?

Average Error: 0.1 → 0.1
Time: 15.0s
Precision: binary32
Cost: 26368

?

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

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 = cbrt(log(Float32(Float32(0.5) / v)))
	return Float32(exp(1)) ^ Float32(Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) - fma(sinTheta_i, Float32(sinTheta_O / v), Float32(Float32(1.0) / v))) + Float32(Float32(0.6931) + Float32(t_0 * (t_0 ^ Float32(2.0)))))
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 := \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\\
{e}^{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + t_0 \cdot {t_0}^{2}\right)\right)}
\end{array}

Error?

Derivation?

  1. Initial program 0.1

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

  2. Simplified0.1

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

    [Start]0.1

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

    +-commutative [=>]0.1

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

    log-div [=>]0.1

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

    metadata-eval [=>]0.1

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

    associate-+l- [=>]0.1

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

    associate-+l- [<=]0.1

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

    metadata-eval [<=]0.1

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

    log-div [<=]0.1

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

    +-commutative [<=]0.1

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

    associate-+l+ [=>]0.1

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Alternatives

Alternative 1
Error0.1
Cost9888
\[{e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{-1}{v}\right)} \]
Alternative 2
Error15.3
Cost3656
\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -3.0000000095132306 \cdot 10^{-29}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{-sinTheta_O}{v}}\\ \end{array} \]
Alternative 3
Error15.3
Cost3656
\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -3.0000000095132306 \cdot 10^{-29}:\\ \;\;\;\;e^{\frac{1}{v} \cdot \left(sinTheta_i \cdot sinTheta_O\right)}\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{-sinTheta_O}{v}}\\ \end{array} \]
Alternative 4
Error17.4
Cost3492
\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -3.0000000095132306 \cdot 10^{-29}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \end{array} \]
Alternative 5
Error0.1
Cost3488
\[\frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
Alternative 6
Error0.7
Cost3424
\[\frac{0.5}{v} \cdot e^{\frac{-1}{v}} \]
Alternative 7
Error19.3
Cost224
\[\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}} \]
Alternative 8
Error19.4
Cost192
\[\frac{-sinTheta_i \cdot sinTheta_O}{v} \]
Alternative 9
Error25.6
Cost160
\[sinTheta_O \cdot \frac{sinTheta_i}{v} \]
Alternative 10
Error25.6
Cost160
\[sinTheta_i \cdot \frac{sinTheta_O}{v} \]
Alternative 11
Error25.6
Cost160
\[\frac{sinTheta_O}{\frac{v}{sinTheta_i}} \]
Alternative 12
Error19.4
Cost160
\[\frac{sinTheta_i \cdot sinTheta_O}{v} \]
Alternative 13
Error29.9
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023025 
(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))))))