Average Error: 0.2 → 0.2
Time: 16.4s
Precision: binary32
Cost: 10144
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\mathsf{fma}\left(v, \log \left(u + \frac{1 - u \cdot u}{\frac{1 + u}{e^{\frac{-2}{v}}}}\right), 1\right) \]
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (fma v (log (+ u (/ (- 1.0 (* u u)) (/ (+ 1.0 u) (exp (/ -2.0 v)))))) 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 fmaf(v, logf((u + ((1.0f - (u * u)) / ((1.0f + u) / expf((-2.0f / v)))))), 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 fma(v, log(Float32(u + Float32(Float32(Float32(1.0) - Float32(u * u)) / Float32(Float32(Float32(1.0) + u) / exp(Float32(Float32(-2.0) / v)))))), Float32(1.0))
end

1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
\mathsf{fma}\left(v, \log \left(u + \frac{1 - u \cdot u}{\frac{1 + u}{e^{\frac{-2}{v}}}}\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. Simplified0.2

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

    [Start]0.2

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

    +-commutative [=>]0.2

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

    fma-def [=>]0.2

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

    +-commutative [=>]0.2

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

    fma-def [=>]0.2

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

  3. Taylor expanded in v around 0 0.2

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

  4. Applied egg-rr0.2

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

  5. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost9952
\[\mathsf{fma}\left(v, \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right), 1\right) \]
Alternative 2
Error0.2
Cost7008
\[1 + v \cdot \log \left(u + \frac{1 - u \cdot u}{\frac{1 + u}{e^{\frac{-2}{v}}}}\right) \]
Alternative 3
Error0.2
Cost6816
\[1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \]
Alternative 4
Error1.4
Cost6688
\[1 + v \cdot \log \left(u + e^{\frac{-2}{v}}\right) \]
Alternative 5
Error3.0
Cost3556
\[\begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\frac{2}{v}\right) \cdot \left(v \cdot u\right) + -1\\ \end{array} \]
Alternative 6
Error3.1
Cost740
\[\begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 2 + -1\right) + \frac{0.5}{v} \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right)\\ \end{array} \]
Alternative 7
Error3.1
Cost740
\[\begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(2 \cdot \frac{u}{v} + \frac{u}{v \cdot v} \cdot \left(2 + \frac{1.3333333333333333}{v}\right)\right) + -1\\ \end{array} \]
Alternative 8
Error3.1
Cost548
\[\begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 2 - \frac{u \cdot \left(-2 + u \cdot 2\right)}{v}\right) + -1\\ \end{array} \]
Alternative 9
Error3.1
Cost420
\[\begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot 2 + \left(-1 + u \cdot \frac{2}{v}\right)\\ \end{array} \]
Alternative 10
Error3.1
Cost356
\[\begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
Alternative 11
Error30.1
Cost32
\[-1 \]
Alternative 12
Error4.4
Cost32
\[1 \]

Error

Reproduce

herbie shell --seed 2022356 
(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))))))))