HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 97.8%

Time: 14.9s

Alternatives: 6

Speedup: 3.1×

• Rival profile vector overflowed, profile may not be complete

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: 92.0% 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

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

FPCore

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

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 / sqrtf(eta))));
	return asinf((h / (eta + (-0.5f * powf(powf(expf(powf(t_0, 2.0f)), t_0), 2.0f)))));
}

Julia

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

Derivation

1. Initial program 89.5%

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

3. Simplified 89.5%

   \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O}{-\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}, eta \cdot eta\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_O around 0 96.9%

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

   1. add-sqr-sqrt 96.9%

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

   2. sqrt-div 96.9%

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

   3. sqrt-pow1 96.0%

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

   4. metadata-eval 96.0%

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

   5. pow1 96.0%

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

   6. sqrt-div 96.0%

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

   7. sqrt-pow1 97.7%

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

   8. metadata-eval 97.7%

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

   9. pow1 97.7%

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

7. Applied egg-rr 97.7%

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

   1. unpow2 97.7%

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

9. Simplified 97.7%

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

    1. add-exp-log 46.0%

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

11. Applied egg-rr 46.0%

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

    1. rem-exp-log 46.0%

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

    2. add-cube-cbrt 46.0%

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

    3. exp-prod 46.0%

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

    4. pow2 46.0%

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

    5. rem-exp-log 46.0%

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

    6. rem-exp-log 46.0%

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

13. Applied egg-rr 46.0%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\color{blue}{\left({\left(e^{{\left(\sqrt[3]{\log \left(\frac{sinTheta\_O}{\sqrt{eta}}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{sinTheta\_O}{\sqrt{eta}}\right)}\right)}\right)}}^{2}}\right) \]
14. 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 (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 89.5%

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

3. Simplified 89.5%

   \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O}{-\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}, eta \cdot eta\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_O around 0 96.9%

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

   1. add-sqr-sqrt 96.9%

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

   2. sqrt-div 96.9%

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

   3. sqrt-pow1 96.0%

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

   4. metadata-eval 96.0%

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

   5. pow1 96.0%

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

   6. sqrt-div 96.0%

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

   7. sqrt-pow1 97.7%

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

   8. metadata-eval 97.7%

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

   9. pow1 97.7%

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

7. Applied egg-rr 97.7%

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

   1. unpow2 97.7%

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

9. Simplified 97.7%

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

    1. add-exp-log 46.0%

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

11. Applied egg-rr 46.0%

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

    1. rem-exp-log 46.0%

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

    2. *-un-lft-identity 46.0%

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

    3. exp-prod 46.0%

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

    4. rem-exp-log 46.0%

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

13. Applied egg-rr 46.0%

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

    1. exp-1-e 46.0%

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

15. Simplified 46.0%

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

Alternative 3: 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 89.5%

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

3. Simplified 89.5%

   \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O}{-\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}, eta \cdot eta\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_O around 0 96.9%

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

   1. add-sqr-sqrt 96.9%

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

   2. sqrt-div 96.9%

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

   3. sqrt-pow1 96.0%

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

   4. metadata-eval 96.0%

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

   5. pow1 96.0%

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

   6. sqrt-div 96.0%

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

   7. sqrt-pow1 97.7%

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

   8. metadata-eval 97.7%

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

   9. pow1 97.7%

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

7. Applied egg-rr 97.7%

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

   1. unpow2 97.7%

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

9. Simplified 97.7%

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

    1. add-exp-log 46.0%

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

11. Applied egg-rr 46.0%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

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 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(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(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(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(eta)) ^ single(2.0))))));
end

Derivation

1. Initial program 89.5%

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

3. Simplified 89.5%

   \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O}{-\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}, eta \cdot eta\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_O around 0 96.9%

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

   1. add-sqr-sqrt 96.9%

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

   2. sqrt-div 96.9%

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

   3. sqrt-pow1 96.0%

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

   4. metadata-eval 96.0%

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

   5. pow1 96.0%

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

   6. sqrt-div 96.0%

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

   7. sqrt-pow1 97.7%

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

   8. metadata-eval 97.7%

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

   9. pow1 97.7%

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

7. Applied egg-rr 97.7%

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

   1. unpow2 97.7%

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

9. Simplified 97.7%

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

Alternative 5: 97.8% accurate, 2.8× speedup

Math

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

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 89.5%

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

3. Simplified 89.5%

   \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O}{-\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}, eta \cdot eta\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_O around 0 96.9%

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

   1. add-sqr-sqrt 96.9%

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

   2. sqrt-div 96.9%

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

   3. sqrt-pow1 96.0%

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

   4. metadata-eval 96.0%

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

   5. pow1 96.0%

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

   6. sqrt-div 96.0%

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

   7. sqrt-pow1 97.7%

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

   8. metadata-eval 97.7%

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

   9. pow1 97.7%

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

7. Applied egg-rr 97.7%

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

   1. unpow2 97.7%

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

9. Simplified 97.7%

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

    1. unpow2 97.7%

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

    2. clear-num 97.7%

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

    3. frac-times 97.7%

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

    4. add-sqr-sqrt 46.0%

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

    5. sqrt-prod 96.0%

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

    6. metadata-eval 96.0%

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

    7. sqrt-prod 96.0%

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

    8. metadata-eval 96.0%

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

    9. div-inv 96.0%

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

    10. /-rgt-identity 96.0%

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

    11. sqrt-prod 46.0%

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

    12. add-sqr-sqrt 97.7%

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

11. Applied egg-rr 97.7%

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

    1. associate-*r/ 97.7%

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

    2. rem-square-sqrt 97.7%

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

13. Simplified 97.7%

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

Alternative 6: 95.4% 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 89.5%

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

3. Simplified 89.5%

   \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O}{-\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}, eta \cdot eta\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_O around 0 94.9%

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

Reproduce

herbie shell --seed 2024155 
(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)))))))))