Average Error: 0.5 → 0.4
Time: 20.2s
Precision: binary32
\[-1 \leq cosTheta_i \land cosTheta_i \leq 1 \land -1 \leq cosTheta_O \land cosTheta_O \leq 1 \land -1 \leq sinTheta_i \land sinTheta_i \leq 1 \land -1 \leq sinTheta_O \land sinTheta_O \leq 1 \land 0.1 < v \land v \leq 1.5707964\]
\[[cosTheta_i, cosTheta_O]=\mathsf{sort}([cosTheta_i, cosTheta_O])\]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\]
↓
\[cosTheta_O \cdot \left(\frac{cosTheta_i}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{\frac{1}{v}}{2 \cdot \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)\]
\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}
↓
cosTheta_O \cdot \left(\frac{cosTheta_i}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{\frac{1}{v}}{2 \cdot \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))
↓
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (*
   (/ cosTheta_i (exp (/ (* sinTheta_i sinTheta_O) v)))
   (/ (/ 1.0 v) (* 2.0 (* v (sinh (/ 1.0 v))))))))
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf(1.0f / v) * 2.0f) * v);
}
↓
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((cosTheta_i / expf((sinTheta_i * sinTheta_O) / v)) * ((1.0f / v) / (2.0f * (v * sinhf(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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Simplified0.4

    \[\leadsto \color{blue}{cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{v \cdot \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)}}\]
  3. Using strategy rm

  4. Applied add-exp-log_binary320.4

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

  5. Applied add-exp-log_binary320.5

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

  6. Applied prod-exp_binary320.5

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

  7. Applied prod-exp_binary320.5

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

  8. Applied add-exp-log_binary320.5

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

  9. Applied prod-exp_binary320.6

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

  10. Simplified0.5

    \[\leadsto cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{e^{\color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v} + \log \left(v \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)\right)}}}\]
  11. Using strategy rm

  12. Applied exp-sum_binary320.5

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

  13. Applied div-inv_binary320.4

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

  14. Applied times-frac_binary320.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

    \[\leadsto cosTheta_O \cdot \left(\frac{cosTheta_i}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{\frac{1}{v}}{2 \cdot \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2021176 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (<= -1.0 cosTheta_i 1.0) (<= -1.0 cosTheta_O 1.0) (<= -1.0 sinTheta_i 1.0) (<= -1.0 sinTheta_O 1.0) (< 0.1 v) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))