HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 16.7s

Precision: binary32

Cost: 10240

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

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

Math

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

↓

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

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
 (*
  (/ (/ 0.5 (pow (exp sinTheta_O) (/ sinTheta_i v))) (sinh (/ 1.0 v)))
  (* (* cosTheta_i (/ (/ 1.0 v) v)) 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 ((0.5f / powf(expf(sinTheta_O), (sinTheta_i / v))) / sinhf((1.0f / v))) * ((cosTheta_i * ((1.0f / v) / v)) * 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 = ((0.5e0 / (exp(sintheta_o) ** (sintheta_i / v))) / sinh((1.0e0 / v))) * ((costheta_i * ((1.0e0 / v) / v)) * 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(Float32(Float32(0.5) / (exp(sinTheta_O) ^ Float32(sinTheta_i / v))) / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(cosTheta_i * Float32(Float32(Float32(1.0) / v) / v)) * 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 = ((single(0.5) / (exp(sinTheta_O) ^ (sinTheta_i / v))) / sinh((single(1.0) / v))) * ((cosTheta_i * ((single(1.0) / v) / v)) * cosTheta_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

   \[\leadsto \color{blue}{\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_i}{v \cdot v} \cdot cosTheta_O\right)} \]
   Proof
   (*.f32 (/.f32 (/.f32 1/2 (pow.f32 (exp.f32 sinTheta_O) (/.f32 sinTheta_i v))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (/.f32 (Rewrite<= metadata-eval (/.f32 1 2)) (pow.f32 (exp.f32 sinTheta_O) (/.f32 sinTheta_i v))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (/.f32 (/.f32 1 2) (Rewrite<= exp-prod_binary32 (exp.f32 (*.f32 sinTheta_O (/.f32 sinTheta_i v))))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (/.f32 (/.f32 1 2) (exp.f32 (Rewrite<= *-commutative_binary32 (*.f32 (/.f32 sinTheta_i v) sinTheta_O)))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (/.f32 (/.f32 1 2) (exp.f32 (Rewrite=> associate-*l/_binary32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (Rewrite<= associate-/r*_binary32 (/.f32 1 (*.f32 2 (exp.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))))) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (Rewrite<= associate-/l/_binary32 (/.f32 (/.f32 1 (exp.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) 2)) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (/.f32 (Rewrite<= exp-neg_binary32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))) 2) (sinh.f32 (/.f32 1 v))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite=> associate-/l/_binary32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2))) (*.f32 (/.f32 cosTheta_i (*.f32 v v)) cosTheta_O)): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2)) (Rewrite<= associate-/r/_binary32 (/.f32 cosTheta_i (/.f32 (*.f32 v v) cosTheta_O)))): 17 points increase in error, 25 points decrease in error
   (*.f32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2)) (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 cosTheta_i cosTheta_O) (*.f32 v v)))): 26 points increase in error, 17 points decrease in error
   (*.f32 (/.f32 (exp.f32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (*.f32 (sinh.f32 (/.f32 1 v)) 2)) (Rewrite<= associate-/l/_binary32 (/.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) v))): 19 points increase in error, 21 points decrease in error
   (Rewrite<= times-frac_binary32 (/.f32 (*.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))): 31 points increase in error, 33 points decrease in error

3. Applied egg-rr 0.5

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

4. Taylor expanded in v around 0 0.5

   \[\leadsto \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\left(\left(-cosTheta_i\right) \cdot \color{blue}{\frac{-1}{{v}^{2}}}\right) \cdot cosTheta_O\right) \]

5. Simplified 0.4

   \[\leadsto \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\left(\left(-cosTheta_i\right) \cdot \color{blue}{\frac{\frac{-1}{v}}{v}}\right) \cdot cosTheta_O\right) \]
   Proof
   (/.f32 (/.f32 -1 v) v): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary32 (/.f32 -1 (*.f32 v v))): 48 points increase in error, 49 points decrease in error
   (/.f32 -1 (Rewrite<= unpow2_binary32 (pow.f32 v 2))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	7104

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

Alternative 2
Error	0.4
Cost	7104

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

Alternative 3
Error	0.5
Cost	7072

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

Alternative 4
Error	0.5
Cost	7008

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

Alternative 5
Error	0.5
Cost	6816

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

Alternative 6
Error	0.6
Cost	3616

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

Alternative 7
Error	13.2
Cost	288

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

Alternative 8
Error	13.3
Cost	224

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

Alternative 9
Error	13.3
Cost	224

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

Alternative 10
Error	13.2
Cost	224

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

Reproduce

herbie shell --seed 2022295 
(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)))