Average Error: 0.1 → 0.1
Time: 18.8s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[{\left({\left({\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) + \frac{-1}{v}}}{v}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \left({\left(\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}\right)}^{0.1111111111111111} \cdot {\left(\sqrt[3]{0.5}\right)}^{0.3333333333333333}\right)\right)}^{3} \]
(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
 (pow
  (*
   (pow
    (pow
     (*
      (cbrt 0.5)
      (cbrt
       (/ (exp (+ (fma cosTheta_O (/ cosTheta_i v) 0.6931) (/ -1.0 v))) v)))
     2.0)
    0.3333333333333333)
   (*
    (pow
     (/ (exp (- 0.6931 (+ (/ 1.0 v) (/ (* sinTheta_i sinTheta_O) v)))) v)
     0.1111111111111111)
    (pow (cbrt 0.5) 0.3333333333333333)))
  3.0))
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 powf((powf(powf((cbrtf(0.5f) * cbrtf((expf((fmaf(cosTheta_O, (cosTheta_i / v), 0.6931f) + (-1.0f / v))) / v))), 2.0f), 0.3333333333333333f) * (powf((expf((0.6931f - ((1.0f / v) + ((sinTheta_i * sinTheta_O) / v)))) / v), 0.1111111111111111f) * powf(cbrtf(0.5f), 0.3333333333333333f))), 3.0f);
}

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end
↓
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(((Float32(cbrt(Float32(0.5)) * cbrt(Float32(exp(Float32(fma(cosTheta_O, Float32(cosTheta_i / v), Float32(0.6931)) + Float32(Float32(-1.0) / v))) / v))) ^ Float32(2.0)) ^ Float32(0.3333333333333333)) * Float32((Float32(exp(Float32(Float32(0.6931) - Float32(Float32(Float32(1.0) / v) + Float32(Float32(sinTheta_i * sinTheta_O) / v)))) / v) ^ Float32(0.1111111111111111)) * (cbrt(Float32(0.5)) ^ Float32(0.3333333333333333)))) ^ Float32(3.0)
end

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)}
↓
{\left({\left({\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) + \frac{-1}{v}}}{v}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \left({\left(\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}\right)}^{0.1111111111111111} \cdot {\left(\sqrt[3]{0.5}\right)}^{0.3333333333333333}\right)\right)}^{3}

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

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 egg-rr0.1

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

  4. Applied egg-rr0.1

    \[\leadsto {\color{blue}{\left({\left({\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)} \cdot \frac{0.5}{v}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)} \cdot \frac{0.5}{v}}}\right)}}^{3} \]

  5. Taylor expanded in cosTheta_O around 0 0.1

    \[\leadsto {\left({\left({\left(\sqrt[3]{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)} \cdot \frac{0.5}{v}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\left({\left(\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}\right)}^{0.1111111111111111} \cdot {\left(\sqrt[3]{0.5}\right)}^{0.3333333333333333}\right)}\right)}^{3} \]

  6. Taylor expanded in sinTheta_i around 0 0.1

    \[\leadsto {\left({\left({\color{blue}{\left({\left(\frac{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}}{v}\right)}^{0.3333333333333333} \cdot \sqrt[3]{0.5}\right)}}^{2}\right)}^{0.3333333333333333} \cdot \left({\left(\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}\right)}^{0.1111111111111111} \cdot {\left(\sqrt[3]{0.5}\right)}^{0.3333333333333333}\right)\right)}^{3} \]

  7. Simplified0.1

    \[\leadsto {\left({\left({\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) + \frac{-1}{v}}}{v}}\right)}}^{2}\right)}^{0.3333333333333333} \cdot \left({\left(\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}\right)}^{0.1111111111111111} \cdot {\left(\sqrt[3]{0.5}\right)}^{0.3333333333333333}\right)\right)}^{3} \]

  8. Final simplification0.1

    \[\leadsto {\left({\left({\left(\sqrt[3]{0.5} \cdot \sqrt[3]{\frac{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) + \frac{-1}{v}}}{v}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \left({\left(\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}\right)}^{0.1111111111111111} \cdot {\left(\sqrt[3]{0.5}\right)}^{0.3333333333333333}\right)\right)}^{3} \]

Reproduce

herbie shell --seed 2022134 
(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))))))