HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.1% → 98.1%

Time: 12.0s

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: 92.1% 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: 98.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 93.2%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
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. *-un-lft-identity 97.9%

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

   3. times-frac 98.5%

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

5. Applied egg-rr 98.5%

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

   1. /-rgt-identity 98.5%

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

   2. add-exp-log 45.5%

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

7. Applied egg-rr 45.5%

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

   1. add-cube-cbrt 45.5%

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

   2. exp-prod 45.5%

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

   3. pow2 45.5%

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

9. Applied egg-rr 45.5%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin
  (/ h (+ eta (* -0.5 (* (/ sinTheta_O_m eta) (pow E (log 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) * powf(((float) M_E), logf(sinTheta_O_m)))))));
}

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(Float32(sinTheta_O_m / eta) * (Float32(exp(1)) ^ log(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) * (single(2.71828182845904523536) ^ log(sinTheta_O_m)))))));
end

Derivation

1. Initial program 93.2%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
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. *-un-lft-identity 97.9%

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

   3. times-frac 98.5%

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

5. Applied egg-rr 98.5%

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

   1. /-rgt-identity 98.5%

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

   2. add-exp-log 45.5%

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

7. Applied egg-rr 45.5%

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

   1. *-un-lft-identity 45.5%

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

   2. exp-prod 45.5%

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

9. Applied egg-rr 45.5%

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

10. Final simplification 45.5%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin
  (/ h (+ eta (* -0.5 (* (/ sinTheta_O_m eta) (exp (log 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) * expf(logf(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) * exp(log(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(Float32(sinTheta_O_m / eta) * exp(log(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) * exp(log(sinTheta_O_m)))))));
end

Derivation

1. Initial program 93.2%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
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. *-un-lft-identity 97.9%

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

   3. times-frac 98.5%

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

5. Applied egg-rr 98.5%

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

   1. /-rgt-identity 98.5%

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

   2. add-exp-log 45.5%

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

7. Applied egg-rr 45.5%

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

8. Final simplification 45.5%

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

Alternative 4: 98.1% accurate, 2.8× speedup

Math

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

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(Float32(-0.5) * 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 93.2%

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

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

   3. times-frac 98.5%

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

5. Applied egg-rr 98.5%

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

   1. unpow2 98.5%

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

7. Simplified 98.5%

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

   1. unpow2 98.5%

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

   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. *-un-lft-identity 97.9%

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

   5. frac-times 98.5%

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

   6. /-rgt-identity 98.5%

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

   7. associate-*r* 98.5%

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

   8. clear-num 98.5%

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

   9. un-div-inv 98.5%

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

9. Applied egg-rr 98.5%

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

Alternative 5: 97.6% accurate, 2.8× speedup

Math

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

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(Float32(sinTheta_O_m * 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 93.2%

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

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

5. Applied egg-rr 97.9%

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

Alternative 6: 95.6% 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 93.2%

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

3. Taylor expanded in eta around inf 96.7%

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

Reproduce

herbie shell --seed 2024136 
(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)))))))))