HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.1% → 97.7%

Time: 4.3s

Alternatives: 5

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
 (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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

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
 (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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|sinTheta\_O\right| \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \left(-\left|sinTheta\_O\right|\right) \cdot \frac{\left|sinTheta\_O\right|}{1}\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (fabs sinTheta_O) 5.000000136226006e-28)
   (asin (/ h eta))
   (asin
    (/
     h
     (sqrt
      (fma eta eta (* (- (fabs sinTheta_O)) (/ (fabs sinTheta_O) 1.0))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if (fabsf(sinTheta_O) <= 5.000000136226006e-28f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(fmaf(eta, eta, (-fabsf(sinTheta_O) * (fabsf(sinTheta_O) / 1.0f))))));
	}
	return tmp;
}

Julia

function code(sinTheta_O, h, eta)
	tmp = Float32(0.0)
	if (abs(sinTheta_O) <= Float32(5.000000136226006e-28))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(fma(eta, eta, Float32(Float32(-abs(sinTheta_O)) * Float32(abs(sinTheta_O) / Float32(1.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 5.00000014e-28

   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

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

      1. lower-/.f32 95.4%

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

   4. Applied rewrites 95.4%

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

   if 5.00000014e-28 < sinTheta_O

   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 sinTheta_O around 0

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

   4. Applied rewrites 91.9%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-/l* N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. lift-*.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lift-/.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-/l* N/A

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

      15. distribute-lft-neg-in N/A

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

   6. Applied rewrites 91.9%

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

Alternative 2: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|sinTheta\_O\right| \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \left|sinTheta\_O\right| \cdot \left|sinTheta\_O\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (fabs sinTheta_O) 5.000000136226006e-28)
   (asin (/ h eta))
   (asin
    (/ h (sqrt (- (* eta eta) (* (fabs sinTheta_O) (fabs sinTheta_O))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if (fabsf(sinTheta_O) <= 5.000000136226006e-28f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta * eta) - (fabsf(sinTheta_O) * fabsf(sinTheta_O))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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
    real(4) :: tmp
    if (abs(sintheta_o) <= 5.000000136226006e-28) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta * eta) - (abs(sintheta_o) * abs(sintheta_o))))))
    end if
    code = tmp
end function

Julia

function code(sinTheta_O, h, eta)
	tmp = Float32(0.0)
	if (abs(sinTheta_O) <= Float32(5.000000136226006e-28))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(Float32(Float32(eta * eta) - Float32(abs(sinTheta_O) * abs(sinTheta_O))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(sinTheta_O, h, eta)
	tmp = single(0.0);
	if (abs(sinTheta_O) <= single(5.000000136226006e-28))
		tmp = asin((h / eta));
	else
		tmp = asin((h / sqrt(((eta * eta) - (abs(sinTheta_O) * abs(sinTheta_O))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 5.00000014e-28

   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

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

      1. lower-/.f32 95.4%

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

   4. Applied rewrites 95.4%

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

   if 5.00000014e-28 < sinTheta_O

   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 sinTheta_O around 0

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

   4. Applied rewrites 91.9%

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

      1. lift-/.f32 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f32 N/A

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

      4. lower-unsound-/.f32 91.2%

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

      5. lift-/.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-/.f32 91.2%

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

   6. Applied rewrites 91.2%

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

   7. Taylor expanded in sinTheta_O around 0

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

   9. Applied rewrites 91.2%

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

       1. lift-/.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. div-flip-rev N/A

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

       4. lower-/.f32 91.9%

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

   11. Applied rewrites 91.9%

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

Alternative 3: 97.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 90.9%

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

6. Applied rewrites 91.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 97.7%

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

8. Applied rewrites 97.7%

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

Alternative 4: 97.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ (fma (* 0.5 h) (/ (* (/ sinTheta_O eta) sinTheta_O) eta) h) eta)))

C

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

Julia

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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 90.9%

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

6. Applied rewrites 91.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 97.7%

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

8. Applied rewrites 97.7%

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

9. Taylor expanded in sinTheta_O around 0

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

11. Applied rewrites 97.7%

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

Alternative 5: 95.4% accurate, 3.5× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

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

   1. lower-/.f32 95.4%

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

4. Applied rewrites 95.4%

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

Reproduce

herbie shell --seed 2025197 
(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)))))))))