Average Error: 0.5 → 0.4
Time: 19.7s
Precision: binary32
\[-1 \leq cosTheta_i \land cosTheta_i \leq 1 \land -1 \leq cosTheta_O \land cosTheta_O \leq 1 \land -1 \leq sinTheta_i \land sinTheta_i \leq 1 \land -1 \leq sinTheta_O \land sinTheta_O \leq 1 \land 0.1 < v \land v \leq 1.5707964\]
\[[cosTheta_i, cosTheta_O]=\mathsf{sort}([cosTheta_i, cosTheta_O])\]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[cosTheta_O \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{1}{v} \cdot cosTheta_i}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)\right) \]
\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 \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{1}{v} \cdot cosTheta_i}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\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
  (expm1
   (log1p
    (/
     (/ (* (/ 1.0 v) cosTheta_i) v)
     (* (exp (/ (* sinTheta_i sinTheta_O) v)) (* 2.0 (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 * expm1f(log1pf((((1.0f / v) * cosTheta_i) / v) / (expf((sinTheta_i * sinTheta_O) / v) * (2.0f * 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. Applied add-sqr-sqrt_binary320.5

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

  4. Applied *-un-lft-identity_binary320.5

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

  5. Applied times-frac_binary320.5

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

  6. Applied times-frac_binary320.5

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Applied expm1-log1p-u_binary320.4

    \[\leadsto cosTheta_O \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{v}}{\sqrt{v}} \cdot \frac{\frac{cosTheta_i}{\sqrt{v}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)\right)} \]

  10. Simplified0.4

    \[\leadsto cosTheta_O \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\frac{\frac{1}{v} \cdot cosTheta_i}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)}\right) \]

  11. Final simplification0.4

    \[\leadsto cosTheta_O \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{1}{v} \cdot cosTheta_i}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)\right) \]

Reproduce

herbie shell --seed 2021225 
(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)))