HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 16.0s

Precision: binary32

Cost: 10368

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

Math

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

↓

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

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 sinTheta_O) v)))
   (* cosTheta_i (* cosTheta_O (/ 1.0 v))))
  (* v (- (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 (expf(-((sinTheta_i * sinTheta_O) / v)) * (cosTheta_i * (cosTheta_O * (1.0f / v)))) / (v * (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 = (exp(-((sintheta_i * sintheta_o) / v)) * (costheta_i * (costheta_o * (1.0e0 / v)))) / (v * (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(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(cosTheta_i * Float32(cosTheta_O * Float32(Float32(1.0) / v)))) / Float32(v * 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 = (exp(-((sinTheta_i * sinTheta_O) / v)) * (cosTheta_i * (cosTheta_O * (single(1.0) / v)))) / (v * (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. Applied egg-rr 0.4

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

3. Taylor expanded in v around 0 0.4

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

4. Simplified 0.4

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{1}{v}\right)\right)}{\color{blue}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]
   Proof
   (*.f32 v (-.f32 (exp.f32 (/.f32 1 v)) (exp.f32 (/.f32 -1 v)))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (-.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 v (-.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 v (-.f32 (exp.f32 (/.f32 1 v)) (Rewrite<= rec-exp_binary32 (/.f32 1 (exp.f32 (/.f32 1 v)))))): 4 points increase in error, 4 points decrease in error
   (*.f32 v (Rewrite=> sub-neg_binary32 (+.f32 (exp.f32 (/.f32 1 v)) (neg.f32 (/.f32 1 (exp.f32 (/.f32 1 v))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (Rewrite=> +-commutative_binary32 (+.f32 (neg.f32 (/.f32 1 (exp.f32 (/.f32 1 v)))) (exp.f32 (/.f32 1 v))))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (+.f32 (Rewrite=> neg-sub0_binary32 (-.f32 0 (/.f32 1 (exp.f32 (/.f32 1 v))))) (exp.f32 (/.f32 1 v)))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (Rewrite=> associate-+l-_binary32 (-.f32 0 (-.f32 (/.f32 1 (exp.f32 (/.f32 1 v))) (exp.f32 (/.f32 1 v)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (Rewrite=> sub0-neg_binary32 (neg.f32 (-.f32 (/.f32 1 (exp.f32 (/.f32 1 v))) (exp.f32 (/.f32 1 v)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (Rewrite<= sub0-neg_binary32 (-.f32 0 (-.f32 (/.f32 1 (exp.f32 (/.f32 1 v))) (exp.f32 (/.f32 1 v)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (Rewrite<= associate-+l-_binary32 (+.f32 (-.f32 0 (/.f32 1 (exp.f32 (/.f32 1 v)))) (exp.f32 (/.f32 1 v))))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (+.f32 (Rewrite<= neg-sub0_binary32 (neg.f32 (/.f32 1 (exp.f32 (/.f32 1 v))))) (exp.f32 (/.f32 1 v)))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (Rewrite<= +-commutative_binary32 (+.f32 (exp.f32 (/.f32 1 v)) (neg.f32 (/.f32 1 (exp.f32 (/.f32 1 v))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 v (Rewrite<= sub-neg_binary32 (-.f32 (exp.f32 (/.f32 1 v)) (/.f32 1 (exp.f32 (/.f32 1 v)))))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	7104

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

Alternative 2
Error	0.5
Cost	7072

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

Alternative 3
Error	0.5
Cost	7008

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

Alternative 4
Error	0.4
Cost	7008

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

Alternative 5
Error	0.5
Cost	6880

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

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	13.1
Cost	288

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

Alternative 8
Error	13.1
Cost	224

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

Alternative 9
Error	13.1
Cost	224

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

Reproduce

herbie shell --seed 2022300 
(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)))