Average Error: 0.1 → 0.1
Time: 1.1min
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\frac{e^{{\left(\sqrt[3]{0.6931 + \log 0.5}\right)}^{3} - \frac{1}{v}}}{v} \]
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)}
↓
\frac{e^{{\left(\sqrt[3]{0.6931 + \log 0.5}\right)}^{3} - \frac{1}{v}}}{v}

(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
 (/ (exp (- (pow (cbrt (+ 0.6931 (log 0.5))) 3.0) (/ 1.0 v))) 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 expf((powf(cbrtf((0.6931f + logf(0.5f))), 3.0f) - (1.0f / v))) / 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. Taylor expanded in v around 0 0.1

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

  3. Taylor expanded in sinTheta_i around 0 0.1

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

  4. Simplified0.1

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

  5. Applied egg-rr0.1

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

  6. Taylor expanded in cosTheta_i around 0 0.1

    \[\leadsto \frac{e^{{\left(\sqrt[3]{\color{blue}{0.6931 + \log 0.5}}\right)}^{3} - \frac{1}{v}}}{v} \]

  7. Final simplification0.1

    \[\leadsto \frac{e^{{\left(\sqrt[3]{0.6931 + \log 0.5}\right)}^{3} - \frac{1}{v}}}{v} \]

Reproduce

herbie shell --seed 2022125 
(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))))))