Average Error: 0.1 → 0.1
Time: 15.4s
Precision: binary32
Cost: 13152
\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[{\left({\left(\frac{e^{0.6931 + \frac{-1}{v}}}{v}\right)}^{0.3333333333333333} \cdot \sqrt[3]{0.5}\right)}^{3} \]
(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 (/ (exp (+ 0.6931 (/ -1.0 v))) v) 0.3333333333333333) (cbrt 0.5))
  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((expf((0.6931f + (-1.0f / v))) / v), 0.3333333333333333f) * cbrtf(0.5f)), 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(exp(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))) / v) ^ Float32(0.3333333333333333)) * cbrt(Float32(0.5))) ^ 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(\frac{e^{0.6931 + \frac{-1}{v}}}{v}\right)}^{0.3333333333333333} \cdot \sqrt[3]{0.5}\right)}^{3}

Error

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. Applied egg-rr0.1

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

  3. Taylor expanded in sinTheta_i around 0 0.1

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

  4. Taylor expanded in cosTheta_i around 0 0.1

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

  5. Final simplification0.1

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

Alternatives

Alternative 1
Error0.1
Cost7136
\[e^{\left(0.6931 + \left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right)} \]
Alternative 2
Error0.1
Cost6688
\[\frac{\left(0.5 \cdot e^{\frac{-1}{v}}\right) \cdot e^{0.6931}}{v} \]
Alternative 3
Error0.1
Cost3488
\[\frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
Alternative 4
Error0.6
Cost3296
\[e^{\frac{-1}{v}} \]
Alternative 5
Error29.9
Cost32
\[1 \]

Error

Reproduce

herbie shell --seed 2022298 
(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))))))