HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.6% → 98.7%

Time: 12.6s

Alternatives: 5

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: 91.6% 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, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, sinTheta\_O \cdot \left(-sinTheta\_O\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_O) 0.0)
   (asin (/ h eta))
   (asin (/ h (sqrt (fma eta eta (* sinTheta_O (- sinTheta_O))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(fmaf(eta, eta, (sinTheta_O * -sinTheta_O)))));
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_O sinTheta_O) < 0.0

   1. Initial program 90.8%

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

   3. Taylor expanded in eta around inf

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

      1. lower-/.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   1. Initial program 99.5%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower--.f32 99.5

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

   5. Applied rewrites 99.5%

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

      1. difference-of-squares N/A

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

      2. lift-*.f32 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lift-*.f32 N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-neg.f32 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, sinTheta\_O \cdot \left(-sinTheta\_O\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ (* h (sqrt (/ 1.0 (+ eta sinTheta_O)))) (sqrt (- eta sinTheta_O)))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf(((h * sqrtf((1.0f / (eta + sinTheta_O)))) / sqrtf((eta - 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((1.0e0 / (eta + sintheta_o)))) / sqrt((eta - sintheta_o))))
end function

Julia

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

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin(((h * sqrt((single(1.0) / (eta + sinTheta_O)))) / sqrt((eta - sinTheta_O))));
end

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. difference-of-squares N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower--.f32 94.8

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

5. Applied rewrites 94.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

   5. pow1/2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. lift-+.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f32 N/A

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

   14. pow1/2 N/A

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

   15. lower-sqrt.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in h around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-+.f32 98.5

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

10. Applied rewrites 98.5%

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

Alternative 3: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\left(eta + sinTheta\_O\right) \cdot \left(eta - sinTheta\_O\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_O) 0.0)
   (asin (/ h eta))
   (asin (/ h (sqrt (* (+ eta sinTheta_O) (- eta sinTheta_O)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta + sinTheta_O) * (eta - 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 ((sintheta_o * sintheta_o) <= 0.0e0) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta + sintheta_o) * (eta - sintheta_o)))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_O sinTheta_O) < 0.0

   1. Initial program 90.8%

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

   3. Taylor expanded in eta around inf

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

      1. lower-/.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   1. Initial program 99.5%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower--.f32 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (* (sqrt (- eta sinTheta_O)) (sqrt (+ eta sinTheta_O))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (sqrtf((eta - sinTheta_O)) * sqrtf((eta + 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 - sintheta_o)) * sqrt((eta + sintheta_o)))))
end function

Julia

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

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (sqrt((eta - sinTheta_O)) * sqrt((eta + sinTheta_O)))));
end

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. difference-of-squares N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower--.f32 94.8

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

5. Applied rewrites 94.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. pow1/2 N/A

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

   6. pow1/2 N/A

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

   7. lower-*.f32 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f32 98.5

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

   12. lift-+.f32 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f32 98.5

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

7. Applied rewrites 98.5%

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

8. Final simplification 98.5%

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

Alternative 5: 95.5% 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 94.9%

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

3. Taylor expanded in eta around inf

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

   1. lower-/.f32 94.5

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

5. Applied rewrites 94.5%

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

Reproduce

herbie shell --seed 2024214 
(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)))))))))