by-step derivation

4. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

6. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Applied rewrites 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}}{\left\lfloor w\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

10. Applied rewrites 42.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}}{\left\lfloor w\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right)\right)}} \cdot dY.u\\ \end{array} \]
11. Add Preprocessing

Alternative 7: 42.6% accurate, 1.9× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_2 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_3 := \mathsf{fma}\left(t\_2, t\_2, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\\ \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(t\_3, \mathsf{fma}\left(t\_0, t\_0, t\_1 \cdot t\_1\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(t\_3, \mathsf{fma}\left(t\_1 \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* dX.v (floor h)))
       (t_1 (* dX.u (floor w)))
       (t_2 (* dY.v (floor h)))
       (t_3 (fma t_2 t_2 (* (* (* dY.u dY.u) (floor w)) (floor w)))))
  (if 0
    (* (/ dX.u (sqrt (fmax t_3 (fma t_0 t_0 (* t_1 t_1))))) (floor w))
    (*
     (/
      (floor w)
      (sqrt
       (fmax
        t_3
        (fma
         (* t_1 dX.u)
         (floor w)
         (* (* (* (floor h) (floor h)) dX.v) dX.v)))))
     dY.u))))

C

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 = dX_46_v * floorf(h);
	float t_1 = dX_46_u * floorf(w);
	float t_2 = dY_46_v * floorf(h);
	float t_3 = fmaf(t_2, t_2, (((dY_46_u * dY_46_u) * floorf(w)) * floorf(w)));
	float tmp;
	if (0.0f) {
		tmp = (dX_46_u / sqrtf(fmaxf(t_3, fmaf(t_0, t_0, (t_1 * t_1))))) * floorf(w);
	} else {
		tmp = (floorf(w) / sqrtf(fmaxf(t_3, fmaf((t_1 * dX_46_u), floorf(w), (((floorf(h) * floorf(h)) * dX_46_v) * dX_46_v))))) * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_v * floor(h))
	t_1 = Float32(dX_46_u * floor(w))
	t_2 = Float32(dY_46_v * floor(h))
	t_3 = fma(t_2, t_2, Float32(Float32(Float32(dY_46_u * dY_46_u) * floor(w)) * floor(w)))
	tmp = Float32(0.0)
	if (Float32(0.0))
		tmp = Float32(Float32(dX_46_u / sqrt(fmax(t_3, fma(t_0, t_0, Float32(t_1 * t_1))))) * floor(w));
	else
		tmp = Float32(Float32(floor(w) / sqrt(fmax(t_3, fma(Float32(t_1 * dX_46_u), floor(w), Float32(Float32(Float32(floor(h) * floor(h)) * dX_46_v) * dX_46_v))))) * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
2. Step-by-step derivation

3. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

4. Taylor expanded in undef-var around zero

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

6. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]
9. Step-by-step derivation

10. Applied rewrites 42.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]
11. Add Preprocessing

Alternative 8: 42.6% accurate, 1.9× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_1 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_2 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_3 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_4 := \mathsf{fma}\left(t\_3, t\_3, t\_0 \cdot t\_0\right)\\ \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(t\_4, \mathsf{fma}\left(t\_1, t\_1, t\_2 \cdot t\_2\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(t\_4, \mathsf{fma}\left(t\_2 \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* dY.u (floor w)))
       (t_1 (* dX.v (floor h)))
       (t_2 (* dX.u (floor w)))
       (t_3 (* dY.v (floor h)))
       (t_4 (fma t_3 t_3 (* t_0 t_0))))
  (if 0
    (* (/ dX.u (sqrt (fmax t_4 (fma t_1 t_1 (* t_2 t_2))))) (floor w))
    (*
     (/
      (floor w)
      (sqrt
       (fmax
        t_4
        (fma
         (* t_2 dX.u)
         (floor w)
         (* (* (* (floor h) (floor h)) dX.v) dX.v)))))
     dY.u))))

