Average Error: 0.1 → 0.1
Time: 12.3s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\left(e^{\log \left({\left(\sqrt[3]{\frac{1}{\left|\sqrt[3]{v}\right|}}\right)}^{4}\right)} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{v} \cdot \sqrt[3]{v}}}\right) \cdot \left(\frac{0.5}{\sqrt[3]{v}} \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]
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)}
↓
\left(e^{\log \left({\left(\sqrt[3]{\frac{1}{\left|\sqrt[3]{v}\right|}}\right)}^{4}\right)} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{v} \cdot \sqrt[3]{v}}}\right) \cdot \left(\frac{0.5}{\sqrt[3]{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)

(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 (log (pow (cbrt (/ 1.0 (fabs (cbrt v)))) 4.0)))
   (cbrt (/ 1.0 (* (cbrt v) (cbrt v)))))
  (*
   (/ 0.5 (cbrt v))
   (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) {
	return (expf(logf(powf(cbrtf(1.0f / fabsf(cbrtf(v))), 4.0f))) * cbrtf(1.0f / (cbrtf(v) * cbrtf(v)))) * ((0.5f / cbrtf(v)) * 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-cube-cbrt_binary320.1

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

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

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

  5. Applied times-frac_binary320.1

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

  6. Applied associate-*l*_binary320.1

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

  7. Applied add-cube-cbrt_binary320.1

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

  8. Applied add-exp-log_binary320.1

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

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

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

  10. Applied cbrt-prod_binary320.1

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

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

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

  12. Applied cbrt-prod_binary320.1

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

  13. Applied swap-sqr_binary320.1

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

  14. Simplified0.1

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

  15. Simplified0.1

    \[\leadsto \left(e^{\log \left(1 \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{\left|\sqrt[3]{v}\right|}}\right)}^{4}}\right)} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{v} \cdot \sqrt[3]{v}}}\right) \cdot \left(\frac{0.5}{\sqrt[3]{v}} \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  16. Final simplification0.1

    \[\leadsto \left(e^{\log \left({\left(\sqrt[3]{\frac{1}{\left|\sqrt[3]{v}\right|}}\right)}^{4}\right)} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{v} \cdot \sqrt[3]{v}}}\right) \cdot \left(\frac{0.5}{\sqrt[3]{v}} \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

Reproduce

herbie shell --seed 2022020 
(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))))))