HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 22.9s

Precision: binary32

Cost: 7104

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

\[ \begin{array}{c}[cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \end{array} \]

Math

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

↓

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

↓

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (*
   (/ (exp (* sinTheta_O (/ sinTheta_i (- v)))) v)
   (/ cosTheta_i (* 2.0 (/ (sinh (/ 1.0 v)) (/ 1.0 v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

↓

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((expf((sinTheta_O * (sinTheta_i / -v))) / v) * (cosTheta_i / (2.0f * (sinhf((1.0f / 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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 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 = costheta_o * ((exp((sintheta_o * (sintheta_i / -v))) / v) * (costheta_i / (2.0e0 * (sinh((1.0e0 / v)) / (1.0e0 / v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

↓

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(exp(Float32(sinTheta_O * Float32(sinTheta_i / Float32(-v)))) / v) * Float32(cosTheta_i / Float32(Float32(2.0) * Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(1.0) / v))))))
end

MATLAB

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

↓

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

Derivation

1. Initial program 0.5

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

2. Simplified 0.4

   \[\leadsto \color{blue}{cosTheta_O \cdot \frac{e^{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \cdot \frac{cosTheta_i}{v}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
   Proof

   [Start] 0.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-47 [=>] 0.4	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-44 [=>] 0.4	\[ \frac{\color{blue}{cosTheta_O \cdot \left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-2 [=>] 0.4	\[ \frac{\color{blue}{\left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i}{v}\right) \cdot cosTheta_O}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-47 [=>] 0.4	\[ \color{blue}{cosTheta_O \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]
   rational_best-simplify-9 [=>] 0.4	\[ cosTheta_O \cdot \frac{e^{\color{blue}{\frac{\frac{sinTheta_i \cdot sinTheta_O}{v}}{-1}}} \cdot \frac{cosTheta_i}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-2 [=>] 0.4	\[ cosTheta_O \cdot \frac{e^{\frac{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}{-1}} \cdot \frac{cosTheta_i}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-47 [=>] 0.4	\[ cosTheta_O \cdot \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{-1}} \cdot \frac{cosTheta_i}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-47 [=>] 0.4	\[ cosTheta_O \cdot \frac{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot \frac{sinTheta_i}{-1}}} \cdot \frac{cosTheta_i}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-8 [=>] 0.4	\[ cosTheta_O \cdot \frac{e^{\frac{sinTheta_O}{v} \cdot \color{blue}{\left(-sinTheta_i\right)}} \cdot \frac{cosTheta_i}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   rational_best-simplify-2 [=>] 0.4	\[ cosTheta_O \cdot \frac{e^{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \cdot \frac{cosTheta_i}{v}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

3. Applied egg-rr 0.4

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

4. Applied egg-rr 0.4

   \[\leadsto cosTheta_O \cdot \left(\frac{e^{sinTheta_O \cdot \frac{sinTheta_i}{-v}}}{v} \cdot \frac{cosTheta_i}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}} + 0}}\right) \]

5. Simplified 0.4

   \[\leadsto cosTheta_O \cdot \left(\frac{e^{sinTheta_O \cdot \frac{sinTheta_i}{-v}}}{v} \cdot \frac{cosTheta_i}{\color{blue}{2 \cdot \frac{\sinh \left(\frac{1}{v}\right)}{\frac{1}{v}}}}\right) \]
   Proof

   [Start] 0.4	\[ cosTheta_O \cdot \left(\frac{e^{sinTheta_O \cdot \frac{sinTheta_i}{-v}}}{v} \cdot \frac{cosTheta_i}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}} + 0}\right) \]
   rational_best-simplify-3 [=>] 0.4	\[ cosTheta_O \cdot \left(\frac{e^{sinTheta_O \cdot \frac{sinTheta_i}{-v}}}{v} \cdot \frac{cosTheta_i}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}}\right) \]
   rational_best-simplify-47 [=>] 0.4	\[ cosTheta_O \cdot \left(\frac{e^{sinTheta_O \cdot \frac{sinTheta_i}{-v}}}{v} \cdot \frac{cosTheta_i}{\color{blue}{2 \cdot \frac{\sinh \left(\frac{1}{v}\right)}{\frac{1}{v}}}}\right) \]

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	7040

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

Alternative 2
Error	0.4
Cost	7008

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

Alternative 3
Error	0.4
Cost	7008

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

Alternative 4
Error	0.5
Cost	6880

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

Alternative 5
Error	9.1
Cost	3876

\[\begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\left(cosTheta_i \cdot \frac{cosTheta_O}{v \cdot \left(e^{\frac{1}{v}} - 1\right)}\right) \cdot \frac{1}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O}{e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}} \cdot \frac{cosTheta_i}{v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333}\\ \end{array} \]

Alternative 6
Error	10.2
Cost	3680

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

Alternative 7
Error	13.4
Cost	288

\[\left(0.5 \cdot \left(cosTheta_i \cdot cosTheta_O\right)\right) \cdot \frac{1}{v} \]

Alternative 8
Error	13.4
Cost	224

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

Alternative 9
Error	13.4
Cost	224

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

Alternative 10
Error	13.4
Cost	224

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

Alternative 11
Error	13.4
Cost	224

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

Alternative 12
Error	13.4
Cost	224

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

Reproduce

herbie shell --seed 2023096 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))