HairBSDF, Mp, upper

Percentage Accurate: 99.4% → 99.5%

Time: 24.2s

Precision: binary32

Cost: 6944

• Unsound rule application detected in e-graph. Results may not be sound.

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

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

Math

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

↓

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

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 v))
  (/ (/ 1.0 v) (- (exp (/ 1.0 v)) (exp (/ -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 / v)) * ((1.0f / v) / (expf((1.0f / v)) - expf((-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 / v)) * ((1.0e0 / v) / (exp((1.0e0 / v)) - exp(((-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 * Float32(cosTheta_O / v)) * Float32(Float32(Float32(1.0) / v) / Float32(exp(Float32(Float32(1.0) / v)) - exp(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 / v)) * ((single(1.0) / v) / (exp((single(1.0) / v)) - exp((single(-1.0) / v))));
end

Derivation

1. Initial program 99.3%

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

2. Simplified 99.3%

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

   [Start] 99.3%	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 99.3%	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-*r/ [<=] 99.3%	\[ \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]
   *-commutative [=>] 99.3%	\[ \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-*l/ [<=] 99.3%	\[ \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 99.3%	\[ \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
   associate-*r* [=>] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]
   associate-/l/ [<=] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]
   exp-neg [=>] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-/l/ [=>] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-/r* [=>] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   metadata-eval [=>] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-*l/ [<=] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   *-commutative [=>] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   exp-prod [=>] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Applied egg-rr 99.4%

   \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{e^{\left(\log 0.5 - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) - \log \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)}} \]
   Step-by-step derivation

   [Start] 99.3%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   add-exp-log [=>] 99.4%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{e^{\log \left(\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}\right)}} \]
   log-div [=>] 99.4%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\log \left(\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}\right) - \log \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)}} \]
   log-div [=>] 99.4%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\left(\log 0.5 - \log \left({\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}\right)\right)} - \log \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)} \]
   pow-exp [=>] 99.4%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\left(\log 0.5 - \log \color{blue}{\left(e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right)}\right) - \log \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)} \]
   add-log-exp [<=] 99.4%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\left(\log 0.5 - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) - \log \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)} \]

4. Taylor expanded in sinTheta_O around 0 99.4%

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

5. Simplified 99.5%

   \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
   Step-by-step derivation

   [Start] 99.4%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\log 0.5 - \log \left(0.5 \cdot \left(v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)\right)} \]
   log-prod [=>] 99.4%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\log 0.5 - \color{blue}{\left(\log 0.5 + \log \left(v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)\right)}} \]
   associate--r+ [=>] 99.4%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\left(\log 0.5 - \log 0.5\right) - \log \left(v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   exp-diff [=>] 99.5%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{e^{\log 0.5 - \log 0.5}}{e^{\log \left(v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}}} \]
   +-inverses [=>] 99.5%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{\color{blue}{0}}}{e^{\log \left(v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   1-exp [<=] 99.5%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{1}}{e^{\log \left(v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   sub-neg [=>] 99.5%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{e^{\log \left(v \cdot \color{blue}{\left(e^{\frac{1}{v}} + \left(-\frac{1}{e^{\frac{1}{v}}}\right)\right)}\right)}} \]
   distribute-lft-in [=>] 99.5%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{e^{\log \color{blue}{\left(v \cdot e^{\frac{1}{v}} + v \cdot \left(-\frac{1}{e^{\frac{1}{v}}}\right)\right)}}} \]
   rec-exp [=>] 99.5%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{e^{\log \left(v \cdot e^{\frac{1}{v}} + v \cdot \left(-\color{blue}{e^{-\frac{1}{v}}}\right)\right)}} \]
   distribute-neg-frac [=>] 99.5%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{e^{\log \left(v \cdot e^{\frac{1}{v}} + v \cdot \left(-e^{\color{blue}{\frac{-1}{v}}}\right)\right)}} \]
   metadata-eval [=>] 99.5%	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{e^{\log \left(v \cdot e^{\frac{1}{v}} + v \cdot \left(-e^{\frac{\color{blue}{-1}}{v}}\right)\right)}} \]

6. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6944

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

Alternative 2
Accuracy	99.5%
Cost	6880

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

Alternative 3
Accuracy	86.1%
Cost	3680

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

Alternative 4
Accuracy	86.1%
Cost	3616

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

Alternative 5
Accuracy	80.7%
Cost	224

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

Alternative 6
Accuracy	80.7%
Cost	224

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

Alternative 7
Accuracy	81.2%
Cost	224

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

Reproduce

herbie shell --seed 2023256 
(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)))