HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.9% → 99.7%

Time: 6.2s

Alternatives: 6

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: 91.9% 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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - sinTheta\_O \cdot sinTheta\_O}\\ \sin^{-1} \left(\frac{h}{\frac{\sqrt{\mathsf{fma}\left(\frac{-sinTheta\_O}{eta}, sinTheta\_O, eta \cdot t\_0\right)}}{\sqrt{t\_0 \cdot eta}} \cdot \left|eta\right|}\right) \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))
   (asin
    (/
     h
     (*
      (/
       (sqrt (fma (/ (- sinTheta_O) eta) sinTheta_O (* eta t_0)))
       (sqrt (* t_0 eta)))
      (fabs eta))))))

C

float code(float sinTheta_O, float h, float eta) {
	float t_0 = sqrtf((1.0f - (sinTheta_O * sinTheta_O)));
	return asinf((h / ((sqrtf(fmaf((-sinTheta_O / eta), sinTheta_O, (eta * t_0))) / sqrtf((t_0 * eta))) * fabsf(eta))));
}

Julia

function code(sinTheta_O, h, eta)
	t_0 = sqrt(Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O)))
	return asin(Float32(h / Float32(Float32(sqrt(fma(Float32(Float32(-sinTheta_O) / eta), sinTheta_O, Float32(eta * t_0))) / sqrt(Float32(t_0 * eta))) * abs(eta))))
end

Derivation

1. Initial program 91.9%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-to-mult N/A

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

   4. sqrt-prod N/A

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

   5. lift-*.f32 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 93.2%

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

5. Applied rewrites 99.7%

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

7. Applied rewrites 99.7%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-lft-identity 99.7

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.7

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

9. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 91.9%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-to-mult N/A

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

   4. sqrt-prod N/A

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

   5. lift-*.f32 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 93.2%

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

5. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 91.9%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-to-mult N/A

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

   4. sqrt-prod N/A

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

   5. lift-*.f32 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 93.2%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 99.7

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

5. Applied rewrites 99.7%

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

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 91.9%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-to-mult N/A

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

   4. sqrt-prod N/A

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

   5. lift-*.f32 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 93.2%

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in sinTheta_O around 0

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

8. Applied rewrites 99.6%

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

Alternative 5: 95.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 91.9%

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

   1. lift--.f32 N/A

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

   2. sub-flip N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-/.f32 91.9

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

3. Applied rewrites 91.9%

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

4. Taylor expanded in sinTheta_O around 0

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

   1. lower-*.f32 91.8

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

6. Applied rewrites 91.8%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. sum-to-mult N/A

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

   5. sqrt-prod N/A

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

   6. pow1/2 N/A

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

   7. lift-*.f32 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. rem-exp-log N/A

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

   10. exp-fabs N/A

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

   11. rem-exp-log N/A

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

   12. lower-*.f32 N/A

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

8. Applied rewrites 93.1%

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

Alternative 6: 93.1% accurate, 3.6× 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

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 91.9%

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

2. Taylor expanded in eta around inf

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

   1. lower-/.f32 95.7

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

4. Applied rewrites 95.7%

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

Reproduce

herbie shell --seed 2025156 
(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)))))))))