HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 27.9s

Precision: binary32

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

\[[cosTheta_i, cosTheta_O]=\mathsf{sort}([cosTheta_i, cosTheta_O])\]

Math

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

↓

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

FPCore

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

↓

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (*
   cosTheta_i
   (/
    (/ -1.0 v)
    (fma
     (fma sinTheta_i sinTheta_O v)
     (exp (/ -1.0 v))
     (- (* (fma sinTheta_i sinTheta_O v) (exp (/ 1.0 v)))))))))

C

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

↓

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * (cosTheta_i * ((-1.0f / v) / fmaf(fmaf(sinTheta_i, sinTheta_O, v), expf(-1.0f / v), -(fmaf(sinTheta_i, sinTheta_O, v) * expf(1.0f / 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 0.5

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

2. Simplified 0.4

   \[\leadsto \color{blue}{cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{v \cdot \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)}} \]

3. Taylor expanded in sinTheta_i around 0 0.5

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

4. Simplified 0.5

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

5. Applied *-un-lft-identity_binary32 0.5

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

6. Applied div-inv_binary32 0.4

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

7. Applied times-frac_binary32 0.4

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

8. Applied div-inv_binary32 0.4

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

9. Applied div-inv_binary32 0.4

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

10. Applied distribute-rgt-out_binary32 0.4

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

11. Applied cancel-sign-sub-inv_binary32 0.4

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

12. Applied frac-2neg_binary32 0.4

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

13. Simplified 0.4

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

14. Simplified 0.4

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

15. Final simplification 0.4

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

Reproduce

herbie shell --seed 2021273 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))