HairBSDF, Mp, upper

Average Accuracy: 98.6% → 98.8%

Time: 25.0s

Precision: binary32

Cost: 10240

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

Math

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

↓

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

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_i (pow (exp sinTheta_i) (/ sinTheta_O v)))
  (/ (* (/ cosTheta_O v) (/ 1.0 v)) (* 2.0 (sinh (/ 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_i / powf(expf(sinTheta_i), (sinTheta_O / v))) * (((cosTheta_O / v) * (1.0f / v)) / (2.0f * sinhf((1.0f / v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

↓

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i / (exp(sintheta_i) ** (sintheta_o / v))) * (((costheta_o / v) * (1.0e0 / v)) / (2.0e0 * sinh((1.0e0 / v))))
end function

Julia

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

↓

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i / (exp(sinTheta_i) ^ Float32(sinTheta_O / v))) * Float32(Float32(Float32(cosTheta_O / v) * Float32(Float32(1.0) / v)) / Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

↓

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / (exp(sinTheta_i) ^ (sinTheta_O / v))) * (((cosTheta_O / v) * (single(1.0) / v)) / (single(2.0) * sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.6%

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

2. Simplified 98.6%

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

   [Start] 98.6	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-*l/ [<=] 98.5	\[ \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
   times-frac [<=] 98.6	\[ \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]
   exp-neg [=>] 98.6	\[ \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]
   associate-*l/ [=>] 98.6	\[ \frac{\color{blue}{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]
   *-lft-identity [=>] 98.6	\[ \frac{\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]
   associate-/l/ [=>] 98.6	\[ \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]
   associate-*l* [=>] 98.6	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)} \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   associate-*l* [=>] 98.6	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(2 \cdot v\right) \cdot v\right)\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   *-commutative [=>] 98.6	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot 2\right)} \cdot v\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   *-commutative [=>] 98.6	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}} \]
   associate-*l/ [<=] 98.6	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

3. Applied egg-rr 98.6%

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

   [Start] 98.6	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   *-commutative [=>] 98.6	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i} \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)}} \]
   times-frac [=>] 98.6	\[ \color{blue}{\frac{cosTheta_i}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \cdot \frac{cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)}} \]
   *-commutative [=>] 98.6	\[ \frac{cosTheta_i}{e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}} \cdot \frac{cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)} \]
   exp-prod [=>] 98.6	\[ \frac{cosTheta_i}{\color{blue}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}} \cdot \frac{cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)} \]
   associate-*r* [=>] 98.6	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \frac{cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot v}} \]
   *-commutative [=>] 98.6	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \frac{cosTheta_O}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)}} \]

4. Applied egg-rr 98.8%

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

   [Start] 98.6	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \frac{cosTheta_O}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \]
   *-un-lft-identity [=>] 98.6	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \frac{\color{blue}{1 \cdot cosTheta_O}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \]
   times-frac [=>] 98.8	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]
   *-commutative [=>] 98.8	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \left(\frac{1}{v} \cdot \frac{cosTheta_O}{\color{blue}{\left(v \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)}}\right) \]
   associate-*l* [=>] 98.8	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \left(\frac{1}{v} \cdot \frac{cosTheta_O}{\color{blue}{v \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}}\right) \]

5. Applied egg-rr 98.8%

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

   [Start] 98.8	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \left(\frac{1}{v} \cdot \frac{cosTheta_O}{v \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}\right) \]
   *-commutative [=>] 98.8	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \color{blue}{\left(\frac{cosTheta_O}{v \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)} \cdot \frac{1}{v}\right)} \]
   associate-/r* [=>] 98.8	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \left(\color{blue}{\frac{\frac{cosTheta_O}{v}}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \cdot \frac{1}{v}\right) \]
   associate-*l/ [=>] 98.8	\[ \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \cdot \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot \frac{1}{v}}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \]

6. Final simplification 98.8%

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

Alternatives

Alternative 1
Accuracy	98.8%
Cost	10240

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

Alternative 2
Accuracy	98.7%
Cost	7104

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

Alternative 3
Accuracy	98.3%
Cost	7008

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

Alternative 4
Accuracy	98.6%
Cost	7008

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

Alternative 5
Accuracy	98.6%
Cost	7008

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

Alternative 6
Accuracy	98.6%
Cost	7008

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

Alternative 7
Accuracy	98.3%
Cost	6880

\[\frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]

Alternative 8
Accuracy	73.1%
Cost	3844

\[\begin{array}{l} t_0 := cosTheta_i \cdot \frac{cosTheta_O}{v}\\ \mathbf{if}\;v \leq 0.49000000953674316:\\ \;\;\;\;t_0 \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	72.2%
Cost	3780

\[\begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}}\\ \end{array} \]

Alternative 10
Accuracy	68.6%
Cost	3680

\[\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)} \]

Alternative 11
Accuracy	68.6%
Cost	3616

\[\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{e^{\frac{1}{v}} + -1} \]

Alternative 12
Accuracy	61.2%
Cost	800

\[\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\left(\left(1 + \frac{1}{v}\right) + \frac{0.5}{v \cdot v}\right) + \left(-1 - \frac{-1}{v}\right)} \]

Alternative 13
Accuracy	61.2%
Cost	544

\[\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\frac{2}{v} + \frac{0.5}{v \cdot v}} \]

Alternative 14
Accuracy	59.1%
Cost	288

\[\frac{1}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O \cdot 0.5}} \]

Alternative 15
Accuracy	58.5%
Cost	224

\[0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \]

Alternative 16
Accuracy	58.5%
Cost	224

\[0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}} \]

Alternative 17
Accuracy	58.5%
Cost	224

\[0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]

Alternative 18
Accuracy	59.0%
Cost	224

\[\frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \]

Reproduce

herbie shell --seed 2023151 
(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)))