\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right)\\ t_1 := \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\\ t_2 := \sqrt[3]{e^{t_0 - t_1}}\\ \frac{0.5}{v} \cdot \left(t_2 \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(1, t_0, -t_1\right)}} \cdot t_2\right)\right) \end{array} \]

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 := \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right)\\
t_1 := \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\\
t_2 := \sqrt[3]{e^{t_0 - t_1}}\\
\frac{0.5}{v} \cdot \left(t_2 \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(1, t_0, -t_1\right)}} \cdot t_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 (fma cosTheta_O (/ cosTheta_i v) 0.6931))
        (t_1 (fma sinTheta_i (/ sinTheta_O v) (/ 1.0 v)))
        (t_2 (cbrt (exp (- t_0 t_1)))))
   (* (/ 0.5 v) (* t_2 (* (cbrt (exp (fma 1.0 t_0 (- t_1)))) t_2)))))

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 = fmaf(cosTheta_O, (cosTheta_i / v), 0.6931f);
	float t_1 = fmaf(sinTheta_i, (sinTheta_O / v), (1.0f / v));
	float t_2 = cbrtf(expf(t_0 - t_1));
	return (0.5f / v) * (t_2 * (cbrtf(expf(fmaf(1.0f, t_0, -t_1))) * t_2));
}

Derivation

Initial program 0.1
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
Simplified0.1
\[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}} \]
Applied add-cube-cbrt_binary320.1
\[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)} \]
Applied *-un-lft-identity_binary320.1
\[\leadsto \frac{0.5}{v} \cdot \left(\left(\sqrt[3]{e^{\color{blue}{1 \cdot \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right)} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right) \]
Applied fma-neg_binary320.1
\[\leadsto \frac{0.5}{v} \cdot \left(\left(\sqrt[3]{e^{\color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right), -\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)}}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right) \]
Final simplification0.1
\[\leadsto \frac{0.5}{v} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}} \cdot \left(\sqrt[3]{e^{\mathsf{fma}\left(1, \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right), -\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}}\right)\right) \]

Reproduce

herbie shell --seed 2021280 
(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))))))

Error

Derivation

Reproduce