HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.9% → 98.6%

Time: 4.9s

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: 91.9% 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: 98.6% accurate, 1.2× 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

\[\begin{array}{l} \mathbf{if}\;\left|sinTheta\_O\right| \leq 5.499347374382884 \cdot 10^{-24}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \left(-\left|sinTheta\_O\right|\right) \cdot \left|sinTheta\_O\right|\right)}}\right)\\ \end{array} \]

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)))
  (if (<= (fabs sinTheta_O) 5.499347374382884e-24)
  (asin (/ h eta))
  (asin
   (/
    h
    (sqrt
     (fma eta eta (* (- (fabs sinTheta_O)) (fabs sinTheta_O))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if (fabsf(sinTheta_O) <= 5.499347374382884e-24f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(fmaf(eta, eta, (-fabsf(sinTheta_O) * fabsf(sinTheta_O))))));
	}
	return tmp;
}

Julia

function code(sinTheta_O, h, eta)
	tmp = Float32(0.0)
	if (abs(sinTheta_O) <= Float32(5.499347374382884e-24))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(fma(eta, eta, Float32(Float32(-abs(sinTheta_O)) * abs(sinTheta_O))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 5.49934737e-24

   1. Initial program 91.9%

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

   2. Taylor expanded in eta around inf

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

   4. Applied rewrites 95.1%

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

   if 5.49934737e-24 < sinTheta_O

   1. Initial program 91.9%

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

   2. Taylor expanded in sinTheta_O around 0

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

   4. Applied rewrites 91.6%

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

   6. Applied rewrites 91.6%

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

   7. Taylor expanded in sinTheta_O around 0

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

   9. Applied rewrites 91.6%

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

   11. Applied rewrites 91.6%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \left(-sinTheta\_O\right) \cdot sinTheta\_O\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.7% accurate, 1.5× 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

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

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

C

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

Julia

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in sinTheta_O around 0

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

4. Applied rewrites 90.3%

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

6. Applied rewrites 96.0%

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

8. Applied rewrites 97.7%

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

Alternative 3: 95.1% 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 91.9%

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

2. Taylor expanded in eta around inf

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

4. Applied rewrites 95.1%

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

Reproduce

herbie shell --seed 2026084 
(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)))))))))