HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 97.8%

Time: 14.0s

Alternatives: 3

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: 92.0% 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, 1.0× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \sin^{-1} \left(\frac{h}{eta + sinTheta\_O\_m \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{sinTheta\_O\_m}{eta}\right)\right)}\right) \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin
  (/
   h
   (+ eta (* sinTheta_O_m (log1p (expm1 (* -0.5 (/ 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 + (sinTheta_O_m * log1pf(expm1f((-0.5f * (sinTheta_O_m / eta))))))));
}

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	return asin(Float32(h / Float32(eta + Float32(sinTheta_O_m * log1p(expm1(Float32(Float32(-0.5) * Float32(sinTheta_O_m / eta))))))))
end

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in sinTheta_O around 0 96.6%

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

   1. unpow2 96.6%

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

   2. *-un-lft-identity 96.6%

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

   3. times-frac 97.3%

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

5. Applied egg-rr 97.3%

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

   1. /-rgt-identity 97.3%

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

   2. associate-*r* 97.3%

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

   3. clear-num 97.3%

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

   4. un-div-inv 97.3%

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

7. Applied egg-rr 97.3%

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

   1. associate-/r/ 97.3%

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

   2. *-commutative 97.3%

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

9. Simplified 97.3%

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

    1. log1p-expm1-u 97.3%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{sinTheta\_O \cdot -0.5}{eta}\right)\right)} \cdot sinTheta\_O}\right) \]

    2. *-commutative 97.3%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{-0.5 \cdot sinTheta\_O}}{eta}\right)\right) \cdot sinTheta\_O}\right) \]

    3. *-un-lft-identity 97.3%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-0.5 \cdot sinTheta\_O}{\color{blue}{1 \cdot eta}}\right)\right) \cdot sinTheta\_O}\right) \]

    4. times-frac 97.3%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{-0.5}{1} \cdot \frac{sinTheta\_O}{eta}}\right)\right) \cdot sinTheta\_O}\right) \]

    5. metadata-eval 97.3%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-0.5} \cdot \frac{sinTheta\_O}{eta}\right)\right) \cdot sinTheta\_O}\right) \]

11. Applied egg-rr 97.3%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{sinTheta\_O}{eta}\right)\right)} \cdot sinTheta\_O}\right) \]

12. Final simplification 97.3%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + sinTheta\_O \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{sinTheta\_O}{eta}\right)\right)}\right) \]
13. Add Preprocessing

Alternative 2: 97.8% accurate, 2.8× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin (/ h (+ eta (* sinTheta_O_m (/ (* -0.5 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 + (sinTheta_O_m * ((-0.5f * 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 + (sintheta_o_m * (((-0.5e0) * 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(sinTheta_O_m * Float32(Float32(Float32(-0.5) * sinTheta_O_m) / eta)))))
end

MATLAB

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

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in sinTheta_O around 0 96.6%

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

   1. unpow2 96.6%

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

   2. *-un-lft-identity 96.6%

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

   3. times-frac 97.3%

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

5. Applied egg-rr 97.3%

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

   1. /-rgt-identity 97.3%

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

   2. associate-*r* 97.3%

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

   3. clear-num 97.3%

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

   4. un-div-inv 97.3%

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

7. Applied egg-rr 97.3%

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

   1. associate-/r/ 97.3%

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

   2. *-commutative 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 3: 95.3% 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 92.2%

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

3. Taylor expanded in eta around inf 94.3%

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

4. Final simplification 94.3%

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

Reproduce

herbie shell --seed 2024034 
(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)))))))))