Average Error: 0.1 → 0.1
Time: 13.4s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := v \cdot \sqrt{v}\\ t_1 := cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)\\ \frac{0.5}{v} \cdot \left(\sqrt{e^{0.6931 + \frac{t_1}{v}}} \cdot \left(\sqrt{e^{0.6931}} \cdot \sqrt{e^{\sqrt[3]{\frac{{\left(\sqrt[3]{t_1}\right)}^{6}}{t_0}} \cdot \sqrt[3]{\frac{t_1}{t_0}}}}\right)\right) \end{array} \]
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 := v \cdot \sqrt{v}\\
t_1 := cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)\\
\frac{0.5}{v} \cdot \left(\sqrt{e^{0.6931 + \frac{t_1}{v}}} \cdot \left(\sqrt{e^{0.6931}} \cdot \sqrt{e^{\sqrt[3]{\frac{{\left(\sqrt[3]{t_1}\right)}^{6}}{t_0}} \cdot \sqrt[3]{\frac{t_1}{t_0}}}}\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 (* v (sqrt v)))
        (t_1 (- (* cosTheta_i cosTheta_O) (fma sinTheta_i sinTheta_O 1.0))))
   (*
    (/ 0.5 v)
    (*
     (sqrt (exp (+ 0.6931 (/ t_1 v))))
     (*
      (sqrt (exp 0.6931))
      (sqrt
       (exp (* (cbrt (/ (pow (cbrt t_1) 6.0) t_0)) (cbrt (/ t_1 t_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 = v * sqrtf(v);
	float t_1 = (cosTheta_i * cosTheta_O) - fmaf(sinTheta_i, sinTheta_O, 1.0f);
	return (0.5f / v) * (sqrtf(expf(0.6931f + (t_1 / v))) * (sqrtf(expf(0.6931f)) * sqrtf(expf(cbrtf(powf(cbrtf(t_1), 6.0f) / t_0) * cbrtf(t_1 / t_0)))));
}

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

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

  4. Simplified0.1

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

  5. Simplified0.1

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

  6. Applied add-cbrt-cube_binary320.1

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

  7. Applied add-cbrt-cube_binary320.1

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

  8. Applied cbrt-undiv_binary320.1

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

  9. Applied exp-sum_binary320.1

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

  10. Applied sqrt-prod_binary320.1

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

  11. Simplified0.1

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

  12. Applied add-sqr-sqrt_binary320.1

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

  13. Applied unpow-prod-down_binary320.1

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

  14. Applied add-cube-cbrt_binary320.1

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

  15. Applied unpow-prod-down_binary320.1

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

  16. Applied times-frac_binary320.1

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

  17. Applied cbrt-prod_binary320.1

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

  18. Simplified0.1

    \[\leadsto \frac{0.5}{v} \cdot \left(\sqrt{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}} \cdot \left(\sqrt{e^{0.6931}} \cdot \sqrt{e^{\color{blue}{\sqrt[3]{\frac{{\left(\sqrt[3]{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}\right)}^{6}}{v \cdot \sqrt{v}}}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}\right)}^{3}}{{\left(\sqrt{v}\right)}^{3}}}}}\right)\right) \]

  19. Simplified0.1

    \[\leadsto \frac{0.5}{v} \cdot \left(\sqrt{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}} \cdot \left(\sqrt{e^{0.6931}} \cdot \sqrt{e^{\sqrt[3]{\frac{{\left(\sqrt[3]{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}\right)}^{6}}{v \cdot \sqrt{v}}} \cdot \color{blue}{\sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v \cdot \sqrt{v}}}}}}\right)\right) \]

  20. Final simplification0.1

    \[\leadsto \frac{0.5}{v} \cdot \left(\sqrt{e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}}} \cdot \left(\sqrt{e^{0.6931}} \cdot \sqrt{e^{\sqrt[3]{\frac{{\left(\sqrt[3]{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}\right)}^{6}}{v \cdot \sqrt{v}}} \cdot \sqrt[3]{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v \cdot \sqrt{v}}}}}\right)\right) \]

Reproduce

herbie shell --seed 2022077 
(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))))))