HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 8.4s

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

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

↓

\[{\left({\left(\sqrt{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{0.3333333333333333}\right)}^{3} \cdot \frac{0.5}{v} \]

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
 (*
  (pow
   (*
    (pow
     (sqrt (exp (+ 0.6931 (/ (fma cosTheta_i cosTheta_O -1.0) v))))
     0.3333333333333333)
    (pow
     (sqrt
      (exp
       (-
        (fma cosTheta_O (/ cosTheta_i v) 0.6931)
        (fma sinTheta_i (/ sinTheta_O v) (/ 1.0 v)))))
     0.3333333333333333))
   3.0)
  (/ 0.5 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) {
	return powf((powf(sqrtf(expf((0.6931f + (fmaf(cosTheta_i, cosTheta_O, -1.0f) / v)))), 0.3333333333333333f) * powf(sqrtf(expf((fmaf(cosTheta_O, (cosTheta_i / v), 0.6931f) - fmaf(sinTheta_i, (sinTheta_O / v), (1.0f / v))))), 0.3333333333333333f)), 3.0f) * (0.5f / v);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

↓

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32((Float32((sqrt(exp(Float32(Float32(0.6931) + Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v)))) ^ Float32(0.3333333333333333)) * (sqrt(exp(Float32(fma(cosTheta_O, Float32(cosTheta_i / v), Float32(0.6931)) - fma(sinTheta_i, Float32(sinTheta_O / v), Float32(Float32(1.0) / v))))) ^ Float32(0.3333333333333333))) ^ Float32(3.0)) * Float32(Float32(0.5) / v))
end

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

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

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

   \[\leadsto {\color{blue}{\left({\left(\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{0.3333333333333333}\right)}}^{3} \cdot \frac{0.5}{v} \]

5. Taylor expanded in sinTheta_i around 0 0.1

   \[\leadsto {\left({\left(\sqrt{\color{blue}{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{0.3333333333333333}\right)}^{3} \cdot \frac{0.5}{v} \]

6. Simplified 0.1

   \[\leadsto {\left({\left(\sqrt{\color{blue}{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{0.3333333333333333}\right)}^{3} \cdot \frac{0.5}{v} \]

7. Final simplification 0.1

   \[\leadsto {\left({\left(\sqrt{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)}^{0.3333333333333333}\right)}^{3} \cdot \frac{0.5}{v} \]

Reproduce

herbie shell --seed 2022159 
(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))))))