Average Error: 0.1 → 0.1
Time: 13.5s
Precision: binary32
\[-1 \leq cosTheta_i \land cosTheta_i \leq 1 \land -1 \leq cosTheta_O \land cosTheta_O \leq 1 \land -1 \leq sinTheta_i \land sinTheta_i \leq 1 \land -1 \leq sinTheta_O \land sinTheta_O \leq 1 \land -1.5707964 \leq v \land v \leq 0.1\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}\]
↓
\[\frac{0.5}{v} \cdot \left(\sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}} \cdot \left(\left(\left(\frac{sinTheta_i \cdot sinTheta_O}{v} \cdot -0.3333333333333333 + 1\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} + 0.6931\right) - \frac{1}{v}}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}}\right)\right)\]
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{0.5}{v} \cdot \left(\sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}} \cdot \left(\left(\left(\frac{sinTheta_i \cdot sinTheta_O}{v} \cdot -0.3333333333333333 + 1\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} + 0.6931\right) - \frac{1}{v}}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}}\right)\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
 (*
  (/ 0.5 v)
  (*
   (cbrt
    (exp
     (+
      0.6931
      (+
       (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) v)
       (/ -1.0 v)))))
   (*
    (*
     (+ (* (/ (* sinTheta_i sinTheta_O) v) -0.3333333333333333) 1.0)
     (cbrt (exp (- (+ (/ (* cosTheta_i cosTheta_O) v) 0.6931) (/ 1.0 v)))))
    (cbrt
     (exp
      (+
       0.6931
       (+
        (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_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 (0.5f / v) * (cbrtf(expf(0.6931f + ((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + (-1.0f / v)))) * ((((((sinTheta_i * sinTheta_O) / v) * -0.3333333333333333f) + 1.0f) * cbrtf(expf((((cosTheta_i * cosTheta_O) / v) + 0.6931f) - (1.0f / v)))) * cbrtf(expf(0.6931f + ((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_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

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

  4. Applied add-cube-cbrt_binary320.1

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

  5. Simplified0.1

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

  6. Simplified0.1

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

  7. Taylor expanded around 0 0.1

    \[\leadsto \frac{0.5}{v} \cdot \left(\left(\color{blue}{\left({\left(e^{\left(\frac{cosTheta_O \cdot cosTheta_i}{v} + 0.6931\right) - \frac{1}{v}}\right)}^{0.3333333333333333} - 0.3333333333333333 \cdot \left(\frac{sinTheta_i \cdot sinTheta_O}{v} \cdot {\left(e^{\left(\frac{cosTheta_O \cdot cosTheta_i}{v} + 0.6931\right) - \frac{1}{v}}\right)}^{0.3333333333333333}\right)\right)} \cdot \sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}}\right)\]

  8. Simplified0.1

    \[\leadsto \frac{0.5}{v} \cdot \left(\left(\color{blue}{\left(\left(\frac{sinTheta_i \cdot sinTheta_O}{v} \cdot -0.3333333333333333 + 1\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} + 0.6931\right) - \frac{1}{v}}}\right)} \cdot \sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}}\right)\]

  9. Final simplification0.1

    \[\leadsto \frac{0.5}{v} \cdot \left(\sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}} \cdot \left(\left(\left(\frac{sinTheta_i \cdot sinTheta_O}{v} \cdot -0.3333333333333333 + 1\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} + 0.6931\right) - \frac{1}{v}}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \frac{-1}{v}\right)}}\right)\right)\]

Reproduce

herbie shell --seed 2021175 
(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))))))