HairBSDF, sample_f, cosTheta

Average Error: 0.1 → 0.1

Time: 16.0s

Precision: binary32

Cost: 6880

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

Math

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

↓

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

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 (/ 1.0 (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))))))

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 / (1.0f / logf((u + (expf((-2.0f / v)) * (1.0f - u))))));
}

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 / (1.0e0 / log((u + (exp(((-2.0e0) / v)) * (1.0e0 - u))))))
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 / Float32(Float32(1.0) / log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)))))))
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 / (single(1.0) / log((u + (exp((single(-2.0) / v)) * (single(1.0) - u))))));
end

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   rational.json-simplify-50 [=>] 0.1	\[ 1 + v \cdot \log \left(u + \color{blue}{e^{\frac{-2}{v}} \cdot \left(1 - u\right)}\right) \]
   rational.json-simplify-63 [=>] 0.2	\[ 1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \color{blue}{\left(\left(1 + 1\right) - \left(1 + u\right)\right)}\right) \]
   rational.json-simplify-42 [=>] 0.2	\[ 1 + v \cdot \log \left(u + \color{blue}{\left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} - e^{\frac{-2}{v}} \cdot \left(1 + u\right)\right)}\right) \]
   rational.json-simplify-3 [=>] 0.2	\[ 1 + v \cdot \log \color{blue}{\left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} + \left(u - e^{\frac{-2}{v}} \cdot \left(1 + u\right)\right)\right)} \]
   rational.json-simplify-1 [=>] 0.2	\[ 1 + v \cdot \log \left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} + \left(u - e^{\frac{-2}{v}} \cdot \color{blue}{\left(u + 1\right)}\right)\right) \]
   rational.json-simplify-56 [=>] 0.2	\[ 1 + v \cdot \log \left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} + \left(u - e^{\frac{-2}{v}} \cdot \color{blue}{\left(u - -1\right)}\right)\right) \]
   rational.json-simplify-42 [=>] 0.2	\[ 1 + v \cdot \log \left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} + \left(u - \color{blue}{\left(u \cdot e^{\frac{-2}{v}} - e^{\frac{-2}{v}} \cdot -1\right)}\right)\right) \]
   rational.json-simplify-50 [<=] 0.2	\[ 1 + v \cdot \log \left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} + \left(u - \left(\color{blue}{e^{\frac{-2}{v}} \cdot u} - e^{\frac{-2}{v}} \cdot -1\right)\right)\right) \]
   rational.json-simplify-31 [<=] 0.2	\[ 1 + v \cdot \log \left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} + \left(u - \left(e^{\frac{-2}{v}} \cdot u - \color{blue}{\left(-e^{\frac{-2}{v}}\right)}\right)\right)\right) \]
   rational.json-simplify-6 [=>] 0.1	\[ 1 + v \cdot \log \left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} + \color{blue}{\left(\left(-e^{\frac{-2}{v}}\right) - \left(e^{\frac{-2}{v}} \cdot u - u\right)\right)}\right) \]
   rational.json-simplify-12 [=>] 0.1	\[ 1 + v \cdot \log \color{blue}{\left(\left(\left(1 + 1\right) \cdot e^{\frac{-2}{v}} + \left(-e^{\frac{-2}{v}}\right)\right) - \left(e^{\frac{-2}{v}} \cdot u - u\right)\right)} \]

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	6816

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

Alternative 2
Error	2.8
Cost	3808

\[1 + \frac{1}{v \cdot \left(0.25 \cdot \left(u - u \cdot \frac{1}{e^{\frac{-2}{v}}}\right)\right) - 0.5} \]

Alternative 3
Error	2.8
Cost	3684

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

Alternative 4
Error	2.9
Cost	548

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{u + u}{v}\right) + \left(-8 - \left(u \cdot -2 + -6\right)\right)\\ \end{array} \]

Alternative 5
Error	2.9
Cost	484

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{u + u}{v} + -7\right) - \left(u \cdot -2 + -6\right)\\ \end{array} \]

Alternative 6
Error	30.2
Cost	32

\[-1 \]

Alternative 7
Error	4.0
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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))))))))