Average Error: 0.1 → 0.1
Time: 16.8s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\log \left(v \cdot 2\right)}\\ t_1 := t_0 \cdot \left(t_0 \cdot t_0\right)\\ e^{\mathsf{fma}\left(1, 0.6931 + \left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \frac{-1}{v}\right), -t_1\right)} \cdot e^{\mathsf{fma}\left(-t_0, {\left({\left(\log 2 + \log v\right)}^{2}\right)}^{0.3333333333333333}, t_1\right)} \end{array} \]
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(v \cdot 2\right)}\\
t_1 := t_0 \cdot \left(t_0 \cdot t_0\right)\\
e^{\mathsf{fma}\left(1, 0.6931 + \left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \frac{-1}{v}\right), -t_1\right)} \cdot e^{\mathsf{fma}\left(-t_0, {\left({\left(\log 2 + \log v\right)}^{2}\right)}^{0.3333333333333333}, t_1\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 (* v 2.0)))) (t_1 (* t_0 (* t_0 t_0))))
   (*
    (exp
     (fma
      1.0
      (+
       0.6931
       (+
        (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
        (/ -1.0 v)))
      (- t_1)))
    (exp
     (fma
      (- t_0)
      (pow (pow (+ (log 2.0) (log v)) 2.0) 0.3333333333333333)
      t_1)))))
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((v * 2.0f)));
	float t_1 = t_0 * (t_0 * t_0);
	return expf(fmaf(1.0f, (0.6931f + ((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) + (-1.0f / v))), -t_1)) * expf(fmaf(-t_0, powf(powf((logf(2.0f) + logf(v)), 2.0f), 0.3333333333333333f), t_1));
}

Error

Bits error versus cosTheta_i

Bits error versus cosTheta_O

Bits error versus sinTheta_i

Bits error versus sinTheta_O

Bits error versus v

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. Applied add-exp-log_binary320.1

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

  3. Simplified0.1

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

  4. Applied add-cube-cbrt_binary320.1

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

  5. Applied *-un-lft-identity_binary320.1

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

  6. Applied prod-diff_binary320.1

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

  7. Applied exp-sum_binary320.1

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

  8. Taylor expanded in v around 0 0.1

    \[\leadsto e^{\mathsf{fma}\left(1, 0.6931 + \left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \frac{-1}{v}\right), -\sqrt[3]{\log \left(v \cdot 2\right)} \cdot \left(\sqrt[3]{\log \left(v \cdot 2\right)} \cdot \sqrt[3]{\log \left(v \cdot 2\right)}\right)\right)} \cdot e^{\mathsf{fma}\left(-\sqrt[3]{\log \left(v \cdot 2\right)}, \color{blue}{{\left({\left(\log 2 + \log v\right)}^{2}\right)}^{0.3333333333333333}}, \sqrt[3]{\log \left(v \cdot 2\right)} \cdot \left(\sqrt[3]{\log \left(v \cdot 2\right)} \cdot \sqrt[3]{\log \left(v \cdot 2\right)}\right)\right)} \]

  9. Final simplification0.1

    \[\leadsto e^{\mathsf{fma}\left(1, 0.6931 + \left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \frac{-1}{v}\right), -\sqrt[3]{\log \left(v \cdot 2\right)} \cdot \left(\sqrt[3]{\log \left(v \cdot 2\right)} \cdot \sqrt[3]{\log \left(v \cdot 2\right)}\right)\right)} \cdot e^{\mathsf{fma}\left(-\sqrt[3]{\log \left(v \cdot 2\right)}, {\left({\left(\log 2 + \log v\right)}^{2}\right)}^{0.3333333333333333}, \sqrt[3]{\log \left(v \cdot 2\right)} \cdot \left(\sqrt[3]{\log \left(v \cdot 2\right)} \cdot \sqrt[3]{\log \left(v \cdot 2\right)}\right)\right)} \]

Reproduce

herbie shell --seed 2022125 
(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))))))