HairBSDF, Mp, lower

Average Accuracy: 99.6% → 99.5%

Time: 16.0s

Precision: binary32

Cost: 9888

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

Math

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

↓

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

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
 (* (exp 0.6931) (exp (+ (log (/ 0.5 v)) (/ -1.0 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 expf(0.6931f) * expf((logf((0.5f / v)) + (-1.0f / 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 = exp(0.6931e0) * exp((log((0.5e0 / v)) + ((-1.0e0) / 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(exp(Float32(0.6931)) * exp(Float32(log(Float32(Float32(0.5) / v)) + Float32(Float32(-1.0) / 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 = exp(single(0.6931)) * exp((log((single(0.5) / v)) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.6%

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

2. Simplified 99.5%

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

   [Start] 99.6	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   associate-+l+ [=>] 99.5	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]
   sub-neg [=>] 99.5	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [=>] 99.5	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [<=] 99.5	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   sub-neg [<=] 99.5	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate--l- [=>] 99.5	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/l* [=>] 99.5	\[ e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/r* [=>] 99.5	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]
   metadata-eval [=>] 99.5	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Taylor expanded in sinTheta_i around 0 99.5%

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

4. Taylor expanded in v around inf 99.4%

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

5. Simplified 99.5%

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

   [Start] 99.4	\[ e^{\left(0.6931 + \left(\log \left(\frac{1}{v}\right) + \log 0.5\right)\right) - \frac{1}{v}} \]
   +-commutative [=>] 99.4	\[ e^{\left(0.6931 + \color{blue}{\left(\log 0.5 + \log \left(\frac{1}{v}\right)\right)}\right) - \frac{1}{v}} \]
   log-rec [=>] 99.7	\[ e^{\left(0.6931 + \left(\log 0.5 + \color{blue}{\left(-\log v\right)}\right)\right) - \frac{1}{v}} \]
   sub-neg [<=] 99.7	\[ e^{\left(0.6931 + \color{blue}{\left(\log 0.5 - \log v\right)}\right) - \frac{1}{v}} \]
   log-div [<=] 99.5	\[ e^{\left(0.6931 + \color{blue}{\log \left(\frac{0.5}{v}\right)}\right) - \frac{1}{v}} \]

6. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}} \]
   associate--l+ [=>] 99.5	\[ e^{\color{blue}{0.6931 + \left(\log \left(\frac{0.5}{v}\right) - \frac{1}{v}\right)}} \]
   exp-sum [=>] 99.5	\[ \color{blue}{e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right) - \frac{1}{v}}} \]
   sub-neg [=>] 99.5	\[ e^{0.6931} \cdot e^{\color{blue}{\log \left(\frac{0.5}{v}\right) + \left(-\frac{1}{v}\right)}} \]
   distribute-neg-frac [=>] 99.5	\[ e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right) + \color{blue}{\frac{-1}{v}}} \]
   metadata-eval [=>] 99.5	\[ e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right) + \frac{\color{blue}{-1}}{v}} \]

7. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6688

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

Alternative 2
Accuracy	99.5%
Cost	3488

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

Alternative 3
Accuracy	97.9%
Cost	3424

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

Alternative 4
Accuracy	97.9%
Cost	3296

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

Alternative 5
Accuracy	69.9%
Cost	288

\[-1 + \left(1 + sinTheta_O \cdot \frac{sinTheta_i}{v}\right) \]

Alternative 6
Accuracy	38.6%
Cost	224

\[\frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}} \]

Alternative 7
Accuracy	38.3%
Cost	192

\[\frac{sinTheta_O \cdot sinTheta_i}{-v} \]

Alternative 8
Accuracy	20.2%
Cost	160

\[sinTheta_i \cdot \frac{sinTheta_O}{v} \]

Alternative 9
Accuracy	38.3%
Cost	160

\[\frac{sinTheta_O \cdot sinTheta_i}{v} \]

Alternative 10
Accuracy	6.4%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023146 
(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))))))