HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 19.4s

Precision: binary32

Cost: 7008

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

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

Math

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

↓

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

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
 (/
  (/ (* (/ 0.5 v) cosTheta_i) (sinh (/ 1.0 v)))
  (* (/ v cosTheta_O) (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 (((0.5f / v) * cosTheta_i) / sinhf((1.0f / v))) / ((v / cosTheta_O) * 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 = (((0.5e0 / v) * costheta_i) / sinh((1.0e0 / v))) / ((v / costheta_o) * 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(Float32(Float32(Float32(0.5) / v) * cosTheta_i) / sinh(Float32(Float32(1.0) / v))) / Float32(Float32(v / cosTheta_O) * exp(Float32(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 = (((single(0.5) / v) * cosTheta_i) / sinh((single(1.0) / v))) / ((v / cosTheta_O) * exp(((sinTheta_i / v) * sinTheta_O)));
end

Derivation

1. Initial program 0.5

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

2. Simplified 0.5

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\sinh \left(\frac{1}{v}\right) \cdot {\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}}{v}} \]
   Proof
   (*.f32 (*.f32 cosTheta_i (/.f32 cosTheta_O v)) (/.f32 (/.f32 1/2 (*.f32 (sinh.f32 (/.f32 1 v)) (pow.f32 (exp.f32 (/.f32 sinTheta_i v)) sinTheta_O))) v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i)) (/.f32 (/.f32 1/2 (*.f32 (sinh.f32 (/.f32 1 v)) (pow.f32 (exp.f32 (/.f32 sinTheta_i v)) sinTheta_O))) v)): 0 points increase in error, 12 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (/.f32 (/.f32 (Rewrite<= metadata-eval (/.f32 1 2)) (*.f32 (sinh.f32 (/.f32 1 v)) (pow.f32 (exp.f32 (/.f32 sinTheta_i v)) sinTheta_O))) v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (/.f32 (/.f32 (/.f32 1 2) (*.f32 (sinh.f32 (/.f32 1 v)) (Rewrite<= exp-prod_binary32 (exp.f32 (*.f32 (/.f32 sinTheta_i v) sinTheta_O))))) v)): 5 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (/.f32 (/.f32 (/.f32 1 2) (*.f32 (sinh.f32 (/.f32 1 v)) (exp.f32 (Rewrite=> associate-*l/_binary32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))))) v)): 0 points increase in error, 5 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (/.f32 (Rewrite<= associate-/r*_binary32 (/.f32 1 (*.f32 2 (*.f32 (sinh.f32 (/.f32 1 v)) (exp.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))))) v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (/.f32 (/.f32 1 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 2 (sinh.f32 (/.f32 1 v))) (exp.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))))) v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (/.f32 (/.f32 1 (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (sinh.f32 (/.f32 1 v)) 2)) (exp.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))) v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (/.f32 (Rewrite<= associate-/l/_binary32 (/.f32 (/.f32 1 (exp.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2))) v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (/.f32 (/.f32 (Rewrite<= exp-neg_binary32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))) (*.f32 (sinh.f32 (/.f32 1 v)) 2)) v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 (/.f32 cosTheta_O v) cosTheta_i) (Rewrite<= associate-/r*_binary32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (*.f32 (sinh.f32 (/.f32 1 v)) 2) v)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite=> associate-*l/_binary32 (/.f32 (*.f32 cosTheta_O cosTheta_i) v)) (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (*.f32 (sinh.f32 (/.f32 1 v)) 2) v))): 15 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 cosTheta_i cosTheta_O)) v) (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (*.f32 (sinh.f32 (/.f32 1 v)) 2) v))): 0 points increase in error, 15 points decrease in error
   (Rewrite=> associate-*r/_binary32 (/.f32 (*.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))) (*.f32 (*.f32 (sinh.f32 (/.f32 1 v)) 2) v))): 10 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (/.f32 (*.f32 cosTheta_i cosTheta_O) v))) (*.f32 (*.f32 (sinh.f32 (/.f32 1 v)) 2) v)): 0 points increase in error, 10 points decrease in error

3. Applied egg-rr 0.4

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

4. Simplified 0.4

   \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{\frac{-0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}{-{\left(e^{\frac{sinTheta_i}{v}}\right)}^{sinTheta_O}}} \]
   Proof
   (*.f32 (*.f32 cosTheta_i (/.f32 cosTheta_O v)) (/.f32 (/.f32 (/.f32 -1/2 v) (sinh.f32 (/.f32 1 v))) (neg.f32 (pow.f32 (exp.f32 (/.f32 sinTheta_i v)) sinTheta_O)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 cosTheta_i (/.f32 cosTheta_O v)) (/.f32 (/.f32 (/.f32 (Rewrite<= metadata-eval (*.f32 1 -1/2)) v) (sinh.f32 (/.f32 1 v))) (neg.f32 (pow.f32 (exp.f32 (/.f32 sinTheta_i v)) sinTheta_O)))): 0 points increase in error, 5 points decrease in error
   (*.f32 (*.f32 cosTheta_i (/.f32 cosTheta_O v)) (/.f32 (/.f32 (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 1 v) -1/2)) (sinh.f32 (/.f32 1 v))) (neg.f32 (pow.f32 (exp.f32 (/.f32 sinTheta_i v)) sinTheta_O)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 cosTheta_i (/.f32 cosTheta_O v)) (Rewrite<= associate-/r*_binary32 (/.f32 (*.f32 (/.f32 1 v) -1/2) (*.f32 (sinh.f32 (/.f32 1 v)) (neg.f32 (pow.f32 (exp.f32 (/.f32 sinTheta_i v)) sinTheta_O)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 cosTheta_i (/.f32 cosTheta_O v)) (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 (/.f32 1 v) (*.f32 (sinh.f32 (/.f32 1 v)) (neg.f32 (pow.f32 (exp.f32 (/.f32 sinTheta_i v)) sinTheta_O)))) -1/2))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 0.4

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

6. Applied egg-rr 0.4

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

7. Applied egg-rr 0.4

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

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	7008

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

Alternative 2
Error	0.4
Cost	7008

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

Alternative 3
Error	0.4
Cost	7008

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

Alternative 4
Error	0.5
Cost	6880

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

Alternative 5
Error	0.5
Cost	3680

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

Alternative 6
Error	0.6
Cost	3616

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

Alternative 7
Error	0.6
Cost	3616

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

Alternative 8
Error	0.6
Cost	3616

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

Alternative 9
Error	0.6
Cost	3616

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

Alternative 10
Error	11.5
Cost	480

\[\frac{cosTheta_i}{0.3333333333333333 \cdot \frac{1}{v \cdot cosTheta_O} + \frac{v}{cosTheta_O} \cdot 2} \]

Alternative 11
Error	11.5
Cost	416

\[\frac{cosTheta_i}{\frac{v}{cosTheta_O} \cdot 2 + \frac{0.3333333333333333}{v \cdot cosTheta_O}} \]

Alternative 12
Error	13.2
Cost	288

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

Alternative 13
Error	13.4
Cost	224

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

Alternative 14
Error	13.4
Cost	224

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

Alternative 15
Error	13.2
Cost	224

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

Reproduce

herbie shell --seed 2022343 
(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)))