HairBSDF, Mp, lower

?

Percentage Accurate: 99.8% → 99.8%
Time: 25.2s
Precision: binary32
Cost: 32640

?

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := 0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}\\ {\left(\sqrt{{\left({e}^{\left(\sqrt[3]{{t_0}^{2}}\right)}\right)}^{\left(\sqrt[3]{t_0}\right)}}\right)}^{2} \cdot \frac{0.5}{v} \end{array} \]
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))
↓
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 0.6931 (/ (fma cosTheta_i cosTheta_O -1.0) v))))
   (*
    (pow (sqrt (pow (pow E (cbrt (pow t_0 2.0))) (cbrt t_0))) 2.0)
    (/ 0.5 v))))
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}
↓
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 0.6931f + (fmaf(cosTheta_i, cosTheta_O, -1.0f) / v);
	return powf(sqrtf(powf(powf(((float) M_E), cbrtf(powf(t_0, 2.0f))), cbrtf(t_0))), 2.0f) * (0.5f / v);
}

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end
↓
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(0.6931) + Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v))
	return Float32((sqrt(((Float32(exp(1)) ^ cbrt((t_0 ^ Float32(2.0)))) ^ cbrt(t_0))) ^ Float32(2.0)) * Float32(Float32(0.5) / v))
end

e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}
↓
\begin{array}{l}
t_0 := 0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}\\
{\left(\sqrt{{\left({e}^{\left(\sqrt[3]{{t_0}^{2}}\right)}\right)}^{\left(\sqrt[3]{t_0}\right)}}\right)}^{2} \cdot \frac{0.5}{v}
\end{array}

Local Percentage Accuracy vs ?

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 9 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 99.9%

    \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  2. Simplified99.9%

    \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]
    Step-by-step derivation

    [Start]99.9%

    \[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

    exp-sum [=>]99.9%

    \[ \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

  3. Taylor expanded in sinTheta_i around 0 99.9%

    \[\leadsto \color{blue}{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}} \cdot \frac{0.5}{v} \]

  4. Applied egg-rr100.0%

    \[\leadsto \color{blue}{{\left(\sqrt{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}}\right)}^{2}} \cdot \frac{0.5}{v} \]
    Step-by-step derivation

    [Start]99.9%

    \[ e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}} \cdot \frac{0.5}{v} \]

    add-sqr-sqrt [=>]100.0%

    \[ \color{blue}{\left(\sqrt{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}} \cdot \sqrt{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}}\right)} \cdot \frac{0.5}{v} \]

    pow2 [=>]100.0%

    \[ \color{blue}{{\left(\sqrt{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}}\right)}^{2}} \cdot \frac{0.5}{v} \]

    associate--l+ [=>]100.0%

    \[ {\left(\sqrt{e^{\color{blue}{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{1}{v}\right)}}}\right)}^{2} \cdot \frac{0.5}{v} \]

    sub-div [=>]100.0%

    \[ {\left(\sqrt{e^{0.6931 + \color{blue}{\frac{cosTheta_i \cdot cosTheta_O - 1}{v}}}}\right)}^{2} \cdot \frac{0.5}{v} \]

  5. Applied egg-rr100.0%

    \[\leadsto {\left(\sqrt{\color{blue}{{\left({e}^{\left(\sqrt[3]{{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}\right)}^{2}}\right)}\right)}^{\left(\sqrt[3]{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)}}}\right)}^{2} \cdot \frac{0.5}{v} \]
    Step-by-step derivation

    [Start]100.0%

    \[ {\left(\sqrt{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}}\right)}^{2} \cdot \frac{0.5}{v} \]

