?

Average Error: 0.38% → 0.26%
Time: 18.1s
Precision: binary32
Cost: 16416

?

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\frac{e^{\left(\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v} + 0.6931\right) - \log \left(\sqrt{v} \cdot 2\right)}}{\sqrt{v}} \]
(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
 (/
  (exp
   (-
    (+ (/ (fma cosTheta_i cosTheta_O -1.0) v) 0.6931)
    (log (* (sqrt v) 2.0))))
  (sqrt 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) {
	return expf((((fmaf(cosTheta_i, cosTheta_O, -1.0f) / v) + 0.6931f) - logf((sqrtf(v) * 2.0f)))) / sqrtf(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)
	return Float32(exp(Float32(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v) + Float32(0.6931)) - log(Float32(sqrt(v) * Float32(2.0))))) / sqrt(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)}
↓
\frac{e^{\left(\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v} + 0.6931\right) - \log \left(\sqrt{v} \cdot 2\right)}}{\sqrt{v}}

Error?

Derivation?

  1. Initial program 0.38

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

  2. Simplified0.37

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

    [Start]0.38

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

    remove-double-neg [<=]0.38

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

    +-commutative [<=]0.38

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

    log-rec [=>]0.25

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

    distribute-neg-in [<=]0.25

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

    sub-neg [<=]0.25

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

    sub0-neg [<=]0.25

    \[ e^{\color{blue}{0 - \left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]

    associate-+l- [<=]0.25

    \[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

  3. Applied egg-rr0.24

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

  4. Applied egg-rr0.24

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

  5. Taylor expanded in sinTheta_i around 0 0.25

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

  6. Applied egg-rr0.26

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

  7. Final simplification0.26

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

Alternatives

Alternative 1
Error0.26%
Cost13152
\[\frac{\frac{\left(e^{\frac{-1}{v}} \cdot e^{0.6931}\right) \cdot 0.5}{\sqrt{v}}}{\sqrt{v}} \]
Alternative 2
Error0.25%
Cost10144
\[\frac{\frac{0.5 \cdot e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) + \frac{-1}{v}}}{\sqrt{v}}}{\sqrt{v}} \]
Alternative 3
Error0.28%
Cost3552
\[\frac{1}{\frac{v}{0.5 \cdot e^{0.6931 + \frac{-1}{v}}}} \]
Alternative 4
Error55.92%
Cost3492
\[\begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;e^{cosTheta_i \cdot \frac{cosTheta_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]
Alternative 5
Error55.92%
Cost3492
\[\begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;e^{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]
Alternative 6
Error0.39%
Cost3488
\[e^{0.6931 + \frac{-1}{v}} \cdot \frac{0.5}{v} \]
Alternative 7
Error1.93%
Cost3424
\[e^{\frac{-1 + cosTheta_i \cdot cosTheta_O}{v}} \]
Alternative 8
Error80.09%
Cost160
\[cosTheta_i \cdot \frac{cosTheta_O}{v} \]
Alternative 9
Error80.09%
Cost160
\[cosTheta_O \cdot \frac{cosTheta_i}{v} \]
Alternative 10
Error61.86%
Cost160
\[\frac{cosTheta_i \cdot cosTheta_O}{v} \]
Alternative 11
Error93.59%
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023121 
(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))))))