HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.7% → 97.8%

Time: 12.7s

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: 91.7% 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, 0.6× speedup

Math

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

FPCore

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

C

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

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(exp(1)) ^ log(Float32(sinTheta_O_m / sqrt(eta)))) ^ Float32(2.0))))))
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * ((single(2.71828182845904523536) ^ log((sinTheta_O_m / sqrt(eta)))) ^ single(2.0))))));
end

Derivation

1. Initial program 92.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.8%

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

   1. add-sqr-sqrt 97.8%

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

   2. pow2 97.8%

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

   3. div-inv 97.8%

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

   4. sqrt-prod 97.8%

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

   5. sqrt-pow1 98.3%

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

   6. metadata-eval 98.3%

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

   7. pow1 98.3%

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

5. Applied egg-rr 98.3%

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

   1. add-cbrt-cube 96.0%

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

   2. pow1/3 96.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(sinTheta\_O \cdot \color{blue}{{\left(\left(\sqrt{\frac{1}{eta}} \cdot \sqrt{\frac{1}{eta}}\right) \cdot \sqrt{\frac{1}{eta}}\right)}^{0.3333333333333333}}\right)}^{2}}\right) \]

   3. add-sqr-sqrt 96.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(sinTheta\_O \cdot {\left(\color{blue}{\frac{1}{eta}} \cdot \sqrt{\frac{1}{eta}}\right)}^{0.3333333333333333}\right)}^{2}}\right) \]

   4. pow1 96.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(sinTheta\_O \cdot {\left(\color{blue}{{\left(\frac{1}{eta}\right)}^{1}} \cdot \sqrt{\frac{1}{eta}}\right)}^{0.3333333333333333}\right)}^{2}}\right) \]

   5. pow1/2 96.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(sinTheta\_O \cdot {\left({\left(\frac{1}{eta}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{eta}\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2}}\right) \]

   6. pow-prod-up 96.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(sinTheta\_O \cdot {\color{blue}{\left({\left(\frac{1}{eta}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2}}\right) \]

   7. metadata-eval 96.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(sinTheta\_O \cdot {\left({\left(\frac{1}{eta}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2}}\right) \]

7. Applied egg-rr 96.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(sinTheta\_O \cdot \color{blue}{{\left({\left(\frac{1}{eta}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)}^{2}}\right) \]
8. Step-by-step derivation

   1. pow-pow 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(sinTheta\_O \cdot \color{blue}{{\left(\frac{1}{eta}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)}^{2}}\right) \]

   2. metadata-eval 98.3%

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

   3. pow1/2 98.3%

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

   4. sqrt-div 98.3%

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

   5. metadata-eval 98.3%

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

   6. div-inv 98.3%

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

   7. rem-exp-log 49.3%

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

   8. add-exp-log 49.3%

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

   9. *-un-lft-identity 49.3%

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

   10. exp-prod 49.3%

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

   11. rem-exp-log 49.3%

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

9. Applied egg-rr 49.3%

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

    1. exp-1-e 49.3%

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

11. Simplified 49.3%

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

Alternative 2: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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) * (exp(log((sintheta_o_m / sqrt(eta)))) ** 2.0e0)))))
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) * (exp(log(Float32(sinTheta_O_m / sqrt(eta)))) ^ Float32(2.0))))))
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (exp(log((sinTheta_O_m / sqrt(eta)))) ^ single(2.0))))));
end

Derivation

1. Initial program 92.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.8%

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

   1. add-sqr-sqrt 97.8%

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

   2. pow2 97.8%

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

   3. div-inv 97.8%

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

   4. sqrt-prod 97.8%

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

   5. sqrt-pow1 98.3%

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

   6. metadata-eval 98.3%

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

   7. pow1 98.3%

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

5. Applied egg-rr 98.3%

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

   1. add-exp-log 49.3%

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

   2. add-sqr-sqrt 49.3%

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

   3. sqrt-prod 97.8%

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

   4. sqrt-prod 97.8%

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

   5. div-inv 97.8%

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

   6. sqrt-div 97.8%

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

   7. sqrt-prod 49.3%

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

   8. add-sqr-sqrt 49.3%

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

7. Applied egg-rr 49.3%

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

Alternative 3: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 * exp((log(eta) * -0.5e0))) ** 2.0e0)))))
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 * exp(Float32(log(eta) * Float32(-Float32(0.5))))) ^ Float32(2.0))))))
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 * exp((log(eta) * -single(0.5)))) ^ single(2.0))))));
end

Derivation

1. Initial program 92.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.8%

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

   1. add-sqr-sqrt 97.8%

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

   2. pow2 97.8%

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

   3. div-inv 97.8%

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

   4. sqrt-prod 97.8%

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

   5. sqrt-pow1 98.3%

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

   6. metadata-eval 98.3%

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

   7. pow1 98.3%

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

5. Applied egg-rr 98.3%

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

   1. pow1/2 98.3%

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

   2. pow-to-exp 98.3%

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

   3. log-rec 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin (/ h (+ eta (* -0.5 (pow (* sinTheta_O_m (sqrt (/ 1.0 eta))) 2.0))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	return asinf((h / (eta + (-0.5f * powf((sinTheta_O_m * sqrtf((1.0f / eta))), 2.0f)))));
}

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 * sqrt((1.0e0 / eta))) ** 2.0e0)))))
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 * sqrt(Float32(Float32(1.0) / eta))) ^ Float32(2.0))))))
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 * sqrt((single(1.0) / eta))) ^ single(2.0))))));
end

Derivation

1. Initial program 92.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.8%

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

   1. add-sqr-sqrt 97.8%

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

   2. pow2 97.8%

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

   3. div-inv 97.8%

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

   4. sqrt-prod 97.8%

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

   5. sqrt-pow1 98.3%

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

   6. metadata-eval 98.3%

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

   7. pow1 98.3%

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

5. Applied egg-rr 98.3%

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

Alternative 5: 97.8% accurate, 2.8× speedup

Math

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

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 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 / 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 / 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 / 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 / eta))))));
end

Derivation

1. Initial program 92.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 97.8%

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

   1. add-sqr-sqrt 97.8%

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

   2. pow2 97.8%

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

   3. div-inv 97.8%

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

   4. sqrt-prod 97.8%

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

   5. sqrt-pow1 98.3%

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

   6. metadata-eval 98.3%

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

   7. pow1 98.3%

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

5. Applied egg-rr 98.3%

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

   1. unpow2 98.3%

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

   2. swap-sqr 97.8%

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

   3. add-sqr-sqrt 97.8%

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

   4. div-inv 97.8%

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

   5. associate-/l* 98.3%

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

   6. *-commutative 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 6: 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 92.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
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 2024172 
(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)))))))))