Average Error: 0.2 → 0.2
Time: 12.7s
Precision: binary32
Cost: 10080
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[1 + \left(\mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right) + -1\right) \]
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (+ 1.0 (+ (fma v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))) 1.0) -1.0)))
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 + (fmaf(v, logf((u + ((1.0f - u) * expf((-2.0f / v))))), 1.0f) + -1.0f);
}

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(fma(v, log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v))))), Float32(1.0)) + Float32(-1.0)))
end

1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
1 + \left(\mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right) + -1\right)

Error

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-rr0.2

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

  3. Taylor expanded in v around 0 0.2

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

  4. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost9952
\[\mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right) \]
Alternative 2
Error0.2
Cost6816
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
Alternative 3
Error4.2
Cost6752
\[\left(1 + \mathsf{fma}\left(v, \log \left(2 \cdot \frac{u}{v}\right), 1\right)\right) + -1 \]
Alternative 4
Error4.2
Cost6688
\[\mathsf{fma}\left(v, \mathsf{log1p}\left(2 \cdot \frac{u}{v} + -1\right), 1\right) \]
Alternative 5
Error4.2
Cost3488
\[1 + v \cdot \log \left(2 \cdot \frac{u}{v}\right) \]
Alternative 6
Error3.0
Cost676
\[\begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(\frac{u}{v} \cdot \left(2 + \left(\frac{2}{v} + \frac{1.3333333333333333}{v \cdot v}\right)\right)\right) + -1\\ \end{array} \]
Alternative 7
Error4.4
Cost32
\[1 \]

Error

Reproduce

herbie shell --seed 2022312 
(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))))))))