HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 98.7%

Time: 9.2s

Alternatives: 4

Speedup: 1.5×

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

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

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: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_1 := \sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{if}\;eta \cdot eta \leq 5.000001076526666 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(\left(t\_0 + t\_0\right) - t\_1\right) \cdot t\_1}{t\_0 + \cos^{-1} \left(\frac{h}{eta}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (/ (PI) 2.0)) (t_1 (asin (/ h eta))))
   (if (<= (* eta eta) 5.000001076526666e-40)
     (/ (* (- (+ t_0 t_0) t_1) t_1) (+ t_0 (acos (/ h eta))))
     (asin
      (/
       h
       (sqrt
        (-
         (* eta eta)
         (/
          (* sinTheta_O sinTheta_O)
          (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 eta eta) < 5.0000011e-40

   1. Initial program 32.0%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. lower-/.f32 93.7

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

   5. Applied rewrites 93.7%

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

      1. lift-asin.f32 N/A

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

      2. asin-acos N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right)} \]

      3. flip-- N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right) \cdot \cos^{-1} \left(\frac{h}{eta}\right)}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)}} \]

      4. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right) \cdot \cos^{-1} \left(\frac{h}{eta}\right)}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)}} \]

   7. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} - {\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)}} \]
   8. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} - {\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      2. lift-pow.f32 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}} - {\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2}} - {\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      4. lift-pow.f32 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{{\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\frac{h}{eta}\right) \cdot \cos^{-1} \left(\frac{h}{eta}\right)}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      6. difference-of-squares N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right)\right)}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)\right)} \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right)\right)}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

   9. Applied rewrites 93.7%

      \[\leadsto \frac{\color{blue}{\left(\cos^{-1} \left(\frac{h}{eta}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin^{-1} \left(\frac{h}{eta}\right)}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]
   10. Step-by-step derivation

       1. lift-+.f32 N/A

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

       2. +-commutative N/A

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

       3. lift-acos.f32 N/A

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

       4. acos-asin N/A

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

       5. lift-PI.f32 N/A

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

       6. lift-/.f32 N/A

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

       7. lift-asin.f32 N/A

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

       8. associate-+r- N/A

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

       9. lower--.f32 N/A

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

   11. Applied rewrites 93.8%

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

   if 5.0000011e-40 < (*.f32 eta eta)

   1. Initial program 99.8%

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

Alternative 2: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{h}{eta}\right)\\ t_1 := \frac{\mathsf{PI}\left(\right)}{2}\\ \mathbf{if}\;eta \cdot eta \leq 5.000001076526666 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(t\_0 + t\_1\right) \cdot \sin^{-1} \left(\frac{h}{eta}\right)}{t\_1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (acos (/ h eta))) (t_1 (/ (PI) 2.0)))
   (if (<= (* eta eta) 5.000001076526666e-40)
     (/ (* (+ t_0 t_1) (asin (/ h eta))) (+ t_1 t_0))
     (asin
      (/
       h
       (sqrt
        (-
         (* eta eta)
         (/
          (* sinTheta_O sinTheta_O)
          (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 eta eta) < 5.0000011e-40

   1. Initial program 32.0%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. lower-/.f32 93.7

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

   5. Applied rewrites 93.7%

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

      1. lift-asin.f32 N/A

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

      2. asin-acos N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right)} \]

      3. flip-- N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right) \cdot \cos^{-1} \left(\frac{h}{eta}\right)}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)}} \]

      4. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right) \cdot \cos^{-1} \left(\frac{h}{eta}\right)}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)}} \]

   7. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} - {\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)}} \]
   8. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} - {\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      2. lift-pow.f32 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}} - {\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2}} - {\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      4. lift-pow.f32 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{{\cos^{-1} \left(\frac{h}{eta}\right)}^{2}}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\frac{h}{eta}\right) \cdot \cos^{-1} \left(\frac{h}{eta}\right)}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      6. difference-of-squares N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right)\right)}}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)\right)} \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{h}{eta}\right)\right)}{\frac{\mathsf{PI}\left(\right)}{2} + \cos^{-1} \left(\frac{h}{eta}\right)} \]

   9. Applied rewrites 93.7%

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

   if 5.0000011e-40 < (*.f32 eta eta)

   1. Initial program 99.8%

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

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eta \cdot eta \leq 5.000001076526666 \cdot 10^{-40}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (* eta eta) 5.000001076526666e-40)
   (asin (/ h eta))
   (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) {
	float tmp;
	if ((eta * eta) <= 5.000001076526666e-40f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((eta * eta) <= 5.000001076526666e-40) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
    end if
    code = tmp
end function

Julia

function code(sinTheta_O, h, eta)
	tmp = Float32(0.0)
	if (Float32(eta * eta) <= Float32(5.000001076526666e-40))
		tmp = asin(Float32(h / eta));
	else
		tmp = 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
	return tmp
end

MATLAB

function tmp_2 = code(sinTheta_O, h, eta)
	tmp = single(0.0);
	if ((eta * eta) <= single(5.000001076526666e-40))
		tmp = asin((h / eta));
	else
		tmp = asin((h / sqrt(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrt((single(1.0) - (sinTheta_O * sinTheta_O))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 eta eta) < 5.0000011e-40

   1. Initial program 32.0%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. lower-/.f32 93.7

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

   5. Applied rewrites 93.7%

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

   if 5.0000011e-40 < (*.f32 eta eta)

   1. Initial program 99.8%

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

Alternative 4: 95.3% accurate, 1.5× 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 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. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0

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

   1. lower-/.f32 95.2

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

5. Applied rewrites 95.2%

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

Reproduce

herbie shell --seed 2024333 
(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)))))))))