C

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 = dY_46_u * floorf(w);
	float t_1 = dX_46_v * floorf(h);
	float t_2 = dX_46_u * floorf(w);
	float t_3 = dY_46_v * floorf(h);
	float t_4 = fmaf(t_3, t_3, (t_0 * t_0));
	float tmp;
	if (0.0f) {
		tmp = (dX_46_u / sqrtf(fmaxf(t_4, fmaf(t_1, t_1, (t_2 * t_2))))) * floorf(w);
	} else {
		tmp = (floorf(w) / sqrtf(fmaxf(t_4, fmaf((t_2 * dX_46_u), floorf(w), (((floorf(h) * floorf(h)) * dX_46_v) * dX_46_v))))) * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_u * floor(w))
	t_1 = Float32(dX_46_v * floor(h))
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = Float32(dY_46_v * floor(h))
	t_4 = fma(t_3, t_3, Float32(t_0 * t_0))
	tmp = Float32(0.0)
	if (Float32(0.0))
		tmp = Float32(Float32(dX_46_u / sqrt(fmax(t_4, fma(t_1, t_1, Float32(t_2 * t_2))))) * floor(w));
	else
		tmp = Float32(Float32(floor(w) / sqrt(fmax(t_4, fma(Float32(t_2 * dX_46_u), floor(w), Float32(Float32(Float32(floor(h) * floor(h)) * dX_46_v) * dX_46_v))))) * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
2. Step-by-step derivation

3. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

4. Taylor expanded in undef-var around zero

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

6. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]
9. Add Preprocessing

Alternative 9: 42.6% accurate, 1.9× speedup

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_2 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_3 := \mathsf{fma}\left(t\_2, t\_2, t\_0 \cdot t\_0\right)\\ \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(t\_3, \mathsf{fma}\left(dX.v \cdot dX.v, t\_1, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(t\_3, \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(t\_1 \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :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))
  (let* ((t_0 (* dY.u (floor w)))
       (t_1 (* (floor h) (floor h)))
       (t_2 (* dY.v (floor h)))
       (t_3 (fma t_2 t_2 (* t_0 t_0))))
  (if 0
    (*
     (/
      dX.u
      (sqrt
       (fmax
        t_3
        (fma
         (* dX.v dX.v)
         t_1
         (* (* (floor w) (floor w)) (* dX.u dX.u))))))
     (floor w))
    (*
     (/
      (floor w)
      (sqrt
       (fmax
        t_3
        (fma
         (* (* dX.u (floor w)) dX.u)
         (floor w)
         (* (* t_1 dX.v) dX.v)))))
     dY.u))))

C

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 = dY_46_u * floorf(w);
	float t_1 = floorf(h) * floorf(h);
	float t_2 = dY_46_v * floorf(h);
	float t_3 = fmaf(t_2, t_2, (t_0 * t_0));
	float tmp;
	if (0.0f) {
		tmp = (dX_46_u / sqrtf(fmaxf(t_3, fmaf((dX_46_v * dX_46_v), t_1, ((floorf(w) * floorf(w)) * (dX_46_u * dX_46_u)))))) * floorf(w);
	} else {
		tmp = (floorf(w) / sqrtf(fmaxf(t_3, fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), ((t_1 * dX_46_v) * dX_46_v))))) * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_u * floor(w))
	t_1 = Float32(floor(h) * floor(h))
	t_2 = Float32(dY_46_v * floor(h))
	t_3 = fma(t_2, t_2, Float32(t_0 * t_0))
	tmp = Float32(0.0)
	if (Float32(0.0))
		tmp = Float32(Float32(dX_46_u / sqrt(fmax(t_3, fma(Float32(dX_46_v * dX_46_v), t_1, Float32(Float32(floor(w) * floor(w)) * Float32(dX_46_u * dX_46_u)))))) * floor(w));
	else
		tmp = Float32(Float32(floor(w) / sqrt(fmax(t_3, fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(t_1 * dX_46_v) * dX_46_v))))) * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 75.8%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
2. Step-by-step derivation

3. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

4. Taylor expanded in undef-var around zero

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

6. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Applied rewrites 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor , \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]
9. Step-by-step derivation

10. Applied rewrites 42.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;0:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(dX.v \cdot dX.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left\lfloor h\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot dX.v\right)\right)}} \cdot dY.u\\ \end{array} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, u)"
  :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 w) dX.u)) (* (/ 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 w) dY.u))))