HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 34.7s

Precision: binary32

Cost: 7168

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

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

Math

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

↓

\[\left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{2}{v}\right)\right) \cdot \frac{0.5}{\frac{2 \cdot \sinh \left(\frac{1}{v}\right)}{\frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v}}} \]

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 (/ 2.0 v)))
  (/
   0.5
   (/
    (* 2.0 (sinh (/ 1.0 v)))
    (/ (exp (* (/ sinTheta_i v) (- sinTheta_O))) 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_i * (cosTheta_O * (2.0f / v))) * (0.5f / ((2.0f * sinhf((1.0f / v))) / (expf(((sinTheta_i / v) * -sinTheta_O)) / 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_i * (costheta_o * (2.0e0 / v))) * (0.5e0 / ((2.0e0 * sinh((1.0e0 / v))) / (exp(((sintheta_i / v) * -sintheta_o)) / 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(Float32(cosTheta_i * Float32(cosTheta_O * Float32(Float32(2.0) / v))) * Float32(Float32(0.5) / Float32(Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v))) / Float32(exp(Float32(Float32(sinTheta_i / v) * Float32(-sinTheta_O))) / 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_i * (cosTheta_O * (single(2.0) / v))) * (single(0.5) / ((single(2.0) * sinh((single(1.0) / v))) / (exp(((sinTheta_i / v) * -sinTheta_O)) / 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

   \[\leadsto \color{blue}{\frac{\frac{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v}}{2}}{\sinh \left(\frac{1}{v}\right)}} \]
   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.json-simplify-28 [=>] 0.6	\[ \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{v}} \]
   rational.json-simplify-15 [=>] 0.5	\[ \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]
   rational.json-simplify-50 [=>] 0.5	\[ \frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \]
   rational.json-simplify-67 [=>] 0.4	\[ \frac{\color{blue}{\left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} + e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right) \cdot \frac{0.5}{v}}}{2 \cdot \sinh \left(\frac{1}{v}\right)} \]
   rational.json-simplify-43 [<=] 0.5	\[ \frac{\color{blue}{\frac{0.5 \cdot \left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} + e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right)}{v}}}{2 \cdot \sinh \left(\frac{1}{v}\right)} \]
   rational.json-simplify-28 [=>] 0.5	\[ \color{blue}{\frac{\frac{\frac{0.5 \cdot \left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} + e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right)}{v}}{2}}{\sinh \left(\frac{1}{v}\right)}} \]

3. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{2}{v}\right)\right) \cdot \frac{0.5}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot \frac{v}{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}}} \]

4. Simplified 0.4

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{2}{v}\right)\right) \cdot \frac{0.5}{\frac{2 \cdot \sinh \left(\frac{1}{v}\right)}{\frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v}}}} \]
   Proof

   [Start] 0.4	\[ \left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{2}{v}\right)\right) \cdot \frac{0.5}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot \frac{v}{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}} \]
   rational.json-simplify-41 [=>] 0.4	\[ \left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{2}{v}\right)\right) \cdot \frac{0.5}{\color{blue}{\frac{2 \cdot \sinh \left(\frac{1}{v}\right)}{\frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v}}}} \]

5. Final simplification 0.4

   \[\leadsto \left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{2}{v}\right)\right) \cdot \frac{0.5}{\frac{2 \cdot \sinh \left(\frac{1}{v}\right)}{\frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v}}} \]

Alternatives

Alternative 1
Error	0.4
Cost	7040

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

Alternative 2
Error	0.4
Cost	7040

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

Alternative 3
Error	0.4
Cost	7040

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

Alternative 4
Error	0.4
Cost	7040

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

Alternative 5
Error	0.5
Cost	6784

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

Alternative 6
Error	2.3
Cost	3680

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

Alternative 7
Error	2.3
Cost	3680

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

Alternative 8
Error	10.1
Cost	3616

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

Alternative 9
Error	13.1
Cost	288

\[\frac{\frac{0.25}{v}}{\frac{\frac{0.5}{cosTheta_O}}{cosTheta_i}} \]

Alternative 10
Error	13.1
Cost	288

\[\frac{\frac{1}{v}}{\frac{2}{cosTheta_i \cdot cosTheta_O}} \]

Alternative 11
Error	13.3
Cost	224

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

Alternative 12
Error	13.3
Cost	224

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

Alternative 13
Error	13.3
Cost	224

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

Alternative 14
Error	13.2
Cost	224

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

Alternative 15
Error	13.2
Cost	224

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

Alternative 16
Error	14.2
Cost	160

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

Alternative 17
Error	14.2
Cost	160

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

Reproduce

herbie shell --seed 2023073 
(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)))