HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.9% → 98.0%

Time: 16.3s

Alternatives: 6

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: 91.9% 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: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{2}\right)\right) \cdot t_0} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (cbrt (log (* sinTheta_O (/ sinTheta_O eta))))))
   (asin
    (/
     h
     (+
      eta
      (*
       -0.5
       (*
        (exp (* (expm1 (log1p (pow t_0 2.0))) t_0))
        (sqrt (/ 1.0 (- 1.0 (* sinTheta_O sinTheta_O)))))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float t_0 = cbrtf(logf((sinTheta_O * (sinTheta_O / eta))));
	return asinf((h / (eta + (-0.5f * (expf((expm1f(log1pf(powf(t_0, 2.0f))) * t_0)) * sqrtf((1.0f / (1.0f - (sinTheta_O * sinTheta_O)))))))));
}

Julia

function code(sinTheta_O, h, eta)
	t_0 = cbrt(log(Float32(sinTheta_O * Float32(sinTheta_O / eta))))
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(exp(Float32(expm1(log1p((t_0 ^ Float32(2.0)))) * t_0)) * sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O)))))))))
end

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in eta around inf 97.1%

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

   1. unpow2 97.1%

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

   2. unpow2 97.1%

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

4. Simplified 97.1%

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

   1. add-exp-log 97.1%

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

   2. associate-/l* 97.6%

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

6. Applied egg-rr 97.6%

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

   1. add-cube-cbrt 97.6%

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

   2. associate-/r/ 97.6%

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

   3. associate-/r/ 97.6%

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

   4. associate-/r/ 97.6%

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

8. Applied egg-rr 97.6%

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

   1. expm1-log1p-u 97.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}\right)\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

   2. pow2 97.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}\right)}^{2}}\right)\right) \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

   3. *-commutative 97.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\log \color{blue}{\left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}}\right)}^{2}\right)\right) \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

10. Applied egg-rr 97.6%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\right)}^{2}\right)\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

11. Final simplification 97.6%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\right)}^{2}\right)\right) \cdot \sqrt[3]{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

Alternative 2: 98.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (sqrt (/ 1.0 (- 1.0 (* sinTheta_O sinTheta_O))))
      (exp
       (*
        (cbrt (log (* sinTheta_O (/ sinTheta_O eta))))
        (pow (cbrt (log (/ sinTheta_O (/ eta sinTheta_O)))) 2.0)))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sqrtf((1.0f / (1.0f - (sinTheta_O * sinTheta_O)))) * expf((cbrtf(logf((sinTheta_O * (sinTheta_O / eta)))) * powf(cbrtf(logf((sinTheta_O / (eta / sinTheta_O)))), 2.0f))))))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O)))) * exp(Float32(cbrt(log(Float32(sinTheta_O * Float32(sinTheta_O / eta)))) * (cbrt(log(Float32(sinTheta_O / Float32(eta / sinTheta_O)))) ^ Float32(2.0)))))))))
end

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in eta around inf 97.1%

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

   1. unpow2 97.1%

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

   2. unpow2 97.1%

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

4. Simplified 97.1%

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

   1. add-exp-log 97.1%

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

   2. associate-/l* 97.6%

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

6. Applied egg-rr 97.6%

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

   1. add-cube-cbrt 97.6%

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

   2. associate-/r/ 97.6%

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

   3. associate-/r/ 97.6%

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

   4. associate-/r/ 97.6%

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

8. Applied egg-rr 97.6%

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

   1. expm1-log1p-u 97.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}\right)\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

   2. expm1-udef 97.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}\right)} - 1\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

   3. pow2 97.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}\right)}^{2}}\right)} - 1\right) \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

   4. *-commutative 97.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\left(e^{\mathsf{log1p}\left({\left(\sqrt[3]{\log \color{blue}{\left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}}\right)}^{2}\right)} - 1\right) \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

10. Applied egg-rr 97.6%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt[3]{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\right)}^{2}\right)} - 1\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]
11. Step-by-step derivation

    1. expm1-def 97.6%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\right)}^{2}\right)\right)} \cdot \sqrt[3]{\log \left(\frac{sinTheta_O}{eta} \cdot sinTheta_O\right)}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \]

    2. expm1-log1p 97.6%

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

    3. associate-*r/ 97.1%

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

    4. associate-/l* 97.6%

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

12. Simplified 97.6%

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

13. Final simplification 97.6%

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

Alternative 3: 98.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (sqrt (/ 1.0 (- 1.0 (* sinTheta_O sinTheta_O))))
      (exp (log (/ sinTheta_O (/ eta sinTheta_O))))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sqrtf((1.0f / (1.0f - (sinTheta_O * sinTheta_O)))) * expf(logf((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) * (sqrt((1.0e0 / (1.0e0 - (sintheta_o * sintheta_o)))) * exp(log((sintheta_o / (eta / sintheta_o)))))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in eta around inf 97.1%

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

   1. unpow2 97.1%

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

   2. unpow2 97.1%

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

4. Simplified 97.1%

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

   1. add-exp-log 97.1%

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

   2. associate-/l* 97.6%

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

6. Applied egg-rr 97.6%

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

7. Final simplification 97.6%

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

Alternative 4: 98.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (/
      sinTheta_O
      (* eta (/ (sqrt (- 1.0 (* sinTheta_O sinTheta_O))) sinTheta_O))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sinTheta_O / (eta * (sqrtf((1.0f - (sinTheta_O * 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 / (eta + ((-0.5e0) * (sintheta_o / (eta * (sqrt((1.0e0 - (sintheta_o * sintheta_o))) / sintheta_o)))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in eta around inf 97.1%

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

   1. unpow2 97.1%

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

   2. unpow2 97.1%

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

4. Simplified 97.1%

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

   1. *-un-lft-identity 97.1%

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

   2. associate-*l/ 97.1%

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

   3. sqrt-div 97.1%

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

   4. metadata-eval 97.1%

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

   5. div-inv 97.1%

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

   6. associate-/l* 97.1%

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

6. Applied egg-rr 97.1%

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

   1. *-lft-identity 97.1%

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

   2. associate-/l/ 97.6%

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

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

Alternative 5: 97.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sinTheta_O * (sinTheta_O / 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 + ((-0.5e0) * (sintheta_o * (sintheta_o / eta))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in sinTheta_O around 0 97.1%

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

   1. associate-*r/ 97.1%

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

   2. unpow2 97.1%

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

4. Simplified 97.1%

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

5. Taylor expanded in sinTheta_O around 0 97.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{-0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}}\right) \]
6. 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. associate-*l/ 97.5%

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

   3. *-commutative 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 6: 95.5% 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 90.6%

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

2. Taylor expanded in eta around inf 95.2%

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

3. Final simplification 95.2%

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

Reproduce

herbie shell --seed 2023268 
(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)))))))))