HairBSDF, Mp, upper

Average Error: 1.44% → 1.18%

Time: 19.1s

Precision: binary32

Cost: 7168

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

Math

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

↓

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

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
 (/
  (*
   (exp (/ (* sinTheta_i (- sinTheta_O)) v))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (/ (* (sinh (/ 1.0 v)) 2.0) (/ 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 (expf(((sinTheta_i * -sinTheta_O) / v)) * ((1.0f / v) * (cosTheta_i * cosTheta_O))) / ((sinhf((1.0f / v)) * 2.0f) / (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 = (exp(((sintheta_i * -sintheta_o) / v)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / ((sinh((1.0e0 / v)) * 2.0e0) / (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(Float32(exp(Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v)) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) / 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 = (exp(((sinTheta_i * -sinTheta_O) / v)) * ((single(1.0) / v) * (cosTheta_i * cosTheta_O))) / ((sinh((single(1.0) / v)) * single(2.0)) / (single(1.0) / v));
end

Derivation

1. Initial program 1.44

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

2. Applied egg-rr 1.27

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

3. Applied egg-rr 1.18

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

4. Final simplification 1.18

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

Alternatives

Alternative 1
Error	1.16%
Cost	7168

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

Alternative 2
Error	1.27%
Cost	7104

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

Alternative 3
Error	1.27%
Cost	7072

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

Alternative 4
Error	1.41%
Cost	7040

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

Alternative 5
Error	1.44%
Cost	7008

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

Alternative 6
Error	1.49%
Cost	6944

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

Alternative 7
Error	1.5%
Cost	6944

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

Alternative 8
Error	1.65%
Cost	6880

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

Alternative 9
Error	1.65%
Cost	6880

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

Alternative 10
Error	30.6%
Cost	3812

\[\begin{array}{l} \mathbf{if}\;v \leq 0.36000001430511475:\\ \;\;\;\;\frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} + \left(2 + \frac{-2}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_i \cdot \frac{cosTheta_O}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}\\ \end{array} \]

Alternative 11
Error	32.65%
Cost	3744

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

Alternative 12
Error	35.48%
Cost	3488

\[cosTheta_i \cdot \frac{cosTheta_O}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \]

Alternative 13
Error	35.49%
Cost	1056

\[\begin{array}{l} t_0 := \frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\\ \frac{cosTheta_i \cdot cosTheta_O}{t_0} - \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot t_0} \end{array} \]

Alternative 14
Error	35.49%
Cost	416

\[\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} \]

Alternative 15
Error	40.64%
Cost	288

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

Alternative 16
Error	41.23%
Cost	224

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

Alternative 17
Error	40.8%
Cost	224

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

Reproduce

herbie shell --seed 2023104 
(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)))