Average Error: 0.1 → 0.1
Time: 16.3s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\sqrt{0.5}}\\ t_1 := \frac{\sqrt{0.5}}{\sqrt{v}}\\ t_1 \cdot \left(\left(\left({\left(\sqrt[3]{\sqrt[3]{t_0}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt[3]{\frac{t_0}{\sqrt{v}}}}\right)}^{8}\right) \cdot \sqrt[3]{\sqrt[3]{t_1}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \end{array} \]
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{\sqrt{0.5}}\\
t_1 := \frac{\sqrt{0.5}}{\sqrt{v}}\\
t_1 \cdot \left(\left(\left({\left(\sqrt[3]{\sqrt[3]{t_0}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt[3]{\frac{t_0}{\sqrt{v}}}}\right)}^{8}\right) \cdot \sqrt[3]{\sqrt[3]{t_1}}\right) \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)}\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 (sqrt (sqrt 0.5))) (t_1 (/ (sqrt 0.5) (sqrt v))))
   (*
    t_1
    (*
     (*
      (* (pow (cbrt (cbrt t_0)) 8.0) (pow (cbrt (cbrt (/ t_0 (sqrt v)))) 8.0))
      (cbrt (cbrt t_1)))
     (exp
      (-
       (fma cosTheta_O (/ cosTheta_i v) 0.6931)
       (fma sinTheta_i (/ sinTheta_O v) (/ 1.0 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 = sqrtf(sqrtf(0.5f));
	float t_1 = sqrtf(0.5f) / sqrtf(v);
	return t_1 * (((powf(cbrtf(cbrtf(t_0)), 8.0f) * powf(cbrtf(cbrtf(t_0 / sqrtf(v))), 8.0f)) * cbrtf(cbrtf(t_1))) * expf(fmaf(cosTheta_O, (cosTheta_i / v), 0.6931f) - fmaf(sinTheta_i, (sinTheta_O / v), (1.0f / v))));
}

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. Simplified0.1

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

  3. Applied add-sqr-sqrt_binary320.1

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

  4. Applied add-sqr-sqrt_binary320.1

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

  5. Applied times-frac_binary320.1

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

  6. Applied associate-*l*_binary320.1

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

  7. Applied add-cube-cbrt_binary320.1

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

  8. Applied add-cube-cbrt_binary320.1

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

  9. Applied associate-*r*_binary320.1

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

  10. Simplified0.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right)}^{8}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

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

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left({\left(\sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{\color{blue}{1 \cdot v}}}}}\right)}^{8} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  12. Applied sqrt-prod_binary320.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left({\left(\sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\color{blue}{\sqrt{1} \cdot \sqrt{v}}}}}\right)}^{8} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  13. Applied add-sqr-sqrt_binary320.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left({\left(\sqrt[3]{\sqrt[3]{\frac{\color{blue}{\sqrt{\sqrt{0.5}} \cdot \sqrt{\sqrt{0.5}}}}{\sqrt{1} \cdot \sqrt{v}}}}\right)}^{8} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  14. Applied times-frac_binary320.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{1}} \cdot \frac{\sqrt{\sqrt{0.5}}}{\sqrt{v}}}}}\right)}^{8} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  15. Applied cbrt-prod_binary320.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left({\left(\sqrt[3]{\color{blue}{\sqrt[3]{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{1}}} \cdot \sqrt[3]{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{v}}}}}\right)}^{8} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  16. Applied cbrt-prod_binary320.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left({\color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{1}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{v}}}}\right)}}^{8} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  17. Applied unpow-prod-down_binary320.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left(\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{1}}}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt[3]{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{v}}}}\right)}^{8}\right)} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  18. Simplified0.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sqrt{\sqrt{0.5}}}}\right)}^{8}} \cdot {\left(\sqrt[3]{\sqrt[3]{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{v}}}}\right)}^{8}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  19. Final simplification0.1

    \[\leadsto \frac{\sqrt{0.5}}{\sqrt{v}} \cdot \left(\left(\left({\left(\sqrt[3]{\sqrt[3]{\sqrt{\sqrt{0.5}}}}\right)}^{8} \cdot {\left(\sqrt[3]{\sqrt[3]{\frac{\sqrt{\sqrt{0.5}}}{\sqrt{v}}}}\right)}^{8}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{0.5}}{\sqrt{v}}}}\right) \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

Reproduce

herbie shell --seed 2021313 
(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))))))