HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 15.6s

Precision: binary32

Cost: 13152

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

↓

\[0.5 \cdot \frac{\frac{e^{\frac{-1}{v}}}{\sqrt{v}} \cdot e^{0.6931}}{\sqrt{v}} \]

FPCore

(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 (/ (* (/ (exp (/ -1.0 v)) (sqrt v)) (exp 0.6931)) (sqrt v))))

C

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 * (((expf((-1.0f / v)) / sqrtf(v)) * expf(0.6931f)) / sqrtf(v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

↓

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 * (((exp(((-1.0e0) / v)) / sqrt(v)) * exp(0.6931e0)) / sqrt(v))
end function

Julia

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(0.5) * Float32(Float32(Float32(exp(Float32(Float32(-1.0) / v)) / sqrt(v)) * exp(Float32(0.6931))) / sqrt(v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

↓

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (((exp((single(-1.0) / v)) / sqrt(v)) * exp(single(0.6931))) / sqrt(v));
end

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. Simplified 0.1

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

   [Start] 0.1	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   remove-double-neg [<=] 0.1	\[ e^{\color{blue}{\left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   +-commutative [<=] 0.1	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)}} \]
   log-rec [=>] 0.1	\[ e^{\color{blue}{\left(-\log \left(2 \cdot v\right)\right)} + \left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)} \]
   distribute-neg-in [<=] 0.1	\[ e^{\color{blue}{-\left(\log \left(2 \cdot v\right) + \left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)}} \]
   sub-neg [<=] 0.1	\[ e^{-\color{blue}{\left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]
   exp-neg [=>] 0.1	\[ \color{blue}{\frac{1}{e^{\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}}} \]
   exp-diff [=>] 0.1	\[ \frac{1}{\color{blue}{\frac{e^{\log \left(2 \cdot v\right)}}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}}}} \]

3. Applied egg-rr 0.1

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

4. Taylor expanded in sinTheta_O around 0 0.1

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

5. Taylor expanded in cosTheta_i around 0 0.1

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

6. Simplified 0.1

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

   [Start] 0.1	\[ 0.5 \cdot e^{0.6931 - \left(\frac{1}{v} + \log v\right)} \]
   sub-neg [=>] 0.1	\[ 0.5 \cdot e^{\color{blue}{0.6931 + \left(-\left(\frac{1}{v} + \log v\right)\right)}} \]
   exp-sum [=>] 0.1	\[ 0.5 \cdot \color{blue}{\left(e^{0.6931} \cdot e^{-\left(\frac{1}{v} + \log v\right)}\right)} \]
   distribute-neg-in [=>] 0.1	\[ 0.5 \cdot \left(e^{0.6931} \cdot e^{\color{blue}{\left(-\frac{1}{v}\right) + \left(-\log v\right)}}\right) \]
   +-commutative [<=] 0.1	\[ 0.5 \cdot \left(e^{0.6931} \cdot e^{\color{blue}{\left(-\log v\right) + \left(-\frac{1}{v}\right)}}\right) \]
   sub-neg [<=] 0.1	\[ 0.5 \cdot \left(e^{0.6931} \cdot e^{\color{blue}{\left(-\log v\right) - \frac{1}{v}}}\right) \]
   exp-diff [=>] 0.2	\[ 0.5 \cdot \left(e^{0.6931} \cdot \color{blue}{\frac{e^{-\log v}}{e^{\frac{1}{v}}}}\right) \]
   log-rec [<=] 0.2	\[ 0.5 \cdot \left(e^{0.6931} \cdot \frac{e^{\color{blue}{\log \left(\frac{1}{v}\right)}}}{e^{\frac{1}{v}}}\right) \]
   rem-exp-log [=>] 0.2	\[ 0.5 \cdot \left(e^{0.6931} \cdot \frac{\color{blue}{\frac{1}{v}}}{e^{\frac{1}{v}}}\right) \]
   associate-/l/ [=>] 0.1	\[ 0.5 \cdot \left(e^{0.6931} \cdot \color{blue}{\frac{1}{e^{\frac{1}{v}} \cdot v}}\right) \]
   associate-/r* [=>] 0.1	\[ 0.5 \cdot \left(e^{0.6931} \cdot \color{blue}{\frac{\frac{1}{e^{\frac{1}{v}}}}{v}}\right) \]
   rec-exp [=>] 0.1	\[ 0.5 \cdot \left(e^{0.6931} \cdot \frac{\color{blue}{e^{-\frac{1}{v}}}}{v}\right) \]
   distribute-neg-frac [=>] 0.1	\[ 0.5 \cdot \left(e^{0.6931} \cdot \frac{e^{\color{blue}{\frac{-1}{v}}}}{v}\right) \]
   metadata-eval [=>] 0.1	\[ 0.5 \cdot \left(e^{0.6931} \cdot \frac{e^{\frac{\color{blue}{-1}}{v}}}{v}\right) \]

7. Applied egg-rr 0.1

   \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{e^{\frac{-1}{v}}}{\sqrt{v}} \cdot e^{0.6931}}{\sqrt{v}}} \]

8. Final simplification 0.1

   \[\leadsto 0.5 \cdot \frac{\frac{e^{\frac{-1}{v}}}{\sqrt{v}} \cdot e^{0.6931}}{\sqrt{v}} \]

Alternatives

Alternative 1
Error	0.1
Cost	13120

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

Alternative 2
Error	0.1
Cost	6688

\[0.5 \cdot e^{0.6931 - \left(\frac{1}{v} + \log v\right)} \]

Alternative 3
Error	0.1
Cost	6688

\[0.5 \cdot \left(e^{0.6931} \cdot \frac{e^{\frac{-1}{v}}}{v}\right) \]

Alternative 4
Error	0.2
Cost	3488

\[\frac{0.5}{v} \cdot e^{\frac{-1}{v} + 0.6931} \]

Alternative 5
Error	0.1
Cost	3488

\[\frac{0.5}{\frac{v}{e^{0.6931 + \frac{-1}{v}}}} \]

Alternative 6
Error	0.7
Cost	3424

\[\frac{0.5}{\frac{v}{e^{\frac{-1}{v}}}} \]

Alternative 7
Error	0.7
Cost	3360

\[0.5 \cdot e^{\frac{-1}{v}} \]

Alternative 8
Error	25.5
Cost	224

\[0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \]

Alternative 9
Error	19.6
Cost	224

\[0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]

Alternative 10
Error	29.9
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023057 
(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))))))