HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 98.5%

Time: 4.1s

Alternatives: 4

Speedup: 3.6×

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

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: 92.0% 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

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: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O\_m \leq 8.000000156331851 \cdot 10^{-25}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(-sinTheta\_O\_m, sinTheta\_O\_m, eta \cdot eta\right)}}\right)\\ \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (if (<= sinTheta_O_m 8.000000156331851e-25)
   (asin (/ h eta))
   (asin (/ h (sqrt (fma (- sinTheta_O_m) sinTheta_O_m (* eta eta)))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	float tmp;
	if (sinTheta_O_m <= 8.000000156331851e-25f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(fmaf(-sinTheta_O_m, sinTheta_O_m, (eta * eta)))));
	}
	return tmp;
}

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	tmp = Float32(0.0)
	if (sinTheta_O_m <= Float32(8.000000156331851e-25))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(fma(Float32(-sinTheta_O_m), sinTheta_O_m, Float32(eta * eta)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 8.00000016e-25

   1. Initial program 92.0%

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

   2. Taylor expanded in eta around inf

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

      1. lower-/.f32 95.2

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

   4. Applied rewrites 95.2%

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

   if 8.00000016e-25 < sinTheta_O

   1. Initial program 92.0%

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

   2. Taylor expanded in sinTheta_O around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-fma.f32 N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f32 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 91.8

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

   6. Applied rewrites 91.8%

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

Alternative 2: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O\_m \leq 8.000000156331851 \cdot 10^{-25}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - sinTheta\_O\_m \cdot sinTheta\_O\_m}}\right)\\ \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (if (<= sinTheta_O_m 8.000000156331851e-25)
   (asin (/ h eta))
   (asin (/ h (sqrt (- (* eta eta) (* sinTheta_O_m sinTheta_O_m)))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	float tmp;
	if (sinTheta_O_m <= 8.000000156331851e-25f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta * eta) - (sinTheta_O_m * sinTheta_O_m)))));
	}
	return tmp;
}

Fortran

sinTheta_O_m =     private
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_m, h, eta)
use fmin_fmax_functions
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    real(4) :: tmp
    if (sintheta_o_m <= 8.000000156331851e-25) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta * eta) - (sintheta_o_m * sintheta_o_m)))))
    end if
    code = tmp
end function

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	tmp = Float32(0.0)
	if (sinTheta_O_m <= Float32(8.000000156331851e-25))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(Float32(Float32(eta * eta) - Float32(sinTheta_O_m * sinTheta_O_m)))));
	end
	return tmp
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp_2 = code(sinTheta_O_m, h, eta)
	tmp = single(0.0);
	if (sinTheta_O_m <= single(8.000000156331851e-25))
		tmp = asin((h / eta));
	else
		tmp = asin((h / sqrt(((eta * eta) - (sinTheta_O_m * sinTheta_O_m)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 8.00000016e-25

   1. Initial program 92.0%

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

   2. Taylor expanded in eta around inf

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

      1. lower-/.f32 95.2

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

   4. Applied rewrites 95.2%

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

   if 8.00000016e-25 < sinTheta_O

   1. Initial program 92.0%

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

   2. Taylor expanded in sinTheta_O around 0

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

      1. pow2 N/A

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

      2. lift-*.f32 91.8

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

   4. Applied rewrites 91.8%

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

Alternative 3: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(sinTheta\_O\_m \cdot \frac{sinTheta\_O\_m}{\left|eta\right|}, -0.5, \left|eta\right|\right)}\right) \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin
  (/ h (fma (* sinTheta_O_m (/ sinTheta_O_m (fabs eta))) -0.5 (fabs eta)))))

C

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

Julia

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

Derivation

1. Initial program 92.0%

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

2. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. pow2 N/A

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

   12. lift-*.f32 90.5

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

4. Applied rewrites 90.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. pow2 N/A

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

   8. lower-/.f32 N/A

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

   9. pow2 N/A

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

   10. rem-sqrt-square N/A

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

   11. lower-fabs.f32 90.6

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

   12. lift-sqrt.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. rem-sqrt-square N/A

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

   15. lower-fabs.f32 97.8

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

6. Applied rewrites 97.8%

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

Alternative 4: 95.2% accurate, 3.6× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta) :precision binary32 (asin (/ h eta)))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	return asinf((h / eta));
}

Fortran

sinTheta_O_m =     private
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_m, h, eta)
use fmin_fmax_functions
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / eta))
end function

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	return asin(Float32(h / eta))
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	tmp = asin((h / eta));
end

Derivation

1. Initial program 92.0%

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

2. Taylor expanded in eta around inf

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

   1. lower-/.f32 95.2

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

4. Applied rewrites 95.2%

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

Reproduce

herbie shell --seed 2025139 
(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)))))))))