HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 20.7s

Precision: binary32

Cost: 10240

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

\[ \begin{array}{c}[cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \end{array} \]

Math

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

↓

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

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 (* 2.0 (pow (exp (/ sinTheta_i v)) sinTheta_O)))
   (sinh (/ 1.0 v)))
  (* (/ 1.0 v) (/ cosTheta_O 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 / (2.0f * powf(expf((sinTheta_i / v)), sinTheta_O))) / sinhf((1.0f / v))) * ((1.0f / v) * (cosTheta_O / 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 / (2.0e0 * (exp((sintheta_i / v)) ** sintheta_o))) / sinh((1.0e0 / v))) * ((1.0e0 / v) * (costheta_o / 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(Float32(cosTheta_i / Float32(Float32(2.0) * (exp(Float32(sinTheta_i / v)) ^ sinTheta_O))) / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_O / 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 / (single(2.0) * (exp((sinTheta_i / v)) ^ sinTheta_O))) / sinh((single(1.0) / v))) * ((single(1.0) / v) * (cosTheta_O / v));
end

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

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

   [Start] 0.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-*r/ [=>] 0.5	\[ \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-/l/ [=>] 0.4	\[ \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]
   associate-*l* [=>] 0.4	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot v\right)}} \]
   sinh-def [=>] 0.4	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\color{blue}{\frac{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}{2}} \cdot 2\right) \cdot \left(v \cdot v\right)} \]
   associate-*l/ [=>] 0.4	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\color{blue}{\frac{\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot 2}{2}} \cdot \left(v \cdot v\right)} \]
   associate-/l* [=>] 0.4	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\color{blue}{\frac{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}{\frac{2}{2}}} \cdot \left(v \cdot v\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\frac{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}{\color{blue}{1}} \cdot \left(v \cdot v\right)} \]
   /-rgt-identity [=>] 0.4	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\color{blue}{\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)} \cdot \left(v \cdot v\right)} \]
   associate-*r* [=>] 0.4	\[ \frac{\color{blue}{\left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot cosTheta_i\right) \cdot cosTheta_O}}{\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)} \]
   times-frac [=>] 0.4	\[ \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot cosTheta_i}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}} \cdot \frac{cosTheta_O}{v \cdot v}} \]

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	7104

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

Alternative 2
Error	0.4
Cost	7104

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

Alternative 3
Error	0.4
Cost	7104

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

Alternative 4
Error	0.4
Cost	7040

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

Alternative 5
Error	0.4
Cost	7008

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

Alternative 6
Error	0.5
Cost	6944

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

Alternative 7
Error	0.5
Cost	3616

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

Alternative 8
Error	0.5
Cost	3616

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

Alternative 9
Error	0.5
Cost	3616

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

Alternative 10
Error	11.4
Cost	416

\[\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

Alternative 11
Error	13.1
Cost	288

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

Alternative 12
Error	13.1
Cost	288

\[\frac{1}{\frac{v}{\frac{cosTheta_i \cdot cosTheta_O}{2}}} \]

Alternative 13
Error	13.3
Cost	224

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

Alternative 14
Error	13.3
Cost	224

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

Alternative 15
Error	13.3
Cost	224

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

Alternative 16
Error	13.1
Cost	224

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

Reproduce

herbie shell --seed 2023060 
(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)))