HairBSDF, sample_f, cosTheta

?

Percentage Accurate: 99.5% → 99.5%
Time: 17.8s
Precision: binary32
Cost: 6880

?

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-6}{v} \cdot 0.3333333333333333}\right) \]
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (+
  1.0
  (* v (log (+ u (* (- 1.0 u) (exp (* (/ -6.0 v) 0.3333333333333333))))))))
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 * logf((u + ((1.0f - u) * expf(((-6.0f / v) * 0.3333333333333333f))))));
}

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 * log((u + ((1.0e0 - u) * exp((((-6.0e0) / v) * 0.3333333333333333e0))))))
end function

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 * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(Float32(-6.0) / v) * Float32(0.3333333333333333))))))))
end

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 * log((u + ((single(1.0) - u) * exp(((single(-6.0) / v) * single(0.3333333333333333)))))));
end

1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-6}{v} \cdot 0.3333333333333333}\right)

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 11 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.6%

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

  2. Applied egg-rr99.5%

    \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\sqrt[3]{e^{3 \cdot \frac{-2}{v}}}}\right) \]
    Step-by-step derivation

    [Start]99.6

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

    add-cbrt-cube [=>]99.6

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

    add-exp-log [=>]99.5

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

    pow3 [=>]99.5

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

    log-pow [=>]99.5

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

    add-log-exp [<=]99.5

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

  3. Applied egg-rr99.6%

    \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-6}{v} \cdot 0.3333333333333333}}\right) \]
    Step-by-step derivation

    [Start]99.5

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

    pow1/3 [=>]99.5

    \[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{3 \cdot \frac{-2}{v}}\right)}^{0.3333333333333333}}\right) \]

    pow-exp [=>]99.5

    \[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\left(3 \cdot \frac{-2}{v}\right) \cdot 0.3333333333333333}}\right) \]

    associate-*r/ [=>]99.6

    \[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{\frac{3 \cdot -2}{v}} \cdot 0.3333333333333333}\right) \]

    metadata-eval [=>]99.6

    \[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{\color{blue}{-6}}{v} \cdot 0.3333333333333333}\right) \]

  4. Final simplification99.6%

    \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-6}{v} \cdot 0.3333333333333333}\right) \]

Alternatives

Alternative 1
Accuracy99.5%
Cost6816
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
Alternative 2
Accuracy99.2%
Cost6752
\[1 + v \cdot \mathsf{log1p}\left(\left(1 - u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right) \]
Alternative 3
Accuracy91.1%
Cost3748
\[\begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \left(\frac{1}{1 + \mathsf{expm1}\left(\frac{-2}{v}\right)} + -1\right)\right) + -1\\ \end{array} \]
Alternative 4
Accuracy91.1%
Cost3556
\[\begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]
Alternative 5
Accuracy90.9%
Cost740
\[\begin{array}{l} \mathbf{if}\;v \leq 0.14499999582767487:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(2 \cdot \frac{u}{v} + \frac{u}{v \cdot v} \cdot \left(2 + \frac{1.3333333333333333}{v}\right)\right) + -1\\ \end{array} \]
Alternative 6
Accuracy90.8%
Cost676
\[\begin{array}{l} \mathbf{if}\;v \leq 0.14499999582767487:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 2 + -1\right) + \frac{\left(1 - u\right) \cdot \left(\left(1 + u\right) + \left(u + -1\right)\right)}{v}\\ \end{array} \]
Alternative 7
Accuracy90.7%
Cost548
\[\begin{array}{l} \mathbf{if}\;v \leq 0.14499999582767487:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \left(\frac{2}{v} + \frac{2}{v \cdot v}\right)\right) + -1\\ \end{array} \]
Alternative 8
Accuracy90.7%
Cost484
\[\begin{array}{l} \mathbf{if}\;v \leq 0.14499999582767487:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(\frac{u}{v} \cdot \left(2 + \frac{2}{v}\right)\right) + -1\\ \end{array} \]
Alternative 9
Accuracy5.7%
Cost32
\[-1 \]
Alternative 10
Accuracy87.1%
Cost32
\[1 \]

Reproduce?

herbie shell --seed 2023160 
(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))))))))