Average Error: 0.1 → 0.1
Time: 7.6s
Precision: binary32
\[-1 \leq cosTheta_i \land cosTheta_i \leq 1 \land -1 \leq cosTheta_O \land cosTheta_O \leq 1 \land -1 \leq sinTheta_i \land sinTheta_i \leq 1 \land -1 \leq sinTheta_O \land sinTheta_O \leq 1 \land -1.5707964 \leq v \land v \leq 0.1\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{\frac{-1}{\sqrt{v}}}{\sqrt{v}}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right)} \]
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)}
↓
e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{\frac{-1}{\sqrt{v}}}{\sqrt{v}}\right)\right) + \log \left(\frac{1}{v \cdot 2}\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
 (exp
  (+
   (+
    (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
    (+ 0.6931 (/ (/ -1.0 (sqrt v)) (sqrt v))))
   (log (/ 1.0 (* v 2.0))))))
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(((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) + (0.6931f + ((-1.0f / sqrtf(v)) / sqrtf(v)))) + logf(1.0f / (v * 2.0f)));
}

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. Using strategy rm

  3. Applied add-sqr-sqrt_binary320.1

    \[\leadsto e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  4. Applied add-cube-cbrt_binary320.1

    \[\leadsto e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{v} \cdot \sqrt{v}}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  5. Applied times-frac_binary320.1

    \[\leadsto e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{v}} \cdot \frac{\sqrt[3]{1}}{\sqrt{v}}}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  6. Applied cancel-sign-sub-inv_binary320.1

    \[\leadsto e^{\left(\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{v}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{v}}\right)} + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  7. Applied associate-+l+_binary320.1

    \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(\left(-\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{v}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{v}} + 0.6931\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]

  8. Simplified0.1

    \[\leadsto e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \color{blue}{\left(0.6931 + \frac{\frac{-1}{\sqrt{v}}}{\sqrt{v}}\right)}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

  9. Final simplification0.1

    \[\leadsto e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{\frac{-1}{\sqrt{v}}}{\sqrt{v}}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right)} \]

Reproduce

herbie shell --seed 2021206 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :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) (<= -1.5707964 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))))))