HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.6% → 97.7%

Time: 4.6s

Alternatives: 3

Speedup: 3.5×

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

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

FPCore

(FPCore (sinTheta_O h eta)
  :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)))))))))

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)
use fmin_fmax_functions
    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: 91.6% accurate, 1.0× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
  :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)))))))))

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)
use fmin_fmax_functions
    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: 97.7% accurate, 0.8× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
  :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
 (/
  (fma
   h
   (*
    (*
     (/ sinTheta_O (* (sqrt (- 1.0 (* sinTheta_O sinTheta_O))) eta))
     (/ sinTheta_O eta))
    0.5)
   h)
  eta)))

C

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

Julia

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

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in eta around inf

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

4. Applied rewrites 90.5%

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

6. Applied rewrites 91.5%

   \[\leadsto \sin^{-1} \left(\frac{\mathsf{fma}\left(h, \left(sinTheta\_O \cdot \frac{sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O} \cdot \left(eta \cdot eta\right)}\right) \cdot 0.5, h\right)}{eta}\right) \]
7. Step-by-step derivation

8. Applied rewrites 97.7%

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

Alternative 2: 97.7% accurate, 1.1× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

\[\sin^{-1} \left(\frac{\mathsf{fma}\left(h, \left(\frac{sinTheta\_O}{eta} \cdot \frac{sinTheta\_O}{eta}\right) \cdot 0.5, h\right)}{eta}\right) \]

FPCore

(FPCore (sinTheta_O h eta)
  :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
 (/ (fma h (* (* (/ sinTheta_O eta) (/ sinTheta_O eta)) 0.5) h) eta)))

C

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

Julia

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

Derivation

1. Initial program 91.6%

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

2. Taylor expanded in eta around inf

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

4. Applied rewrites 90.5%

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

6. Applied rewrites 91.5%

   \[\leadsto \sin^{-1} \left(\frac{\mathsf{fma}\left(h, \left(sinTheta\_O \cdot \frac{sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O} \cdot \left(eta \cdot eta\right)}\right) \cdot 0.5, h\right)}{eta}\right) \]
7. Step-by-step derivation

8. Applied rewrites 97.7%

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

9. Taylor expanded in sinTheta_O around 0

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

11. Applied rewrites 97.7%

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

Alternative 3: 95.3% accurate, 3.5× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
  :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 eta)))

C

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

Fortran

real(4) function code(sintheta_o, h, eta)
use fmin_fmax_functions
    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 91.6%

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

2. Taylor expanded in eta around inf

   \[\leadsto \sin^{-1} \left(\frac{h}{eta}\right) \]
3. Step-by-step derivation

4. Applied rewrites 95.3%

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

Reproduce

herbie shell --seed 2026086 
(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)))))))))