HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.4% → 97.8%

Time: 13.7s

Alternatives: 5

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: 91.4% 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: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} t_0 := \log \left(\frac{sinTheta\_O\_m}{\sqrt{eta}}\right)\\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left({\left(e^{\sqrt[3]{{t\_0}^{2}}}\right)}^{\left(\sqrt[3]{t\_0}\right)}\right)}^{2}}\right) \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (let* ((t_0 (log (/ sinTheta_O_m (sqrt eta)))))
   (asin
    (/
     h
     (+ eta (* -0.5 (pow (pow (exp (cbrt (pow t_0 2.0))) (cbrt t_0)) 2.0)))))))

C

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

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	t_0 = log(Float32(sinTheta_O_m / sqrt(eta)))
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * ((exp(cbrt((t_0 ^ Float32(2.0)))) ^ cbrt(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. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.9%

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

   1. pow2 97.9%

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

   2. add-sqr-sqrt 97.9%

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

   3. sqrt-div 97.9%

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

   4. sqrt-prod 48.3%

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

   5. add-sqr-sqrt 95.9%

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

   6. sqrt-div 95.9%

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

   7. sqrt-prod 48.6%

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

   8. add-sqr-sqrt 98.3%

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

5. Applied egg-rr 98.3%

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

   1. unpow2 98.3%

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

7. Simplified 98.3%

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

   1. add-exp-log 48.6%

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

9. Applied egg-rr 48.6%

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

    1. rem-exp-log 48.6%

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

    2. add-cube-cbrt 48.6%

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

    3. exp-prod 48.6%

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

    4. cbrt-unprod 48.6%

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

    5. pow2 48.6%

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

    6. rem-exp-log 48.6%

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

    7. rem-exp-log 48.6%

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

11. Applied egg-rr 48.6%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\color{blue}{\left({\left(e^{\sqrt[3]{{\log \left(\frac{sinTheta\_O}{\sqrt{eta}}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \left(\frac{sinTheta\_O}{\sqrt{eta}}\right)}\right)}\right)}}^{2}}\right) \]
12. Add Preprocessing

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(sinTheta\_O\_m \cdot \left(sinTheta\_O\_m \cdot e^{-\log eta}\right)\right)}\right) \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin
  (/
   h
   (+ eta (* -0.5 (* sinTheta_O_m (* sinTheta_O_m (exp (- (log eta))))))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sinTheta_O_m * (sinTheta_O_m * expf(-logf(eta))))))));
}

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / (eta + ((-0.5e0) * (sintheta_o_m * (sintheta_o_m * exp(-log(eta))))))))
end function

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(sinTheta_O_m * Float32(sinTheta_O_m * exp(Float32(-log(eta)))))))))
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (sinTheta_O_m * (sinTheta_O_m * exp(-log(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. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.9%

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

   1. pow2 97.9%

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

   2. add-sqr-sqrt 97.9%

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

   3. sqrt-div 97.9%

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

   4. sqrt-prod 48.3%

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

   5. add-sqr-sqrt 95.9%

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

   6. sqrt-div 95.9%

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

   7. sqrt-prod 48.6%

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

   8. add-sqr-sqrt 98.3%

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

5. Applied egg-rr 98.3%

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

   1. unpow2 98.3%

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

7. Simplified 98.3%

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

   1. unpow2 98.3%

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

   2. frac-times 97.9%

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

   3. add-sqr-sqrt 97.9%

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

   4. div-inv 97.9%

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

   5. associate-*l* 98.3%

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

9. Applied egg-rr 98.3%

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

    1. add-exp-log 98.3%

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

    2. log-rec 98.3%

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

11. Applied egg-rr 98.3%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot \color{blue}{e^{-\log eta}}\right)\right)}\right) \]
12. Add Preprocessing

Alternative 3: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(sinTheta\_O\_m \cdot \left(sinTheta\_O\_m \cdot \frac{1}{eta}\right)\right)}\right) \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin (/ h (+ eta (* -0.5 (* sinTheta_O_m (* sinTheta_O_m (/ 1.0 eta))))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sinTheta_O_m * (sinTheta_O_m * (1.0f / eta)))))));
}

Fortran

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

Julia

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

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (sinTheta_O_m * (sinTheta_O_m * (single(1.0) / 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. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.9%

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

   1. pow2 97.9%

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

   2. add-sqr-sqrt 97.9%

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

   3. sqrt-div 97.9%

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

   4. sqrt-prod 48.3%

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

   5. add-sqr-sqrt 95.9%

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

   6. sqrt-div 95.9%

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

   7. sqrt-prod 48.6%

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

   8. add-sqr-sqrt 98.3%

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

5. Applied egg-rr 98.3%

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

   1. unpow2 98.3%

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

7. Simplified 98.3%

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

   1. unpow2 98.3%

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

   2. frac-times 97.9%

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

   3. add-sqr-sqrt 97.9%

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

   4. div-inv 97.9%

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

   5. associate-*l* 98.3%

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

9. Applied egg-rr 98.3%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \color{blue}{\left(sinTheta\_O \cdot \left(sinTheta\_O \cdot \frac{1}{eta}\right)\right)}}\right) \]
10. Add Preprocessing

Alternative 4: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{sinTheta\_O\_m}{\frac{eta}{sinTheta\_O\_m}}}\right) \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin (/ h (+ eta (* -0.5 (/ sinTheta_O_m (/ eta sinTheta_O_m)))))))

C

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

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / (eta + ((-0.5e0) * (sintheta_o_m / (eta / sintheta_o_m))))))
end function

Julia

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

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (sinTheta_O_m / (eta / sinTheta_O_m))))));
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. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.9%

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

   1. pow2 97.9%

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

   2. add-sqr-sqrt 97.9%

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

   3. sqrt-div 97.9%

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

   4. sqrt-prod 48.3%

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

   5. add-sqr-sqrt 95.9%

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

   6. sqrt-div 95.9%

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

   7. sqrt-prod 48.6%

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

   8. add-sqr-sqrt 98.3%

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

5. Applied egg-rr 98.3%

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

   1. unpow2 98.3%

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

7. Simplified 98.3%

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

   1. unpow2 98.3%

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

   2. clear-num 98.3%

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

   3. frac-times 98.3%

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

   4. metadata-eval 98.3%

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

   5. div-inv 98.3%

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

   6. /-rgt-identity 98.3%

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

9. Applied egg-rr 98.3%

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

    1. associate-*r/ 98.3%

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

    2. rem-square-sqrt 98.3%

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

11. Simplified 98.3%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \color{blue}{\frac{sinTheta\_O}{\frac{eta}{sinTheta\_O}}}}\right) \]
12. Add Preprocessing

Alternative 5: 95.2% accurate, 3.1× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta) :precision binary32 (asin (/ h eta)))

C

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

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / eta))
end function

Julia

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

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, 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. Add Preprocessing

3. Taylor expanded in eta around inf 95.2%

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

Reproduce

herbie shell --seed 2024137 
(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)))))))))