HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 19.1s

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

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}\\ t_1 := \sqrt{t_0}\\ \frac{0.5}{v} \cdot \left(t_0 \cdot \left(\left(t_1 \cdot t_1\right) \cdot \left(t_1 \cdot \sqrt{\sqrt[3]{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}}\right)\right)\right) \end{array} \]

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
         (cbrt
          (exp
           (+
            0.6931
            (/
             (- (* cosTheta_i cosTheta_O) (fma sinTheta_i sinTheta_O 1.0))
             v)))))
        (t_1 (sqrt t_0)))
   (*
    (/ 0.5 v)
    (*
     t_0
     (*
      (* t_1 t_1)
      (*
       t_1
       (sqrt
        (cbrt
         (exp (- 0.6931 (+ (/ 1.0 v) (/ (* sinTheta_i sinTheta_O) v))))))))))))

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 = cbrtf(expf(0.6931f + (((cosTheta_i * cosTheta_O) - fmaf(sinTheta_i, sinTheta_O, 1.0f)) / v)));
	float t_1 = sqrtf(t_0);
	return (0.5f / v) * (t_0 * ((t_1 * t_1) * (t_1 * sqrtf(cbrtf(expf(0.6931f - ((1.0f / v) + ((sinTheta_i * sinTheta_O) / v))))))));
}

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. Simplified 0.1

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

3. Applied add-cube-cbrt_binary32 0.1

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)} \]

4. Simplified 0.1

   \[\leadsto \frac{0.5}{v} \cdot \left(\color{blue}{\left(\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}} \cdot \sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}\right)} \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right) \]

5. Simplified 0.1

   \[\leadsto \frac{0.5}{v} \cdot \left(\left(\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}} \cdot \sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}\right) \cdot \color{blue}{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right) \]

6. Applied add-sqr-sqrt_binary32 0.1

   \[\leadsto \frac{0.5}{v} \cdot \left(\left(\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right)}\right) \cdot \sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}\right) \]

7. Applied add-sqr-sqrt_binary32 0.1

   \[\leadsto \frac{0.5}{v} \cdot \left(\left(\color{blue}{\left(\sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right)} \cdot \left(\sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right)\right) \cdot \sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}\right) \]

8. Applied unswap-sqr_binary32 0.1

   \[\leadsto \frac{0.5}{v} \cdot \left(\color{blue}{\left(\left(\sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right) \cdot \left(\sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right)\right)} \cdot \sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}\right) \]

9. Taylor expanded in cosTheta_i around 0 0.1

   \[\leadsto \frac{0.5}{v} \cdot \left(\left(\left(\sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right) \cdot \left(\sqrt{\sqrt[3]{\color{blue}{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right)\right) \cdot \sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}\right) \]

10. Final simplification 0.1

    \[\leadsto \frac{0.5}{v} \cdot \left(\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}} \cdot \left(\left(\sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}}\right) \cdot \left(\sqrt{\sqrt[3]{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}}} \cdot \sqrt{\sqrt[3]{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}}\right)\right)\right) \]

Reproduce

herbie shell --seed 2021275 
(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))))))