HairBSDF, Mp, upper

Average Accuracy: 98.6% → 98.8%

Time: 20.6s

Precision: binary32

Cost: 7072

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

Math

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

↓

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

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 cosTheta_O)
   (* (/ 0.5 (* v (exp (/ sinTheta_O (/ v sinTheta_i))))) (/ 1.0 v)))
  (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 * cosTheta_O) * ((0.5f / (v * expf((sinTheta_O / (v / sinTheta_i))))) * (1.0f / v))) / 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 * costheta_o) * ((0.5e0 / (v * exp((sintheta_o / (v / sintheta_i))))) * (1.0e0 / v))) / 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(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(Float32(0.5) / Float32(v * exp(Float32(sinTheta_O / Float32(v / sinTheta_i))))) * Float32(Float32(1.0) / v))) / 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 * cosTheta_O) * ((single(0.5) / (v * exp((sinTheta_O / (v / sinTheta_i))))) * (single(1.0) / v))) / 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%

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

   [Start] 98.6	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 98.6	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-*r/ [<=] 98.5	\[ \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]
   *-commutative [=>] 98.5	\[ \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-*l/ [<=] 98.6	\[ \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 98.6	\[ \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
   associate-*r* [=>] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]
   associate-/l/ [<=] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]
   exp-neg [=>] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-/l/ [=>] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-/r* [=>] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   metadata-eval [=>] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-*l/ [<=] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   *-commutative [=>] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   exp-prod [=>] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Applied egg-rr 98.7%

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

   [Start] 98.6	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   *-commutative [=>] 98.6	\[ \color{blue}{\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \]
   associate-/r* [=>] 98.8	\[ \color{blue}{\frac{\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v}}{\sinh \left(\frac{1}{v}\right)}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \]
   associate-*l/ [=>] 98.7	\[ \color{blue}{\frac{\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right)}} \]
   associate-/l/ [=>] 98.7	\[ \frac{\color{blue}{\frac{0.5}{v \cdot {\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]
   pow-exp [=>] 98.7	\[ \frac{\frac{0.5}{v \cdot \color{blue}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]

4. Applied egg-rr 98.7%

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

   [Start] 98.7	\[ \frac{\frac{0.5}{v \cdot e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]
   associate-*r/ [=>] 98.7	\[ \frac{\frac{0.5}{v \cdot e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right)} \]
   associate-*r/ [=>] 98.7	\[ \frac{\color{blue}{\frac{\frac{0.5}{v \cdot e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v}}}{\sinh \left(\frac{1}{v}\right)} \]
   associate-/r* [=>] 98.7	\[ \frac{\frac{\color{blue}{\frac{\frac{0.5}{v}}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v}}{\sinh \left(\frac{1}{v}\right)} \]
   exp-prod [=>] 98.7	\[ \frac{\frac{\frac{\frac{0.5}{v}}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v}}{\sinh \left(\frac{1}{v}\right)} \]

5. Applied egg-rr 98.8%

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

   [Start] 98.7	\[ \frac{\frac{\frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v}}{\sinh \left(\frac{1}{v}\right)} \]
   div-inv [=>] 98.8	\[ \frac{\color{blue}{\left(\frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)\right) \cdot \frac{1}{v}}}{\sinh \left(\frac{1}{v}\right)} \]
   *-commutative [=>] 98.8	\[ \frac{\color{blue}{\left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}\right)} \cdot \frac{1}{v}}{\sinh \left(\frac{1}{v}\right)} \]
   associate-*l* [=>] 98.8	\[ \frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot \left(\frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}} \cdot \frac{1}{v}\right)}}{\sinh \left(\frac{1}{v}\right)} \]
   associate-/l/ [=>] 98.8	\[ \frac{\left(cosTheta_i \cdot cosTheta_O\right) \cdot \left(\color{blue}{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)} \cdot v}} \cdot \frac{1}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]
   *-commutative [=>] 98.8	\[ \frac{\left(cosTheta_i \cdot cosTheta_O\right) \cdot \left(\frac{0.5}{\color{blue}{v \cdot {\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}} \cdot \frac{1}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]
   pow-exp [=>] 98.8	\[ \frac{\left(cosTheta_i \cdot cosTheta_O\right) \cdot \left(\frac{0.5}{v \cdot \color{blue}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}} \cdot \frac{1}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]
   clear-num [=>] 98.8	\[ \frac{\left(cosTheta_i \cdot cosTheta_O\right) \cdot \left(\frac{0.5}{v \cdot e^{sinTheta_O \cdot \color{blue}{\frac{1}{\frac{v}{sinTheta_i}}}}} \cdot \frac{1}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]
   un-div-inv [=>] 98.8	\[ \frac{\left(cosTheta_i \cdot cosTheta_O\right) \cdot \left(\frac{0.5}{v \cdot e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}} \cdot \frac{1}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]

6. Final simplification 98.8%

   \[\leadsto \frac{\left(cosTheta_i \cdot cosTheta_O\right) \cdot \left(\frac{0.5}{v \cdot e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}} \cdot \frac{1}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]

Alternatives

Alternative 1
Accuracy	98.8%
Cost	7008

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

Alternative 2
Accuracy	98.3%
Cost	3616

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

Alternative 3
Accuracy	98.3%
Cost	3616

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

Alternative 4
Accuracy	98.5%
Cost	3616

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

Alternative 5
Accuracy	58.5%
Cost	480

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

Alternative 6
Accuracy	58.0%
Cost	224

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

Alternative 7
Accuracy	58.0%
Cost	224

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

Alternative 8
Accuracy	58.0%
Cost	224

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

Alternative 9
Accuracy	58.5%
Cost	224

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

Reproduce

herbie shell --seed 2023146 
(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)))