HairBSDF, Mp, upper

Average Accuracy: 98.6% → 98.8%

Time: 23.4s

Precision: binary32

Cost: 7168

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

Math

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

↓

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

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
 (*
  (/ (exp (/ (- sinTheta_i) (/ v sinTheta_O))) (* (sinh (/ 1.0 v)) 2.0))
  (* (/ -1.0 v) (* (/ -1.0 v) (* cosTheta_i cosTheta_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 (expf((-sinTheta_i / (v / sinTheta_O))) / (sinhf((1.0f / v)) * 2.0f)) * ((-1.0f / v) * ((-1.0f / v) * (cosTheta_i * cosTheta_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 = (exp((-sintheta_i / (v / sintheta_o))) / (sinh((1.0e0 / v)) * 2.0e0)) * (((-1.0e0) / v) * (((-1.0e0) / v) * (costheta_i * costheta_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(exp(Float32(Float32(-sinTheta_i) / Float32(v / sinTheta_O))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) * Float32(Float32(Float32(-1.0) / v) * Float32(Float32(Float32(-1.0) / v) * Float32(cosTheta_i * cosTheta_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 = (exp((-sinTheta_i / (v / sinTheta_O))) / (sinh((single(1.0) / v)) * single(2.0))) * ((single(-1.0) / v) * ((single(-1.0) / v) * (cosTheta_i * cosTheta_O)));
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 98.5%

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

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

3. Applied egg-rr 98.7%

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

   [Start] 98.5	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]
   frac-2neg [=>] 98.5	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\frac{-\frac{cosTheta_i \cdot cosTheta_O}{v}}{-v}} \]
   div-inv [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(-\frac{cosTheta_i \cdot cosTheta_O}{v}\right) \cdot \frac{1}{-v}\right)} \]
   distribute-neg-frac [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\frac{-cosTheta_i \cdot cosTheta_O}{v}} \cdot \frac{1}{-v}\right) \]
   distribute-rgt-neg-in [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{\color{blue}{cosTheta_i \cdot \left(-cosTheta_O\right)}}{v} \cdot \frac{1}{-v}\right) \]
   metadata-eval [<=] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{cosTheta_i \cdot \left(-cosTheta_O\right)}{v} \cdot \frac{\color{blue}{--1}}{-v}\right) \]
   distribute-neg-frac [<=] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{cosTheta_i \cdot \left(-cosTheta_O\right)}{v} \cdot \color{blue}{\left(-\frac{-1}{-v}\right)}\right) \]
   metadata-eval [<=] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{cosTheta_i \cdot \left(-cosTheta_O\right)}{v} \cdot \left(-\frac{\color{blue}{-1}}{-v}\right)\right) \]
   frac-2neg [<=] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{cosTheta_i \cdot \left(-cosTheta_O\right)}{v} \cdot \left(-\color{blue}{\frac{1}{v}}\right)\right) \]
   distribute-neg-frac [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{cosTheta_i \cdot \left(-cosTheta_O\right)}{v} \cdot \color{blue}{\frac{-1}{v}}\right) \]
   metadata-eval [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{cosTheta_i \cdot \left(-cosTheta_O\right)}{v} \cdot \frac{\color{blue}{-1}}{v}\right) \]

