HairBSDF, gamma for a refracted ray

?

Percentage Accurate: 92.3% → 98.2%
Time: 18.1s
Precision: binary32
Cost: 9952

?

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\sin^{-1} \left(\left({\left(sinTheta_O + eta\right)}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right) \cdot h\right) \]
(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))
↓
(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (* (* (pow (+ sinTheta_O eta) -0.5) (pow (- eta sinTheta_O) -0.5)) h)))
float code(float sinTheta_O, float h, float eta) {
	return asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
}
↓
float code(float sinTheta_O, float h, float eta) {
	return asinf(((powf((sinTheta_O + eta), -0.5f) * powf((eta - sinTheta_O), -0.5f)) * h));
}

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
end function
↓
real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin(((((sintheta_o + eta) ** (-0.5e0)) * ((eta - sintheta_o) ** (-0.5e0))) * h))
end function

function code(sinTheta_O, h, eta)
	return asin(Float32(h / sqrt(Float32(Float32(eta * eta) - Float32(Float32(sinTheta_O * sinTheta_O) / sqrt(Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O))))))))
end
↓
function code(sinTheta_O, h, eta)
	return asin(Float32(Float32((Float32(sinTheta_O + eta) ^ Float32(-0.5)) * (Float32(eta - sinTheta_O) ^ Float32(-0.5))) * h))
end

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / sqrt(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrt((single(1.0) - (sinTheta_O * sinTheta_O))))))));
end
↓
function tmp = code(sinTheta_O, h, eta)
	tmp = asin(((((sinTheta_O + eta) ^ single(-0.5)) * ((eta - sinTheta_O) ^ single(-0.5))) * h));
end

\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right)
↓
\sin^{-1} \left(\left({\left(sinTheta_O + eta\right)}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right) \cdot h\right)

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 4 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 94.5%

    \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]

  2. Applied egg-rr93.9%

    \[\leadsto \sin^{-1} \color{blue}{\left({\left(eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{-0.5} \cdot h\right)} \]
    Step-by-step derivation

    [Start]94.5

    \[ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]

    clear-num [=>]94.0

    \[ \sin^{-1} \color{blue}{\left(\frac{1}{\frac{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}{h}}\right)} \]

    associate-/r/ [=>]93.9

    \[ \sin^{-1} \color{blue}{\left(\frac{1}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}} \cdot h\right)} \]

    pow1/2 [=>]93.9

    \[ \sin^{-1} \left(\frac{1}{\color{blue}{{\left(eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{0.5}}} \cdot h\right) \]

    pow-flip [=>]93.9

    \[ \sin^{-1} \left(\color{blue}{{\left(eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{\left(-0.5\right)}} \cdot h\right) \]

    metadata-eval [=>]93.9

    \[ \sin^{-1} \left({\left(eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{\color{blue}{-0.5}} \cdot h\right) \]

  3. Taylor expanded in sinTheta_O around 0 93.6%

    \[\leadsto \sin^{-1} \left({\color{blue}{\left({eta}^{2} + -1 \cdot {sinTheta_O}^{2}\right)}}^{-0.5} \cdot h\right) \]

  4. Simplified93.6%

    \[\leadsto \sin^{-1} \left({\color{blue}{\left(eta \cdot eta - sinTheta_O \cdot sinTheta_O\right)}}^{-0.5} \cdot h\right) \]
    Step-by-step derivation

    [Start]93.6

    \[ \sin^{-1} \left({\left({eta}^{2} + -1 \cdot {sinTheta_O}^{2}\right)}^{-0.5} \cdot h\right) \]

    unpow2 [=>]93.6

    \[ \sin^{-1} \left({\left(\color{blue}{eta \cdot eta} + -1 \cdot {sinTheta_O}^{2}\right)}^{-0.5} \cdot h\right) \]

    mul-1-neg [=>]93.6

    \[ \sin^{-1} \left({\left(eta \cdot eta + \color{blue}{\left(-{sinTheta_O}^{2}\right)}\right)}^{-0.5} \cdot h\right) \]

    unsub-neg [=>]93.6

    \[ \sin^{-1} \left({\color{blue}{\left(eta \cdot eta - {sinTheta_O}^{2}\right)}}^{-0.5} \cdot h\right) \]

    unpow2 [=>]93.6

    \[ \sin^{-1} \left({\left(eta \cdot eta - \color{blue}{sinTheta_O \cdot sinTheta_O}\right)}^{-0.5} \cdot h\right) \]

  5. Applied egg-rr98.2%

    \[\leadsto \sin^{-1} \left(\color{blue}{\left({\left(eta + sinTheta_O\right)}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right)} \cdot h\right) \]
    Step-by-step derivation

    [Start]93.6

    \[ \sin^{-1} \left({\left(eta \cdot eta - sinTheta_O \cdot sinTheta_O\right)}^{-0.5} \cdot h\right) \]

    difference-of-squares [=>]93.6

    \[ \sin^{-1} \left({\color{blue}{\left(\left(eta + sinTheta_O\right) \cdot \left(eta - sinTheta_O\right)\right)}}^{-0.5} \cdot h\right) \]

    unpow-prod-down [=>]98.2

    \[ \sin^{-1} \left(\color{blue}{\left({\left(eta + sinTheta_O\right)}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right)} \cdot h\right) \]

  6. Simplified98.2%

    \[\leadsto \sin^{-1} \left(\color{blue}{\left({\left(sinTheta_O + eta\right)}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right)} \cdot h\right) \]
    Step-by-step derivation

    [Start]98.2

    \[ \sin^{-1} \left(\left({\left(eta + sinTheta_O\right)}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right) \cdot h\right) \]

    +-commutative [=>]98.2

    \[ \sin^{-1} \left(\left({\color{blue}{\left(sinTheta_O + eta\right)}}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right) \cdot h\right) \]

  7. Final simplification98.2%

    \[\leadsto \sin^{-1} \left(\left({\left(sinTheta_O + eta\right)}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right) \cdot h\right) \]

Alternatives

Alternative 1
Accuracy97.4%
Cost3552
\[\sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{sinTheta_O \cdot sinTheta_O}{eta}}\right) \]
Alternative 2
Accuracy97.8%
Cost3552
\[\sin^{-1} \left(\frac{h}{eta + \frac{-0.5}{\frac{\frac{eta}{sinTheta_O}}{sinTheta_O}}}\right) \]
Alternative 3
Accuracy95.3%
Cost3296
\[\sin^{-1} \left(\frac{h}{eta}\right) \]

Reproduce?

herbie shell --seed 2023164 
(FPCore (sinTheta_O h eta)
  :name "HairBSDF, gamma for a refracted ray"
  :precision binary32
  :pre (and (and (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0)) (and (<= -1.0 h) (<= h 1.0))) (and (<= 0.0 eta) (<= eta 10.0)))
  (asin (/ h (sqrt (- (* eta eta) (/ (* sinTheta_O sinTheta_O) (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))