    *-un-lft-identity [=>]100.0%

    \[ {\left(\sqrt{e^{\color{blue}{1 \cdot \left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right)}}}\right)}^{2} \cdot \frac{0.5}{v} \]

    pow-exp [<=]99.9%

    \[ {\left(\sqrt{\color{blue}{{\left(e^{1}\right)}^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right)}}}\right)}^{2} \cdot \frac{0.5}{v} \]

    e-exp-1 [<=]99.9%

    \[ {\left(\sqrt{{\color{blue}{e}}^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right)}}\right)}^{2} \cdot \frac{0.5}{v} \]

    add-cube-cbrt [=>]99.9%

    \[ {\left(\sqrt{{e}^{\color{blue}{\left(\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}} \cdot \sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right) \cdot \sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)}}}\right)}^{2} \cdot \frac{0.5}{v} \]

    pow-unpow [=>]99.9%

    \[ {\left(\sqrt{\color{blue}{{\left({e}^{\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}} \cdot \sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)}\right)}^{\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)}}}\right)}^{2} \cdot \frac{0.5}{v} \]

    cbrt-unprod [=>]100.0%

    \[ {\left(\sqrt{{\left({e}^{\color{blue}{\left(\sqrt[3]{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right) \cdot \left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right)}\right)}}\right)}^{\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)}}\right)}^{2} \cdot \frac{0.5}{v} \]

    pow2 [=>]100.0%

    \[ {\left(\sqrt{{\left({e}^{\left(\sqrt[3]{\color{blue}{{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}\right)}^{2}}}\right)}\right)}^{\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)}}\right)}^{2} \cdot \frac{0.5}{v} \]

    fma-neg [=>]100.0%

    \[ {\left(\sqrt{{\left({e}^{\left(\sqrt[3]{{\left(0.6931 + \frac{\color{blue}{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}}{v}\right)}^{2}}\right)}\right)}^{\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)}}\right)}^{2} \cdot \frac{0.5}{v} \]

    metadata-eval [=>]100.0%

    \[ {\left(\sqrt{{\left({e}^{\left(\sqrt[3]{{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, \color{blue}{-1}\right)}{v}\right)}^{2}}\right)}\right)}^{\left(\sqrt[3]{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)}}\right)}^{2} \cdot \frac{0.5}{v} \]

  6. Final simplification100.0%

    \[\leadsto {\left(\sqrt{{\left({e}^{\left(\sqrt[3]{{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}\right)}^{2}}\right)}\right)}^{\left(\sqrt[3]{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)}}\right)}^{2} \cdot \frac{0.5}{v} \]

Alternatives

Alternative 1
Accuracy99.8%
Cost32640
\[\begin{array}{l} t_0 := 0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}\\ {\left(\sqrt{{\left({e}^{\left(\sqrt[3]{{t_0}^{2}}\right)}\right)}^{\left(\sqrt[3]{t_0}\right)}}\right)}^{2} \cdot \frac{0.5}{v} \end{array} \]
Alternative 2
Accuracy99.8%
Cost19616
\[\frac{0.5}{v} \cdot {\left({\left(\sqrt[3]{\sqrt{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}}}\right)}^{3}\right)}^{2} \]
Alternative 3
Accuracy99.8%
Cost16448
\[\frac{0.5}{v} \cdot {\left({\left({\left(e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)}^{0.25}\right)}^{2}\right)}^{2} \]
Alternative 4
Accuracy99.8%
Cost9920
\[\frac{0.5}{v} \cdot {\left(\sqrt{e^{0.6931 + \frac{-1}{v}}}\right)}^{2} \]
Alternative 5
Accuracy99.8%
Cost6688
\[\frac{0.5}{v} \cdot {e}^{\left(0.6931 + \frac{-1}{v}\right)} \]
Alternative 6
Accuracy99.8%
Cost3488
\[\frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
Alternative 7
Accuracy99.2%
Cost3424
\[\frac{0.5}{v} \cdot e^{\frac{-1}{v}} \]
Alternative 8
Accuracy4.3%
Cost3360
\[\frac{0.5}{v} \cdot e^{0.6931} \]
Alternative 9
Accuracy4.3%
Cost3360
\[\frac{0.5 \cdot e^{0.6931}}{v} \]

Reproduce?

herbie shell --seed 2023272 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))