HairBSDF, Mp, upper

Average Error: 0.4 → 0.4

Time: 16.1s

Precision: binary32

Cost: 7136

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

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

Math

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

↓

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

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)) 2.0) (sinh (/ 1.0 v)))
  (/ cosTheta_O (/ (/ (- v) cosTheta_i) (/ -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)) / 2.0f) / sinhf((1.0f / v))) * (cosTheta_O / ((-v / cosTheta_i) / (-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)) / 2.0e0) / sinh((1.0e0 / v))) * (costheta_o / ((-v / costheta_i) / ((-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(exp(Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v)) / Float32(2.0)) / sinh(Float32(Float32(1.0) / v))) * Float32(cosTheta_O / Float32(Float32(Float32(-v) / cosTheta_i) / 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)) / single(2.0)) / sinh((single(1.0) / v))) * (cosTheta_O / ((-v / cosTheta_i) / (single(-1.0) / v)));
end

Derivation

1. Initial program 0.4

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

2. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}}{2}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}} \]
   Proof
   (*.f32 (/.f32 (/.f32 (exp.f32 (/.f32 (*.f32 sinTheta_i (neg.f32 sinTheta_O)) v)) 2) (sinh.f32 (/.f32 1 v))) (/.f32 (*.f32 cosTheta_i cosTheta_O) (*.f32 v v))): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (/.f32 (exp.f32 (/.f32 (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 sinTheta_i sinTheta_O))) v)) 2) (sinh.f32 (/.f32 1 v))) (/.f32 (*.f32 cosTheta_i cosTheta_O) (*.f32 v v))): 0 points increase in error, 0 points decrease in error
   (*.f32 (/.f32 (/.f32 (exp.f32 (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)))) 2) (sinh.f32 (/.f32 1 v))) (/.f32 (*.f32 cosTheta_i cosTheta_O) (*.f32 v v))): 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 cosTheta_O) (*.f32 v v))): 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-/l/_binary32 (/.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) v))): 18 points increase in error, 22 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))): 28 points increase in error, 28 points decrease in error

3. Applied egg-rr 0.4

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

4. Applied egg-rr 0.4

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

5. Applied egg-rr 0.4

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	7104

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

Alternative 2
Error	0.4
Cost	7104

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

Alternative 3
Error	0.5
Cost	7040

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

Alternative 4
Error	0.5
Cost	7008

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

Alternative 5
Error	0.5
Cost	6880

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

Alternative 6
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 7
Error	0.8
Cost	6784

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

Alternative 8
Error	9.6
Cost	3940

\[\begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;\frac{1}{e^{\frac{1}{v}} + \left(\frac{1}{v} + -1\right)} \cdot \frac{1}{\frac{v}{cosTheta_i \cdot \frac{cosTheta_O}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}}{v} \cdot \frac{cosTheta_O \cdot cosTheta_i}{2 + \frac{0.3333333333333333}{v \cdot v}}\\ \end{array} \]

Alternative 9
Error	11.3
Cost	3840

\[\frac{e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}}{v} \cdot \frac{cosTheta_O \cdot cosTheta_i}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

Alternative 10
Error	11.3
Cost	416

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

Alternative 11
Error	13.0
Cost	288

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

Alternative 12
Error	13.1
Cost	224

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

Alternative 13
Error	13.1
Cost	224

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

Alternative 14
Error	13.1
Cost	224

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

Alternative 15
Error	13.1
Cost	224

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

Alternative 16
Error	13.0
Cost	224

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

Reproduce

herbie shell --seed 2022329 
(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)))