HairBSDF, Mp, upper

Average Accuracy: 98.5% → 98.7%

Time: 19.3s

Precision: binary32

Cost: 10176

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

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

Math

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

↓

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

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 (sinh (/ 1.0 v)))
  (* (/ cosTheta_O v) (/ (/ 0.5 v) (pow (exp (/ sinTheta_i v)) sinTheta_O)))))

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 / sinhf((1.0f / v))) * ((cosTheta_O / v) * ((0.5f / v) / powf(expf((sinTheta_i / v)), sinTheta_O)));
}

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 / sinh((1.0e0 / v))) * ((costheta_o / v) * ((0.5e0 / v) / (exp((sintheta_i / v)) ** sintheta_o)))
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 / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(cosTheta_O / v) * Float32(Float32(Float32(0.5) / v) / (exp(Float32(sinTheta_i / v)) ^ sinTheta_O))))
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 / sinh((single(1.0) / v))) * ((cosTheta_O / v) * ((single(0.5) / v) / (exp((sinTheta_i / v)) ^ sinTheta_O)));
end

Derivation

1. Initial program 98.5%

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

2. Simplified 98.5%

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

   [Start] 98.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 98.5	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   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* [=>] 98.5	\[ \color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-/r/ [=>] 98.5	\[ \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 98.5	\[ \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-/r* [=>] 98.5	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{v}} \]

3. Applied egg-rr 98.1%

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

   [Start] 98.5	\[ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot {\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v} \]
   associate-*r/ [=>] 98.2	\[ \color{blue}{\frac{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot {\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v}} \]
   clear-num [=>] 98.2	\[ \frac{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{1}{\frac{\sinh \left(\frac{1}{v}\right) \cdot {\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}{0.5}}}}{v} \]
   un-div-inv [=>] 98.2	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot {\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}{0.5}}}}{v} \]
   associate-/l* [=>] 98.2	\[ \frac{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}}}}{v} \]
   associate-/r/ [=>] 98.1	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}}{v} \]

4. Applied egg-rr 51.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{v}}{v} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}\right)} - 1} \]
   Proof

   [Start] 98.1	\[ \frac{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v} \]
   expm1-log1p-u [=>] 98.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v}\right)\right)} \]
   expm1-udef [=>] 51.3	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v}\right)} - 1} \]
   associate-/l* [=>] 51.3	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)}}{\frac{v}{\frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}}}\right)} - 1 \]
   associate-/r/ [=>] 51.3	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)}}{v} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}\right)} - 1 \]
   associate-/l* [=>] 51.3	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{cosTheta_i}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta_O}{v}}}}}{v} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}\right)} - 1 \]
   associate-/r/ [=>] 51.3	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{v}}}{v} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}\right)} - 1 \]

5. Simplified 98.7%

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

   [Start] 51.3	\[ e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{v}}{v} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}\right)} - 1 \]
   expm1-def [=>] 98.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{v}}{v} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}\right)\right)} \]
   expm1-log1p [=>] 98.3	\[ \color{blue}{\frac{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{v}}{v} \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}} \]
   associate-*l/ [=>] 98.1	\[ \color{blue}{\frac{\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v}} \]
   associate-*r/ [<=] 98.5	\[ \color{blue}{\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v}} \]
   associate-*l* [=>] 98.7	\[ \color{blue}{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_O}{v} \cdot \frac{\frac{0.5}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v}\right)} \]
   associate-/l/ [=>] 98.7	\[ \frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_O}{v} \cdot \color{blue}{\frac{0.5}{v \cdot {\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}\right) \]
   associate-/r* [=>] 98.7	\[ \frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_O}{v} \cdot \color{blue}{\frac{\frac{0.5}{v}}{{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}\right) \]

6. Final simplification 98.7%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	7104

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

Alternative 2
Accuracy	98.4%
Cost	6944

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

Alternative 3
Accuracy	98.4%
Cost	3680

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

Alternative 4
Accuracy	67.2%
Cost	3552

\[cosTheta_i \cdot \frac{cosTheta_O}{\left(v \cdot v\right) \cdot \mathsf{expm1}\left(\frac{1}{v}\right)} \]

Alternative 5
Accuracy	67.2%
Cost	3552

\[\frac{cosTheta_i \cdot cosTheta_O}{\left(v \cdot v\right) \cdot \mathsf{expm1}\left(\frac{1}{v}\right)} \]

Alternative 6
Accuracy	75.3%
Cost	3552

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

Alternative 7
Accuracy	57.1%
Cost	224

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

Alternative 8
Accuracy	57.1%
Cost	224

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

Alternative 9
Accuracy	57.1%
Cost	224

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

Alternative 10
Accuracy	57.5%
Cost	224

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

Alternative 11
Accuracy	54.3%
Cost	160

\[2 \cdot \left(cosTheta_i \cdot cosTheta_O\right) \]

Alternative 12
Accuracy	54.3%
Cost	160

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

Alternative 13
Accuracy	54.3%
Cost	160

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

Alternative 14
Accuracy	54.3%
Cost	160

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

Reproduce

herbie shell --seed 2023138 
(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)))