HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 9.0s

Precision: binary32

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

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

Math

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

↓

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

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
 (*
  (/
   (/ 1.0 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
   (pow (exp sinTheta_O) (/ sinTheta_i v)))
  (* (* (/ 1.0 v) (/ cosTheta_i 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 ((1.0f / (expf((1.0f / v)) - expf((-1.0f / v)))) / powf(expf(sinTheta_O), (sinTheta_i / v))) * (((1.0f / v) * (cosTheta_i / 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 = ((1.0e0 / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) / (exp(sintheta_o) ** (sintheta_i / v))) * (((1.0e0 / v) * (costheta_i / 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(1.0) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) / (exp(sinTheta_O) ^ Float32(sinTheta_i / v))) * Float32(Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i / 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(1.0) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) / (exp(sinTheta_O) ^ (sinTheta_i / v))) * (((single(1.0) / v) * (cosTheta_i / v)) * cosTheta_O);
end

Error

Bits error versus cosTheta_i

Bits error versus cosTheta_O

Bits error versus sinTheta_i

Bits error versus sinTheta_O

Bits error versus v

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

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

3. Applied egg-rr 0.4

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

4. Taylor expanded in v around 0 0.4

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

5. Simplified 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022156 
(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)))