HairBSDF, Mp, upper

Average Accuracy: 98.6% → 98.7%

Time: 19.8s

Precision: binary32

Cost: 7104

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

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

Math

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

↓

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

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
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (* (/ cosTheta_i v) cosTheta_O))
  (/ (* (sinh (/ 1.0 v)) 2.0) (/ 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 (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i / v) * cosTheta_O)) / ((sinhf((1.0f / v)) * 2.0f) / (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 = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i / v) * costheta_o)) / ((sinh((1.0e0 / v)) * 2.0e0) / (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(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i / v) * cosTheta_O)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) / 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 = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i / v) * cosTheta_O)) / ((sinh((single(1.0) / v)) * single(2.0)) / (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. Applied egg-rr 98.7%

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

   [Start] 98.6	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   /-rgt-identity [<=] 98.6	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \color{blue}{\frac{v}{1}}} \]
   clear-num [=>] 98.6	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \color{blue}{\frac{1}{\frac{1}{v}}}} \]
   associate-*r/ [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{\frac{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot 1}{\frac{1}{v}}}} \]
   *-commutative [<=] 98.7	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\frac{\color{blue}{1 \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}}{\frac{1}{v}}} \]
   *-un-lft-identity [<=] 98.7	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{\frac{1}{v}}} \]

3. Applied egg-rr 98.7%

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

   [Start] 98.7	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}} \]
   associate-/l* [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}} \]
   associate-/r/ [=>] 98.7	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)}}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}} \]

4. Final simplification 98.7%

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

Alternatives

Alternative 1
Accuracy	98.5%
Cost	7008

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

Alternative 2
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 3
Accuracy	98.4%
Cost	6880

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

Alternative 4
Accuracy	98.4%
Cost	6880

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

Alternative 5
Accuracy	98.4%
Cost	6880

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

Alternative 6
Accuracy	93.1%
Cost	3680

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

Alternative 7
Accuracy	73.1%
Cost	3616

\[\frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} + -0.3333333333333333} \]

Alternative 8
Accuracy	58.9%
Cost	288

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

Alternative 9
Accuracy	58.9%
Cost	288

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

Alternative 10
Accuracy	58.4%
Cost	224

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

Alternative 11
Accuracy	58.4%
Cost	224

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

Alternative 12
Accuracy	58.9%
Cost	224

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

Reproduce

herbie shell --seed 2023131 
(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)))