HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 12.9s

Precision: binary32

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\\ t_1 := -t_0\\ t_2 := 0.6931 + \log \left(\frac{0.5}{v}\right)\\ t_3 := \sqrt{t_2}\\ {e}^{\left(\frac{\mathsf{fma}\left(t_3, t_3, t_1\right) + \mathsf{fma}\left(t_1, 1, t_0\right)}{2}\right)} \cdot {e}^{\left(\frac{t_2 - t_0}{2}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

↓

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (fma sinTheta_i (/ sinTheta_O v) (/ 1.0 v)))
        (t_1 (- t_0))
        (t_2 (+ 0.6931 (log (/ 0.5 v))))
        (t_3 (sqrt t_2)))
   (*
    (pow E (/ (+ (fma t_3 t_3 t_1) (fma t_1 1.0 t_0)) 2.0))
    (pow E (/ (- t_2 t_0) 2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf(1.0f / (2.0f * v)));
}

↓

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = fmaf(sinTheta_i, (sinTheta_O / v), (1.0f / v));
	float t_1 = -t_0;
	float t_2 = 0.6931f + logf(0.5f / v);
	float t_3 = sqrtf(t_2);
	return powf(((float) M_E), ((fmaf(t_3, t_3, t_1) + fmaf(t_1, 1.0f, t_0)) / 2.0f)) * powf(((float) M_E), ((t_2 - t_0) / 2.0f));
}

Error

Bits error versus cosTheta_i

Bits error versus cosTheta_O

Bits error versus sinTheta_i

Bits error versus sinTheta_O

Bits error versus v

Derivation

1. Initial program 0.1

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in cosTheta_i around 0 0.1

   \[\leadsto e^{\color{blue}{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}} \]

3. Simplified 0.1

   \[\leadsto e^{\color{blue}{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}} \]

4. Applied *-un-lft-identity_binary32 0.1

   \[\leadsto e^{\color{blue}{1 \cdot \left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)}} \]

5. Applied exp-prod_binary32 0.1

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)}} \]

6. Simplified 0.1

   \[\leadsto {\color{blue}{e}}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)} \]

7. Applied sqr-pow_binary32 0.1

   \[\leadsto \color{blue}{{e}^{\left(\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}{2}\right)}} \]

8. Applied *-un-lft-identity_binary32 0.1

   \[\leadsto {e}^{\left(\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \color{blue}{1 \cdot \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}{2}\right)} \]

9. Applied add-sqr-sqrt_binary32 0.1

   \[\leadsto {e}^{\left(\frac{\color{blue}{\sqrt{0.6931 + \log \left(\frac{0.5}{v}\right)} \cdot \sqrt{0.6931 + \log \left(\frac{0.5}{v}\right)}} - 1 \cdot \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}{2}\right)} \]

10. Applied prod-diff_binary32 0.1

    \[\leadsto {e}^{\left(\frac{\color{blue}{\mathsf{fma}\left(\sqrt{0.6931 + \log \left(\frac{0.5}{v}\right)}, \sqrt{0.6931 + \log \left(\frac{0.5}{v}\right)}, -\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right) \cdot 1\right) + \mathsf{fma}\left(-\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right), 1, \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right) \cdot 1\right)}}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}{2}\right)} \]

11. Final simplification 0.1

    \[\leadsto {e}^{\left(\frac{\mathsf{fma}\left(\sqrt{0.6931 + \log \left(\frac{0.5}{v}\right)}, \sqrt{0.6931 + \log \left(\frac{0.5}{v}\right)}, -\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \mathsf{fma}\left(-\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right), 1, \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}{2}\right)} \]

Reproduce

herbie shell --seed 2021310 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))