HairBSDF, Mp, upper

Average Error: 0.5 → 0.5

Time: 1.2min

Precision: binary32

Cost: 3808

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

Math

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

↓

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

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 (sinh (/ 1.0 v)))
  (* cosTheta_i (* (+ cosTheta_O cosTheta_O) (* (/ 1.0 v) (/ 0.5 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 (0.5f / sinhf((1.0f / v))) * (cosTheta_i * ((cosTheta_O + cosTheta_O) * ((1.0f / v) * (0.5f / 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 = (0.5e0 / sinh((1.0e0 / v))) * (costheta_i * ((costheta_o + costheta_o) * ((1.0e0 / v) * (0.5e0 / 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(0.5) / sinh(Float32(Float32(1.0) / v))) * Float32(cosTheta_i * Float32(Float32(cosTheta_O + cosTheta_O) * Float32(Float32(Float32(1.0) / v) * Float32(Float32(0.5) / 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 = (single(0.5) / sinh((single(1.0) / v))) * (cosTheta_i * ((cosTheta_O + cosTheta_O) * ((single(1.0) / v) * (single(0.5) / 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

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

   [Start] 0.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-1 [=>] 0.5	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   exponential-simplify-2 [=>] 0.5	\[ \frac{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-55 [=>] 0.5	\[ \frac{\color{blue}{1 \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-1 [=>] 0.5	\[ \frac{\color{blue}{\frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot 1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-7 [=>] 0.5	\[ \frac{\color{blue}{\frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-53 [=>] 0.5	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-1 [=>] 0.5	\[ \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

3. Taylor expanded in v around inf 0.5

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

4. Applied egg-rr 0.5

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

5. Applied egg-rr 0.5

   \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(cosTheta_i \cdot \color{blue}{\left(\left(cosTheta_O + cosTheta_O\right) \cdot \left(\frac{1}{v} \cdot \frac{0.5}{v}\right)\right)}\right) \]

6. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\left(\frac{0.5}{\sinh \left(\frac{1}{v}\right)} + 0\right)} \cdot \left(cosTheta_i \cdot \left(\left(cosTheta_O + cosTheta_O\right) \cdot \left(\frac{1}{v} \cdot \frac{0.5}{v}\right)\right)\right) \]

7. Simplified 0.5

   \[\leadsto \color{blue}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}} \cdot \left(cosTheta_i \cdot \left(\left(cosTheta_O + cosTheta_O\right) \cdot \left(\frac{1}{v} \cdot \frac{0.5}{v}\right)\right)\right) \]
   Proof

   [Start] 0.5	\[ \left(\frac{0.5}{\sinh \left(\frac{1}{v}\right)} + 0\right) \cdot \left(cosTheta_i \cdot \left(\left(cosTheta_O + cosTheta_O\right) \cdot \left(\frac{1}{v} \cdot \frac{0.5}{v}\right)\right)\right) \]
   rational_best-simplify-3 [<=] 0.5	\[ \color{blue}{\left(0 + \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right)} \cdot \left(cosTheta_i \cdot \left(\left(cosTheta_O + cosTheta_O\right) \cdot \left(\frac{1}{v} \cdot \frac{0.5}{v}\right)\right)\right) \]
   rational_best-simplify-6 [=>] 0.5	\[ \color{blue}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}} \cdot \left(cosTheta_i \cdot \left(\left(cosTheta_O + cosTheta_O\right) \cdot \left(\frac{1}{v} \cdot \frac{0.5}{v}\right)\right)\right) \]

8. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	3744

\[\left(cosTheta_i \cdot cosTheta_O\right) \cdot \left(\frac{4}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{0.25}{v} \cdot \frac{0.5}{v}\right)\right) \]

Alternative 2
Error	0.5
Cost	3680

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

Alternative 3
Error	0.5
Cost	3616

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

Alternative 4
Error	0.5
Cost	3616

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

Alternative 5
Error	11.5
Cost	3584

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

Alternative 6
Error	13.3
Cost	352

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

Alternative 7
Error	13.3
Cost	224

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

Alternative 8
Error	13.3
Cost	224

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

Alternative 9
Error	13.3
Cost	224

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

Reproduce

herbie shell --seed 2023099 
(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)))