Average Error: 0.1 → 0.1
Time: 12.3s
Precision: binary32
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)} \cdot \frac{0.5}{v}}\\ t_0 \cdot \left(t_0 \cdot \sqrt[3]{\frac{0.5}{v} \cdot {\left(e^{\sqrt[3]{{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}\right)}^{2}}}\right)}^{\left(\sqrt[3]{0.6931 + \left(\frac{-1}{v} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right)}\right)}}\right) \end{array} \]
(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
 (let* ((t_0
         (cbrt
          (*
           (exp
            (+
             (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) v)
             (+ 0.6931 (/ -1.0 v))))
           (/ 0.5 v)))))
   (*
    t_0
    (*
     t_0
     (cbrt
      (*
       (/ 0.5 v)
       (pow
        (exp
         (cbrt (pow (+ 0.6931 (/ (fma cosTheta_i cosTheta_O -1.0) v)) 2.0)))
        (cbrt (+ 0.6931 (- (/ -1.0 v) (* sinTheta_O (/ sinTheta_i 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) {
	float t_0 = cbrtf((expf(((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + (0.6931f + (-1.0f / v)))) * (0.5f / v)));
	return t_0 * (t_0 * cbrtf(((0.5f / v) * powf(expf(cbrtf(powf((0.6931f + (fmaf(cosTheta_i, cosTheta_O, -1.0f) / v)), 2.0f))), cbrtf((0.6931f + ((-1.0f / v) - (sinTheta_O * (sinTheta_i / v)))))))));
}

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)
	t_0 = cbrt(Float32(exp(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) / v) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)))) * Float32(Float32(0.5) / v)))
	return Float32(t_0 * Float32(t_0 * cbrt(Float32(Float32(Float32(0.5) / v) * (exp(cbrt((Float32(Float32(0.6931) + Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v)) ^ Float32(2.0)))) ^ cbrt(Float32(Float32(0.6931) + Float32(Float32(Float32(-1.0) / v) - Float32(sinTheta_O * Float32(sinTheta_i / v))))))))))
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)}
↓
\begin{array}{l}
t_0 := \sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)} \cdot \frac{0.5}{v}}\\
t_0 \cdot \left(t_0 \cdot \sqrt[3]{\frac{0.5}{v} \cdot {\left(e^{\sqrt[3]{{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}\right)}^{2}}}\right)}^{\left(\sqrt[3]{0.6931 + \left(\frac{-1}{v} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right)}\right)}}\right)
\end{array}

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. Applied egg-rr0.1

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

  3. Applied egg-rr0.1

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

  4. Taylor expanded in sinTheta_i around 0 0.1

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

  5. Simplified0.1

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

  6. Taylor expanded in cosTheta_i around 0 32.0

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2022178 
(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))))))