?

Average Error: 99.4% → 99.3%
Time: 29.1s
Precision: binary32
Cost: 16384.00

?

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[1 + v \cdot {\left(\sqrt[3]{\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3} \]
(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 (pow (cbrt (log (fma (- 1.0 u) (exp (/ -2.0 v)) u))) 3.0))))
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 + (v * powf(cbrtf(logf(fmaf((1.0f - u), expf((-2.0f / v)), u))), 3.0f));
}

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(v * (cbrt(log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u))) ^ Float32(3.0))))
end

1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
1 + v \cdot {\left(\sqrt[3]{\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}

Error?

Derivation?

  1. Initial program 99.4

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

  2. Applied egg-rr99.3

    \[\leadsto 1 + v \cdot \color{blue}{{\left(\sqrt[3]{\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3}} \]

  3. Final simplification99.3

    \[\leadsto 1 + v \cdot {\left(\sqrt[3]{\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}\right)}^{3} \]

Alternatives

Alternative 1
Error99.4%
Cost9984.00
\[1 + v \cdot \log \left(u + e^{\mathsf{log1p}\left(-u\right) + \frac{-2}{v}}\right) \]
Alternative 2
Error99.4%
Cost6816.00
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
Alternative 3
Error94.0%
Cost6496.00
\[\mathsf{fma}\left(v, \log u, 1\right) \]
Alternative 4
Error94.0%
Cost3360.00
\[1 + v \cdot \log u \]
Alternative 5
Error6.2%
Cost32.00
\[-1 \]
Alternative 6
Error86.3%
Cost32.00
\[1 \]

Error

Reproduce?

herbie shell --seed 2023093 
(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))))))))