HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.1% → 98.0%

Time: 29.6s

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: 92.1% 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: 98.0% 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

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

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
   (fma
    (*
     (/ sinTheta_O eta)
     (/ sinTheta_O (* (sqrt (- 1.0 (* sinTheta_O sinTheta_O))) eta)))
    -0.5
    1.0)))))

C

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

Julia

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

Derivation

1. Initial program 92.1%

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

2. Taylor expanded in eta around inf

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

4. Applied rewrites 92.0%

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

6. Applied rewrites 92.0%

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

8. Applied rewrites 98.0%

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

Alternative 2: 98.0% 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

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

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 (fma (* (/ sinTheta_O eta) (/ sinTheta_O eta)) -0.5 1.0)))))

C

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

Julia

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

Derivation

1. Initial program 92.1%

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

2. Taylor expanded in eta around inf

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

4. Applied rewrites 92.0%

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

6. Applied rewrites 92.0%

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in sinTheta_O around 0

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

11. Applied rewrites 98.0%

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

Alternative 3: 95.5% 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 92.1%

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

2. Taylor expanded in eta around inf

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

4. Applied rewrites 95.5%

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

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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)))))))))