HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 18.7s

Precision: binary32

Cost: 7200

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

Math

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

↓

\[\frac{cosTheta_i \cdot cosTheta_O}{\frac{\frac{1}{0.5 \cdot \frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)}}}{\frac{1}{v}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

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_i cosTheta_O)
  (*
   (/ (/ 1.0 (* 0.5 (/ (/ 1.0 v) (sinh (/ 1.0 v))))) (/ 1.0 v))
   (exp (* (/ sinTheta_O v) sinTheta_i)))))

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_i * cosTheta_O) / (((1.0f / (0.5f * ((1.0f / v) / sinhf((1.0f / v))))) / (1.0f / v)) * expf(((sinTheta_O / v) * sinTheta_i)));
}

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_i * costheta_o) / (((1.0e0 / (0.5e0 * ((1.0e0 / v) / sinh((1.0e0 / v))))) / (1.0e0 / v)) * exp(((sintheta_o / v) * sintheta_i)))
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(Float32(cosTheta_i * cosTheta_O) / Float32(Float32(Float32(Float32(1.0) / Float32(Float32(0.5) * Float32(Float32(Float32(1.0) / v) / sinh(Float32(Float32(1.0) / v))))) / Float32(Float32(1.0) / v)) * exp(Float32(Float32(sinTheta_O / v) * sinTheta_i))))
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_i * cosTheta_O) / (((single(1.0) / (single(0.5) * ((single(1.0) / v) / sinh((single(1.0) / v))))) / (single(1.0) / v)) * exp(((sinTheta_O / v) * sinTheta_i)));
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 0.5

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

   [Start] 0.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 0.5	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-/l* [=>] 0.5	\[ \color{blue}{\frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\frac{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}} \]
   associate-/l/ [=>] 0.5	\[ \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot v}} \]
   associate-*l/ [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}} \]
   exp-neg [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\frac{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}} \]
   associate-/r/ [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}{1} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]
   /-rgt-identity [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   associate-*l* [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)} \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   associate-*l* [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(2 \cdot v\right) \cdot v\right)\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   *-commutative [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot 2\right)} \cdot v\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   *-commutative [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}} \]
   associate-*l/ [<=] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.4

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

5. Applied egg-rr 0.4

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

6. Final simplification 0.4

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\frac{\frac{1}{0.5 \cdot \frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)}}}{\frac{1}{v}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

Alternatives

Alternative 1
Error	0.4
Cost	7136

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

Alternative 2
Error	0.4
Cost	7104

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

Alternative 3
Error	0.4
Cost	7040

\[\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i} \cdot \frac{-v}{\frac{\frac{-0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}} \]

Alternative 4
Error	0.5
Cost	6944

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

Alternative 5
Error	0.5
Cost	6880

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

Alternative 6
Error	0.5
Cost	6880

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

Alternative 7
Error	10.1
Cost	3616

\[\frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} + -1} \]

Alternative 8
Error	0.5
Cost	3616

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

Alternative 9
Error	13.2
Cost	352

\[0.5 \cdot \frac{\frac{1}{v}}{\frac{\frac{1}{cosTheta_O}}{cosTheta_i}} \]

Alternative 10
Error	13.2
Cost	288

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

Alternative 11
Error	13.4
Cost	224

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

Alternative 12
Error	13.4
Cost	224

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

Alternative 13
Error	13.2
Cost	224

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

Reproduce

herbie shell --seed 2023053 
(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)))