HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 20.9s

Precision: binary32

Cost: 3744

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

Math

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

↓

\[0.5 \cdot \left(\left(\frac{e^{0.6931 + \frac{-1}{v}}}{v} \cdot 4\right) \cdot \left(v \cdot \frac{0.25}{v}\right)\right) \]

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 (+ 0.6931 (/ -1.0 v))) v) 4.0) (* v (/ 0.25 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((0.6931f + (-1.0f / v))) / v) * 4.0f) * (v * (0.25f / 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((0.6931e0 + ((-1.0e0) / v))) / v) * 4.0e0) * (v * (0.25e0 / 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(0.6931) + Float32(Float32(-1.0) / v))) / v) * Float32(4.0)) * Float32(v * Float32(Float32(0.25) / 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(0.6931) + (single(-1.0) / v))) / v) * single(4.0)) * (v * (single(0.25) / 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

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

   [Start] 0.1	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   exponential-simplify-1 [=>] 0.1	\[ \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]
   rational_best-simplify-66 [=>] 0.1	\[ e^{\left(\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}} - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]
   rational_best-simplify-66 [=>] 0.1	\[ e^{\color{blue}{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v}} + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]
   exponential-simplify-7 [=>] 0.1	\[ e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]
   rational_best-simplify-54 [=>] 0.1	\[ e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]
   metadata-eval [=>] 0.1	\[ e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931} \cdot \frac{\color{blue}{0.5}}{v} \]

3. Taylor expanded in cosTheta_i around 0 0.1

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

4. Simplified 0.1

   \[\leadsto \color{blue}{e^{0.6931 - \frac{1 + sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{0.5}{v}} \]
   Proof

   [Start] 0.1	\[ 0.5 \cdot \frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v} \]
   rational_best-simplify-55 [=>] 0.1	\[ \color{blue}{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)} \cdot \frac{0.5}{v}} \]
   rational_best-simplify-64 [=>] 0.1	\[ e^{0.6931 - \color{blue}{\frac{1 + sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{0.5}{v} \]

5. Taylor expanded in sinTheta_i around 0 0.1

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

6. Applied egg-rr 0.1

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\frac{e^{0.6931 + \frac{-1}{v}}}{v} \cdot 4\right) \cdot \left(v \cdot \frac{0.25}{v}\right)\right)} \]

7. Final simplification 0.1

   \[\leadsto 0.5 \cdot \left(\left(\frac{e^{0.6931 + \frac{-1}{v}}}{v} \cdot 4\right) \cdot \left(v \cdot \frac{0.25}{v}\right)\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	3488

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

Alternative 2
Error	0.6
Cost	3424

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

Alternative 3
Error	20.4
Cost	288

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

Alternative 4
Error	30.5
Cost	96

\[\frac{0.5}{v} \]

Reproduce

herbie shell --seed 2023100 
(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))))))