Average Error: 0.2 → 0.2
Time: 19.1s
Precision: binary32
Cost: 13088
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]
(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 (fma (- 1.0 u) (exp (/ -2.0 v)) u)) 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(fmaf((1.0f - u), expf((-2.0f / v)), u)), 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(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u)), 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(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\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. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost9952
\[1 + v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right) \]
Alternative 2
Error0.2
Cost6816
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
Alternative 3
Error0.3
Cost6752
\[1 + v \cdot \mathsf{log1p}\left(\left(1 - u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right) \]
Alternative 4
Error3.0
Cost1252
\[\begin{array}{l} t_0 := \left(1 - u\right) \cdot -2\\ \mathbf{if}\;v \leq 0.125:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot t_0 + -1}{t_0 + -1} + \frac{0.5}{v} \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right)\\ \end{array} \]
Alternative 5
Error3.0
Cost804
\[\begin{array}{l} \mathbf{if}\;v \leq 0.125:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{v} \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) + \left(1 + \left(1 - u\right) \cdot -2\right)\\ \end{array} \]
Alternative 6
Error3.0
Cost740
\[\begin{array}{l} \mathbf{if}\;v \leq 0.125:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{v} \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) + \left(u \cdot 2 + -1\right)\\ \end{array} \]
Alternative 7
Error3.1
Cost484
\[\begin{array}{l} \mathbf{if}\;v \leq 0.125:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 2 \cdot \frac{u}{v}\right)\\ \end{array} \]
Alternative 8
Error3.1
Cost420
\[\begin{array}{l} \mathbf{if}\;v \leq 0.125:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 2 + -1\right) + 2 \cdot \frac{u}{v}\\ \end{array} \]
Alternative 9
Error3.1
Cost420
\[\begin{array}{l} \mathbf{if}\;v \leq 0.125:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) + -1\\ \end{array} \]
Alternative 10
Error3.3
Cost228
\[\begin{array}{l} \mathbf{if}\;v \leq 0.125:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot 2 + -1\\ \end{array} \]
Alternative 11
Error30.1
Cost32
\[-1 \]
Alternative 12
Error4.3
Cost32
\[1 \]

Error

Reproduce

herbie shell --seed 2022354 
(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))))))))