HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.9% → 98.8%

Time: 4.3s

Alternatives: 6

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
  :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)))))))))

C

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

Fortran

real(4) function code(sintheta_o, h, eta)
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

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

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
  :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)))))))))

C

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

Fortran

real(4) function code(sintheta_o, h, eta)
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: 98.8% accurate, 0.7× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

\[\begin{array}{l} t_0 := -\left(-\left|eta\right|\right)\\ \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(t\_0, t\_0, \frac{\left(-sinTheta\_O\right) \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
  :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)))
  (let* ((t_0 (- (- (fabs eta)))))
  (if (<= (* sinTheta_O sinTheta_O) 0.0)
    (asin (/ h eta))
    (asin
     (/
      h
      (sqrt
       (fma
        t_0
        t_0
        (/
         (* (- sinTheta_O) sinTheta_O)
         (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float t_0 = -(-fabsf(eta));
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(fmaf(t_0, t_0, ((-sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
	}
	return tmp;
}

Julia

function code(sinTheta_O, h, eta)
	t_0 = Float32(-Float32(-abs(eta)))
	tmp = Float32(0.0)
	if (Float32(sinTheta_O * sinTheta_O) <= Float32(0.0))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(fma(t_0, t_0, Float32(Float32(Float32(-sinTheta_O) * sinTheta_O) / sqrt(Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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

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

   4. Applied rewrites 95.4%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   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

   3. Applied rewrites 91.9%

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

Alternative 2: 98.7% accurate, 0.8× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
  :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)))
  (if (<= (* sinTheta_O sinTheta_O) 0.0)
  (asin (/ h eta))
  (asin
   (/
    h
    (sqrt
     (-
      (* eta eta)
      (/
       (* sinTheta_O sinTheta_O)
       (sqrt (- 1.0 (* sinTheta_O sinTheta_O))))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
	}
	return tmp;
}

Fortran

real(4) function code(sintheta_o, h, eta)
use fmin_fmax_functions
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    real(4) :: tmp
    if ((sintheta_o * sintheta_o) <= 0.0e0) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   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

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

   4. Applied rewrites 95.4%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   1. Initial program 91.9%

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

Alternative 3: 98.6% accurate, 0.9× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
  :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)))
  (if (<= (* sinTheta_O sinTheta_O) 0.0)
  (asin (/ h eta))
  (asin
   (/
    h
    (sqrt
     (fma
      (- eta)
      (- eta)
      (/ (* (- sinTheta_O) sinTheta_O) (sqrt 1.0))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(fmaf(-eta, -eta, ((-sinTheta_O * sinTheta_O) / sqrtf(1.0f))))));
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   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

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

   4. Applied rewrites 95.4%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   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 sinTheta_O around 0

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

   4. Applied rewrites 91.7%

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

   6. Applied rewrites 91.7%

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

   8. Applied rewrites 91.7%

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

Alternative 4: 98.6% accurate, 1.0× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
  :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)))
  (if (<= (* sinTheta_O sinTheta_O) 0.0)
  (asin (/ h eta))
  (asin
   (/
    h
    (sqrt
     (fma (/ (- sinTheta_O) (sqrt 1.0)) sinTheta_O (* eta eta)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(fmaf((-sinTheta_O / sqrtf(1.0f)), sinTheta_O, (eta * eta)))));
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   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

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

   4. Applied rewrites 95.4%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   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

   3. Applied rewrites 91.9%

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

   4. Taylor expanded in sinTheta_O around 0

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

   6. Applied rewrites 91.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
  :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)))
  (if (<= (* sinTheta_O sinTheta_O) 0.0)
  (asin (/ h eta))
  (asin
   (/
    h
    (sqrt (- (* eta eta) (/ (* sinTheta_O sinTheta_O) (sqrt 1.0))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf(1.0f))))));
	}
	return tmp;
}

Fortran

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 ((sintheta_o * sintheta_o) <= 0.0e0) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt(1.0e0))))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   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

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

   4. Applied rewrites 95.4%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   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 sinTheta_O around 0

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

   4. Applied rewrites 91.7%

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

Alternative 6: 95.4% accurate, 3.5× speedup

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
  :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 eta)))

C

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

Fortran

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

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

4. Applied rewrites 95.4%

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

Reproduce

herbie shell --seed 2026070 
(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)))))))))