Average Error: 0.1 → 0.1
Time: 9.7s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\sqrt{0.5} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{0.5}}{v}\right)\right) \cdot e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}\right) \]
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)}
↓
\sqrt{0.5} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{0.5}}{v}\right)\right) \cdot e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}\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
 (*
  (sqrt 0.5)
  (*
   (expm1 (log1p (/ (sqrt 0.5) v)))
   (exp (- (+ 0.6931 (/ (* cosTheta_i cosTheta_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 sqrtf(0.5f) * (expm1f(log1pf(sqrtf(0.5f) / v)) * expf((0.6931f + ((cosTheta_i * cosTheta_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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 *-un-lft-identity_binary320.1

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

  4. Applied add-sqr-sqrt_binary320.1

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

  5. Applied times-frac_binary320.1

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

  6. Applied associate-*l*_binary320.1

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

  7. Applied expm1-log1p-u_binary320.1

    \[\leadsto \frac{\sqrt{0.5}}{1} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{0.5}}{v}\right)\right)} \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)}\right) \]

  8. Taylor expanded in sinTheta_i around 0 0.1

    \[\leadsto \frac{\sqrt{0.5}}{1} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{0.5}}{v}\right)\right) \cdot \color{blue}{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}}\right) \]

  9. Final simplification0.1

    \[\leadsto \sqrt{0.5} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{0.5}}{v}\right)\right) \cdot e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}\right) \]

Reproduce

herbie shell --seed 2022082 
(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))))))