HairBSDF, sample_f, cosTheta

Average Error: 0.2 → 0.2

Time: 15.8s

Precision: binary32

Cost: 10048

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

↓

\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{\frac{-1}{v}}\right)}^{2}\right) \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

↓

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (pow (exp (/ -1.0 v)) 2.0)))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

↓

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * powf(expf((-1.0f / v)), 2.0f)))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

↓

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * (exp(((-1.0e0) / v)) ** 2.0e0)))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

↓

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * (exp(Float32(Float32(-1.0) / v)) ^ Float32(2.0)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

↓

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (exp((single(-1.0) / v)) ^ single(2.0))))));
end

Derivation

1. Initial program 0.2

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

2. Applied egg-rr 0.2

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(\sqrt{e^{\frac{-2}{v}}}\right)}^{2}}\right) \]

3. Applied egg-rr 0.2

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{-1}{v}}\right)} - 1\right)}}^{2}\right) \]

4. Simplified 0.2

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\color{blue}{\left(e^{\frac{-1}{v}}\right)}}^{2}\right) \]
   Proof

   [Start] 0.2	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{\mathsf{log1p}\left(e^{\frac{-1}{v}}\right)} - 1\right)}^{2}\right) \]
   expm1-def [=>] 0.2	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{-1}{v}}\right)\right)\right)}}^{2}\right) \]
   expm1-log1p [=>] 0.2	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\color{blue}{\left(e^{\frac{-1}{v}}\right)}}^{2}\right) \]

5. Final simplification 0.2

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{\frac{-1}{v}}\right)}^{2}\right) \]

Alternatives

Alternative 1
Error	0.2
Cost	10016

\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {e}^{\left(\frac{-2}{v}\right)}\right) \]

Alternative 2
Error	0.2
Cost	6816

\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

Alternative 3
Error	2.8
Cost	3556

\[\begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + v \cdot \left(u \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right)\\ \end{array} \]

Alternative 4
Error	2.9
Cost	740

\[\begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + u \cdot 2\right) + \frac{0.5}{v} \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right)\\ \end{array} \]

Alternative 5
Error	2.9
Cost	676

\[\begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + v \cdot \left(u \cdot \left(-1 + \left(1 + \left(\frac{2}{v} + \frac{2}{v \cdot v}\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	2.9
Cost	356

\[\begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]

Alternative 7
Error	3.1
Cost	228

\[\begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]

Alternative 8
Error	30.2
Cost	32

\[-1 \]

Alternative 9
Error	4.0
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023046 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))