HairBSDF, Mp, upper

Average Error: 0.5 → 0.5

Time: 14.7s

Precision: binary32

Cost: 6944

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

Math

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

↓

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

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
 (*
  (* (/ cosTheta_i v) (/ 1.0 v))
  (/ cosTheta_O (- (exp (/ 1.0 v)) (exp (/ -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 ((cosTheta_i / v) * (1.0f / v)) * (cosTheta_O / (expf((1.0f / v)) - expf((-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 = ((costheta_i / v) * (1.0e0 / v)) * (costheta_o / (exp((1.0e0 / v)) - exp(((-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(Float32(cosTheta_i / v) * Float32(Float32(1.0) / v)) * Float32(cosTheta_O / Float32(exp(Float32(Float32(1.0) / v)) - exp(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 = ((cosTheta_i / v) * (single(1.0) / v)) * (cosTheta_O / (exp((single(1.0) / v)) - exp((single(-1.0) / v))));
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)))): 14 points increase in error, 13 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)))): 17 points increase in error, 11 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))): 16 points increase in error, 19 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))): 30 points increase in error, 33 points decrease in error

3. Taylor expanded in sinTheta_O around 0 0.5

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

4. Simplified 0.5

   \[\leadsto \color{blue}{\frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
   Proof
   (*.f32 (/.f32 cosTheta_i (*.f32 v v)) (/.f32 cosTheta_O (-.f32 (exp.f32 (/.f32 1 v)) (exp.f32 (/.f32 -1 v))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 cosTheta_i (Rewrite<= unpow2_binary32 (pow.f32 v 2))) (/.f32 cosTheta_O (-.f32 (exp.f32 (/.f32 1 v)) (exp.f32 (/.f32 -1 v))))): 1 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 cosTheta_i (pow.f32 v 2)) (/.f32 cosTheta_O (-.f32 (exp.f32 (/.f32 1 v)) (exp.f32 (/.f32 (Rewrite<= metadata-eval (neg.f32 1)) v))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 cosTheta_i (pow.f32 v 2)) (/.f32 cosTheta_O (-.f32 (exp.f32 (/.f32 1 v)) (exp.f32 (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 1 v))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 cosTheta_i (pow.f32 v 2)) (/.f32 cosTheta_O (-.f32 (exp.f32 (/.f32 1 v)) (Rewrite<= rec-exp_binary32 (/.f32 1 (exp.f32 (/.f32 1 v))))))): 1 points increase in error, 1 points decrease in error
   (Rewrite<= times-frac_binary32 (/.f32 (*.f32 cosTheta_i cosTheta_O) (*.f32 (pow.f32 v 2) (-.f32 (exp.f32 (/.f32 1 v)) (/.f32 1 (exp.f32 (/.f32 1 v))))))): 43 points increase in error, 20 points decrease in error

5. Applied egg-rr 0.5

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

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	6880

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

Alternative 2
Error	0.5
Cost	6880

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

Alternative 3
Error	0.5
Cost	3616

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

Alternative 4
Error	11.5
Cost	416

\[cosTheta_O \cdot \left(cosTheta_i \cdot \frac{1}{\frac{0.3333333333333333}{v} + v \cdot 2}\right) \]

Alternative 5
Error	11.5
Cost	352

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

Alternative 6
Error	13.8
Cost	224

\[v \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot 3\right)\right) \]

Alternative 7
Error	13.3
Cost	224

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

Alternative 8
Error	13.3
Cost	224

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

Alternative 9
Error	13.2
Cost	224

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

Reproduce

herbie shell --seed 2022294 
(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)))