HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.1% → 98.4%

Time: 15.6s

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: 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: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (sqrtf(fmaf(sinTheta_O, powf(hypotf(1.0f, sinTheta_O), -0.5f), eta)) * sqrtf((eta - (sinTheta_O / sqrtf(hypotf(1.0f, sinTheta_O))))))));
}

Julia

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

Derivation

1. Initial program 90.9%

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

   1. add-sqr-sqrt 90.9%

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

   2. difference-of-squares 90.9%

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

3. Applied egg-rr 90.7%

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

   1. sqrt-prod 98.5%

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

   2. +-commutative 98.5%

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

   3. div-inv 98.5%

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

   4. fma-def 98.5%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\mathsf{fma}\left(sinTheta_O, \frac{1}{\sqrt{\mathsf{hypot}\left(1, sinTheta_O\right)}}, eta\right)}} \cdot \sqrt{eta - \frac{sinTheta_O}{\sqrt{\mathsf{hypot}\left(1, sinTheta_O\right)}}}}\right) \]

   5. pow1/2 98.5%

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

   6. pow-flip 98.5%

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

   7. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

6. Final simplification 98.5%

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

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h \cdot \sqrt{\frac{1}{eta - sinTheta_O \cdot {\left(\frac{1}{1 + sinTheta_O \cdot sinTheta_O}\right)}^{0.25}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   (*
    h
    (sqrt
     (/
      1.0
      (-
       eta
       (* sinTheta_O (pow (/ 1.0 (+ 1.0 (* sinTheta_O sinTheta_O))) 0.25))))))
   (sqrt (fma sinTheta_O (pow (hypot 1.0 sinTheta_O) -0.5) eta)))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf(((h * sqrtf((1.0f / (eta - (sinTheta_O * powf((1.0f / (1.0f + (sinTheta_O * sinTheta_O))), 0.25f)))))) / sqrtf(fmaf(sinTheta_O, powf(hypotf(1.0f, sinTheta_O), -0.5f), eta))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(Float32(h * sqrt(Float32(Float32(1.0) / Float32(eta - Float32(sinTheta_O * (Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sinTheta_O * sinTheta_O))) ^ Float32(0.25))))))) / sqrt(fma(sinTheta_O, (hypot(Float32(1.0), sinTheta_O) ^ Float32(-0.5)), eta))))
end

Derivation

1. Initial program 90.9%

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

   1. add-sqr-sqrt 90.9%

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

   2. difference-of-squares 90.9%

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

3. Applied egg-rr 90.7%

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

   1. *-un-lft-identity 90.7%

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

   2. sqrt-prod 98.5%

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

   3. times-frac 98.1%

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

   4. +-commutative 98.1%

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

   5. div-inv 98.1%

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

   6. fma-def 98.1%

      \[\leadsto \sin^{-1} \left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(sinTheta_O, \frac{1}{\sqrt{\mathsf{hypot}\left(1, sinTheta_O\right)}}, eta\right)}}} \cdot \frac{h}{\sqrt{eta - \frac{sinTheta_O}{\sqrt{\mathsf{hypot}\left(1, sinTheta_O\right)}}}}\right) \]

   7. pow1/2 98.1%

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

   8. pow-flip 98.1%

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

   9. metadata-eval 98.1%

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

5. Applied egg-rr 98.1%

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

   1. associate-*l/ 98.2%

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

   2. *-lft-identity 98.2%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\frac{h}{\sqrt{eta - \frac{sinTheta_O}{\sqrt{\mathsf{hypot}\left(1, sinTheta_O\right)}}}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \]

7. Simplified 98.2%

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\frac{h}{\sqrt{eta - \frac{sinTheta_O}{\sqrt{\mathsf{hypot}\left(1, sinTheta_O\right)}}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right)} \]

8. Taylor expanded in h around 0 98.5%

   \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\sqrt{\frac{1}{eta - {\left(\frac{1}{{sinTheta_O}^{2} + 1}\right)}^{0.25} \cdot sinTheta_O}} \cdot h}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \]
9. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{h \cdot \sqrt{\frac{1}{eta - {\left(\frac{1}{{sinTheta_O}^{2} + 1}\right)}^{0.25} \cdot sinTheta_O}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \]

   2. *-commutative 98.5%

      \[\leadsto \sin^{-1} \left(\frac{h \cdot \sqrt{\frac{1}{eta - \color{blue}{sinTheta_O \cdot {\left(\frac{1}{{sinTheta_O}^{2} + 1}\right)}^{0.25}}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \]

   3. +-commutative 98.5%

      \[\leadsto \sin^{-1} \left(\frac{h \cdot \sqrt{\frac{1}{eta - sinTheta_O \cdot {\left(\frac{1}{\color{blue}{1 + {sinTheta_O}^{2}}}\right)}^{0.25}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \]

   4. unpow2 98.5%

      \[\leadsto \sin^{-1} \left(\frac{h \cdot \sqrt{\frac{1}{eta - sinTheta_O \cdot {\left(\frac{1}{1 + \color{blue}{sinTheta_O \cdot sinTheta_O}}\right)}^{0.25}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \]

10. Simplified 98.5%

    \[\leadsto \sin^{-1} \left(\frac{\color{blue}{h \cdot \sqrt{\frac{1}{eta - sinTheta_O \cdot {\left(\frac{1}{1 + sinTheta_O \cdot sinTheta_O}\right)}^{0.25}}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \]

11. Final simplification 98.5%

    \[\leadsto \sin^{-1} \left(\frac{h \cdot \sqrt{\frac{1}{eta - sinTheta_O \cdot {\left(\frac{1}{1 + sinTheta_O \cdot sinTheta_O}\right)}^{0.25}}}}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(\mathsf{hypot}\left(1, sinTheta_O\right)\right)}^{-0.5}, eta\right)}}\right) \]

Alternative 3: 97.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.9%

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

2. Taylor expanded in sinTheta_O around 0 97.2%

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

   1. unpow2 97.2%

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

   2. associate-*r/ 97.2%

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

4. Simplified 97.2%

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

5. Final simplification 97.2%

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

Alternative 4: 97.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.9%

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

2. Taylor expanded in sinTheta_O around 0 97.2%

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

   1. unpow2 97.2%

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

   2. associate-*r/ 97.2%

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

4. Simplified 97.2%

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

5. Taylor expanded in h around 0 97.2%

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

   1. metadata-eval 97.2%

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

   2. cancel-sign-sub-inv 97.2%

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

   3. *-commutative 97.2%

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

   4. unpow2 97.2%

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

   5. associate-/l* 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 5: 95.0% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta) :precision binary32 (asin (/ h eta)))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / eta));
end

Derivation

1. Initial program 90.9%

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

2. Taylor expanded in eta around inf 95.7%

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

3. Final simplification 95.7%

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

Reproduce

herbie shell --seed 2023194 
(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)))))))))