HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 21.8s

Precision: binary32

Cost: 20000

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\frac{0.5}{v}\right)\\ e^{\left(0.6931 + t_0\right) + \frac{-1}{v}} - \frac{\left(sinTheta_i \cdot e^{0.6931 + \mathsf{log1p}\left(\mathsf{expm1}\left(t_0 + \frac{-1}{v}\right)\right)}\right) \cdot sinTheta_O}{v} \end{array} \]

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 (log (/ 0.5 v))))
   (-
    (exp (+ (+ 0.6931 t_0) (/ -1.0 v)))
    (/
     (*
      (* sinTheta_i (exp (+ 0.6931 (log1p (expm1 (+ t_0 (/ -1.0 v)))))))
      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 = logf((0.5f / v));
	return expf(((0.6931f + t_0) + (-1.0f / v))) - (((sinTheta_i * expf((0.6931f + log1pf(expm1f((t_0 + (-1.0f / v))))))) * sinTheta_O) / 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)
	t_0 = log(Float32(Float32(0.5) / v))
	return Float32(exp(Float32(Float32(Float32(0.6931) + t_0) + Float32(Float32(-1.0) / v))) - Float32(Float32(Float32(sinTheta_i * exp(Float32(Float32(0.6931) + log1p(expm1(Float32(t_0 + Float32(Float32(-1.0) / v))))))) * sinTheta_O) / v))
end

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

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

   [Start] 0.1	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   +-commutative [=>] 0.1	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]
   log-div [=>] 0.1	\[ e^{\color{blue}{\left(\log 1 - \log \left(2 \cdot v\right)\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   metadata-eval [=>] 0.1	\[ e^{\left(\color{blue}{0} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   associate-+l- [=>] 0.1	\[ e^{\color{blue}{0 - \left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]
   associate-+l- [<=] 0.1	\[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]
   metadata-eval [<=] 0.1	\[ e^{\left(\color{blue}{\log 1} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   log-div [<=] 0.1	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   +-commutative [<=] 0.1	\[ e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]
   associate-+l+ [=>] 0.1	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

3. Taylor expanded in sinTheta_i around 0 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}} + -1 \cdot \frac{sinTheta_i \cdot \left(e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}} \cdot sinTheta_O\right)}{v} \]
   associate--l+ [=>] 0.1	\[ e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} + -1 \cdot \frac{sinTheta_i \cdot \left(e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}} \cdot sinTheta_O\right)}{v} \]
   associate-*r/ [<=] 0.1	\[ e^{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \color{blue}{cosTheta_i \cdot \frac{cosTheta_O}{v}}\right) - \frac{1}{v}\right)} + -1 \cdot \frac{sinTheta_i \cdot \left(e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}} \cdot sinTheta_O\right)}{v} \]
   mul-1-neg [=>] 0.1	\[ e^{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + cosTheta_i \cdot \frac{cosTheta_O}{v}\right) - \frac{1}{v}\right)} + \color{blue}{\left(-\frac{sinTheta_i \cdot \left(e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}} \cdot sinTheta_O\right)}{v}\right)} \]

5. Applied egg-rr 0.1

   \[\leadsto e^{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + cosTheta_i \cdot \frac{cosTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(-\frac{\left(sinTheta_i \cdot e^{0.6931 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right)\right)}}\right) \cdot sinTheta_O}{v}\right) \]

6. Taylor expanded in cosTheta_i around 0 0.1

   \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}}} + \left(-\frac{\left(sinTheta_i \cdot e^{0.6931 + \mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right)\right)}\right) \cdot sinTheta_O}{v}\right) \]

7. Taylor expanded in cosTheta_i around 0 0.1

   \[\leadsto e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}} + \left(-\frac{\left(sinTheta_i \cdot e^{0.6931 + \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\log \left(\frac{0.5}{v}\right) - \frac{1}{v}}\right)\right)}\right) \cdot sinTheta_O}{v}\right) \]

8. Final simplification 0.1

   \[\leadsto e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{-1}{v}} - \frac{\left(sinTheta_i \cdot e^{0.6931 + \mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{0.5}{v}\right) + \frac{-1}{v}\right)\right)}\right) \cdot sinTheta_O}{v} \]

Alternatives

Alternative 1
Error	0.1
Cost	10400

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

Alternative 2
Error	0.1
Cost	6880

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

Alternative 3
Error	0.1
Cost	6816

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

Alternative 4
Error	0.1
Cost	3680

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

Alternative 5
Error	0.1
Cost	3616

\[\frac{0.5}{v} \cdot e^{0.6931 + \frac{-1 - sinTheta_i \cdot sinTheta_O}{v}} \]

Alternative 6
Error	0.1
Cost	3488

\[\frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]

Alternative 7
Error	27.9
Cost	3392

\[e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \]

Alternative 8
Error	30.5
Cost	3360

\[\frac{0.5}{v} \cdot e^{0.6931} \]

Alternative 9
Error	30.5
Cost	3360

\[\frac{0.5 \cdot e^{0.6931}}{v} \]

Reproduce

herbie shell --seed 2023083 
(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))))))