4. Applied egg-rr 98.8%

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

   [Start] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{cosTheta_i \cdot \left(-cosTheta_O\right)}{v} \cdot \frac{-1}{v}\right) \]
   frac-2neg [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\frac{-cosTheta_i \cdot \left(-cosTheta_O\right)}{-v}} \cdot \frac{-1}{v}\right) \]
   div-inv [=>] 98.8	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(\left(-cosTheta_i \cdot \left(-cosTheta_O\right)\right) \cdot \frac{1}{-v}\right)} \cdot \frac{-1}{v}\right) \]
   metadata-eval [<=] 98.8	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(\left(-cosTheta_i \cdot \left(-cosTheta_O\right)\right) \cdot \frac{\color{blue}{--1}}{-v}\right) \cdot \frac{-1}{v}\right) \]
   frac-2neg [<=] 98.8	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(\left(-cosTheta_i \cdot \left(-cosTheta_O\right)\right) \cdot \color{blue}{\frac{-1}{v}}\right) \cdot \frac{-1}{v}\right) \]
   distribute-lft-neg-in [<=] 98.8	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(-\left(cosTheta_i \cdot \left(-cosTheta_O\right)\right) \cdot \frac{-1}{v}\right)} \cdot \frac{-1}{v}\right) \]
   *-commutative [=>] 98.8	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(-\color{blue}{\frac{-1}{v} \cdot \left(cosTheta_i \cdot \left(-cosTheta_O\right)\right)}\right) \cdot \frac{-1}{v}\right) \]
   distribute-rgt-neg-in [=>] 98.8	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(\frac{-1}{v} \cdot \left(-cosTheta_i \cdot \left(-cosTheta_O\right)\right)\right)} \cdot \frac{-1}{v}\right) \]
   distribute-rgt-neg-out [=>] 98.8	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(\frac{-1}{v} \cdot \left(-\color{blue}{\left(-cosTheta_i \cdot cosTheta_O\right)}\right)\right) \cdot \frac{-1}{v}\right) \]
   remove-double-neg [=>] 98.8	\[ \frac{e^{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(\frac{-1}{v} \cdot \color{blue}{\left(cosTheta_i \cdot cosTheta_O\right)}\right) \cdot \frac{-1}{v}\right) \]

5. Final simplification 98.8%

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

Alternatives

Alternative 1
Accuracy	98.9%
Cost	7136

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

Alternative 2
Accuracy	98.5%
Cost	7072

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

Alternative 3
Accuracy	98.8%
Cost	7072

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

Alternative 4
Accuracy	98.7%
Cost	7072

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

Alternative 5
Accuracy	98.5%
Cost	7040

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

Alternative 6
Accuracy	98.6%
Cost	7008

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

Alternative 7
Accuracy	98.5%
Cost	7008

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

Alternative 8
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 9
Accuracy	98.5%
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 10
Accuracy	98.4%
Cost	6880

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

Alternative 11
Accuracy	98.4%
Cost	6880

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

Alternative 12
Accuracy	98.4%
Cost	6880

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

Alternative 13
Accuracy	72.9%
Cost	4164

\[\begin{array}{l} \mathbf{if}\;v \leq 0.5099999904632568:\\ \;\;\;\;\frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \cdot \frac{1}{-1 + e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{v \cdot v} \cdot \left(\frac{cosTheta_O \cdot -0.004166666666666667 - cosTheta_O \cdot -0.013888888888888888}{{v}^{3}} + \left(\frac{cosTheta_O}{v} \cdot -0.08333333333333333 + 0.5 \cdot \left(v \cdot cosTheta_O\right)\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	72.9%
Cost	3908

\[\begin{array}{l} \mathbf{if}\;v \leq 0.5099999904632568:\\ \;\;\;\;\frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \cdot \frac{1}{-1 + e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 + \frac{-0.08333333333333333}{v}\right)\right) \cdot \left(\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{v}\right)\\ \end{array} \]

Alternative 15
Accuracy	72.0%
Cost	3812

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

Alternative 16
Accuracy	68.3%
Cost	3680

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

Alternative 17
Accuracy	68.3%
Cost	3616

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

Alternative 18
Accuracy	58.9%
Cost	288

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

Alternative 19
Accuracy	58.3%
Cost	224

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

Alternative 20
Accuracy	58.3%
Cost	224

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

Alternative 21
Accuracy	58.3%
Cost	224

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

Alternative 22
Accuracy	58.8%
Cost	224

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

Reproduce

herbie shell --seed 2023135 
(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)))