HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.5% → 97.9%

Time: 12.7s

Alternatives: 3

Speedup: 3.1×

Specification

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

C

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))))))));
}

Fortran

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

Julia

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

MATLAB

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

Initial Program: 91.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

C

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))))))));
}

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{eta}{sinTheta_O}}\\ \sin^{-1} \left(\frac{h}{eta + \frac{\frac{-0.5 \cdot sinTheta_O}{t_0}}{{t_0}^{2}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (cbrt (/ eta sinTheta_O))))
   (asin (/ h (+ eta (/ (/ (* -0.5 sinTheta_O) t_0) (pow t_0 2.0)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float t_0 = cbrtf((eta / sinTheta_O));
	return asinf((h / (eta + (((-0.5f * sinTheta_O) / t_0) / powf(t_0, 2.0f)))));
}

Julia

function code(sinTheta_O, h, eta)
	t_0 = cbrt(Float32(eta / sinTheta_O))
	return asin(Float32(h / Float32(eta + Float32(Float32(Float32(Float32(-0.5) * sinTheta_O) / t_0) / (t_0 ^ Float32(2.0))))))
end

Derivation

1. Initial program 90.7%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
2. Step-by-step derivation

   1. sqr-neg 90.7%

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

   2. sqr-neg 90.7%

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

   3. sqr-neg 90.7%

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

   4. sqr-neg 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in sinTheta_O around 0 97.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}}\right) \]
5. Step-by-step derivation

   1. unpow2 97.1%

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

   2. *-un-lft-identity 97.1%

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

   3. times-frac 97.7%

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

6. Applied egg-rr 97.7%

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

   1. /-rgt-identity 97.7%

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

   2. associate-*r* 97.7%

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

   3. clear-num 97.7%

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

   4. un-div-inv 97.7%

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

8. Applied egg-rr 97.7%

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

   1. *-un-lft-identity 97.7%

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

   2. add-cube-cbrt 97.7%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{1 \cdot \left(-0.5 \cdot sinTheta_O\right)}{\color{blue}{\left(\sqrt[3]{\frac{eta}{sinTheta_O}} \cdot \sqrt[3]{\frac{eta}{sinTheta_O}}\right) \cdot \sqrt[3]{\frac{eta}{sinTheta_O}}}}}\right) \]

   3. times-frac 97.7%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\frac{1}{\sqrt[3]{\frac{eta}{sinTheta_O}} \cdot \sqrt[3]{\frac{eta}{sinTheta_O}}} \cdot \frac{-0.5 \cdot sinTheta_O}{\sqrt[3]{\frac{eta}{sinTheta_O}}}}}\right) \]

   4. pow2 97.7%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}} \cdot \frac{-0.5 \cdot sinTheta_O}{\sqrt[3]{\frac{eta}{sinTheta_O}}}}\right) \]

10. Applied egg-rr 97.7%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\frac{1}{{\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}} \cdot \frac{-0.5 \cdot sinTheta_O}{\sqrt[3]{\frac{eta}{sinTheta_O}}}}}\right) \]
11. Step-by-step derivation

    1. associate-*l/ 97.7%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\frac{1 \cdot \frac{-0.5 \cdot sinTheta_O}{\sqrt[3]{\frac{eta}{sinTheta_O}}}}{{\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}}}\right) \]

    2. *-lft-identity 97.7%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{\color{blue}{\frac{-0.5 \cdot sinTheta_O}{\sqrt[3]{\frac{eta}{sinTheta_O}}}}}{{\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}}\right) \]

12. Simplified 97.7%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\frac{\frac{-0.5 \cdot sinTheta_O}{\sqrt[3]{\frac{eta}{sinTheta_O}}}}{{\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}}}\right) \]

13. Final simplification 97.7%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + \frac{\frac{-0.5 \cdot sinTheta_O}{\sqrt[3]{\frac{eta}{sinTheta_O}}}}{{\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}}\right) \]

Alternative 2: 97.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + \frac{-0.5 \cdot sinTheta_O}{\frac{eta}{sinTheta_O}}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (+ eta (/ (* -0.5 sinTheta_O) (/ eta sinTheta_O))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + ((-0.5f * sinTheta_O) / (eta / sinTheta_O)))));
}

Fortran

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 / (eta + (((-0.5e0) * sintheta_o) / (eta / sintheta_o)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 90.7%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
2. Step-by-step derivation

   1. sqr-neg 90.7%

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

   2. sqr-neg 90.7%

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

   3. sqr-neg 90.7%

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

   4. sqr-neg 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in sinTheta_O around 0 97.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}}\right) \]
5. Step-by-step derivation

   1. unpow2 97.1%

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

   2. *-un-lft-identity 97.1%

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

   3. times-frac 97.7%

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

6. Applied egg-rr 97.7%

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

   1. /-rgt-identity 97.7%

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

   2. associate-*r* 97.7%

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

   3. clear-num 97.7%

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

   4. un-div-inv 97.7%

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

8. Applied egg-rr 97.7%

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

9. Final simplification 97.7%

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

Alternative 3: 95.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta) :precision binary32 (asin (/ h eta)))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / eta));
}

Fortran

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 / eta))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / eta))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / eta));
end

Derivation

1. Initial program 90.7%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
2. Step-by-step derivation

   1. sqr-neg 90.7%

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

   2. sqr-neg 90.7%

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

   3. sqr-neg 90.7%

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

   4. sqr-neg 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in eta around inf 95.4%

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

5. Final simplification 95.4%

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

Reproduce

herbie shell --seed 2023305 
(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)))))))))