HairBSDF, Mp, lower

Average Accuracy: 99.6% → 99.0%

Time: 23.2s

Precision: binary32

Cost: 13664

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

Math

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

↓

\[\begin{array}{l} t_0 := 0.6931 - \frac{1}{v}\\ \left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + t_0\right) \cdot 2}} \cdot \sqrt[3]{e^{t_0}}\right) \cdot \frac{0.5}{v} \end{array} \]

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 (- 0.6931 (/ 1.0 v))))
   (*
    (*
     (cbrt
      (exp
       (*
        (+ (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) v) t_0)
        2.0)))
     (cbrt (exp t_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) {
	float t_0 = 0.6931f - (1.0f / v);
	return (cbrtf(expf((((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + t_0) * 2.0f))) * cbrtf(expf(t_0))) * (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)
	t_0 = Float32(Float32(0.6931) - Float32(Float32(1.0) / v))
	return Float32(Float32(cbrt(exp(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) / v) + t_0) * Float32(2.0)))) * cbrt(exp(t_0))) * Float32(Float32(0.5) / v))
end

Derivation

1. Initial program 99.6%

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

2. Simplified 99.6%

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

   [Start] 99.6	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   exp-sum [=>] 99.6	\[ \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}\right)}^{2}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}}\right)} \cdot \frac{0.5}{v} \]
   Proof

   [Start] 99.6	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   add-cbrt-cube [=>] 98.6	\[ \color{blue}{\sqrt[3]{\left(e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)}}} \cdot \frac{0.5}{v} \]
   cbrt-prod [=>] 99.0	\[ \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)}}\right)} \cdot \frac{0.5}{v} \]

4. Applied egg-rr 99.0%

   \[\leadsto \left(\color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)\right) \cdot 2}} \cdot 1\right)} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \frac{0.5}{v} \]
   Proof

   [Start] 99.0	\[ \left(\sqrt[3]{{\left(e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}\right)}^{2}} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \frac{0.5}{v} \]
   *-un-lft-identity [=>] 99.0	\[ \left(\color{blue}{\left(1 \cdot \sqrt[3]{{\left(e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}\right)}^{2}}\right)} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \frac{0.5}{v} \]
   *-commutative [=>] 99.0	\[ \left(\color{blue}{\left(\sqrt[3]{{\left(e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}\right)}^{2}} \cdot 1\right)} \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \frac{0.5}{v} \]
   pow-exp [=>] 99.0	\[ \left(\left(\sqrt[3]{\color{blue}{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)\right) \cdot 2}}} \cdot 1\right) \cdot \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)}}\right) \cdot \frac{0.5}{v} \]

5. Taylor expanded in sinTheta_i around 0 99.0%

   \[\leadsto \left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)\right) \cdot 2}} \cdot 1\right) \cdot \sqrt[3]{\color{blue}{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}}}\right) \cdot \frac{0.5}{v} \]

6. Taylor expanded in cosTheta_i around 0 99.0%

   \[\leadsto \left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)\right) \cdot 2}} \cdot 1\right) \cdot \color{blue}{{\left(e^{0.6931 - \frac{1}{v}}\right)}^{0.3333333333333333}}\right) \cdot \frac{0.5}{v} \]

7. Simplified 99.0%

   \[\leadsto \left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)\right) \cdot 2}} \cdot 1\right) \cdot \color{blue}{\sqrt[3]{e^{0.6931 - \frac{1}{v}}}}\right) \cdot \frac{0.5}{v} \]
   Proof

   [Start] 99.0	\[ \left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)\right) \cdot 2}} \cdot 1\right) \cdot {\left(e^{0.6931 - \frac{1}{v}}\right)}^{0.3333333333333333}\right) \cdot \frac{0.5}{v} \]
   unpow1/3 [=>] 99.0	\[ \left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)\right) \cdot 2}} \cdot 1\right) \cdot \color{blue}{\sqrt[3]{e^{0.6931 - \frac{1}{v}}}}\right) \cdot \frac{0.5}{v} \]

8. Final simplification 99.0%

   \[\leadsto \left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 - \frac{1}{v}\right)\right) \cdot 2}} \cdot \sqrt[3]{e^{0.6931 - \frac{1}{v}}}\right) \cdot \frac{0.5}{v} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	10080

\[\frac{0.5}{v} \cdot \left(e^{0.23103333333333334 + \frac{-0.3333333333333333}{v}} \cdot \sqrt[3]{e^{1.3862 + \frac{-2}{v}}}\right) \]

Alternative 2
Accuracy	99.5%
Cost	6688

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

Alternative 3
Accuracy	99.5%
Cost	3552

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

Alternative 4
Accuracy	99.6%
Cost	3488

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

Alternative 5
Accuracy	97.8%
Cost	3296

\[e^{\frac{-1}{v}} \]

Alternative 6
Accuracy	20.3%
Cost	160

\[cosTheta_O \cdot \frac{cosTheta_i}{v} \]

Alternative 7
Accuracy	38.7%
Cost	160

\[\frac{cosTheta_i \cdot cosTheta_O}{v} \]

Alternative 8
Accuracy	6.4%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023151 
(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))))))