Average Error: 0.5 → 0.4
Time: 16.0s
Precision: binary32
\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]
\[[cosTheta_i, cosTheta_O]=\mathsf{sort}([cosTheta_i, cosTheta_O])\]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[cosTheta_O \cdot \left(\frac{cosTheta_i}{v} \cdot \frac{\frac{1}{v}}{\left(2 \cdot \frac{e^{\frac{1}{v}} - {e}^{\left(\frac{-1}{v}\right)}}{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\right) \]
\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}{v} \cdot \frac{\frac{1}{v}}{\left(2 \cdot \frac{e^{\frac{1}{v}} - {e}^{\left(\frac{-1}{v}\right)}}{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\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 v)
   (/
    (/ 1.0 v)
    (*
     (* 2.0 (/ (- (exp (/ 1.0 v)) (pow E (/ -1.0 v))) 2.0))
     (exp (/ (* sinTheta_i sinTheta_O) 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 / v) * ((1.0f / v) / ((2.0f * ((expf(1.0f / v) - powf(((float) M_E), (-1.0f / v))) / 2.0f)) * expf((sinTheta_i * sinTheta_O) / 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. Applied div-inv_binary320.4

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

  4. Applied times-frac_binary320.4

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

  5. Applied sinh-def_binary320.4

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

  6. Simplified0.4

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

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

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

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

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

  9. Applied times-frac_binary320.4

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

  10. Applied exp-prod_binary320.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2021329 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))