HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 15.1s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Initial Program: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\left(0.5 \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right) \cdot 2} \cdot \left(\frac{1}{v} \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O}{v}\right)\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (/
    (exp (/ sinTheta_i (/ v (- sinTheta_O))))
    (* (* 0.5 (- (exp (/ 1.0 v)) (exp (/ -1.0 v)))) 2.0))
   (* (/ 1.0 v) (* cosTheta_i_m (/ cosTheta_O v))))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((expf((sinTheta_i / (v / -sinTheta_O))) / ((0.5f * (expf((1.0f / v)) - expf((-1.0f / v)))) * 2.0f)) * ((1.0f / v) * (cosTheta_i_m * (cosTheta_O / v))));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * ((exp((sintheta_i / (v / -sintheta_o))) / ((0.5e0 * (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * 2.0e0)) * ((1.0e0 / v) * (costheta_i_m * (costheta_o / v))))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(Float32(0.5) * Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) * Float32(2.0))) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i_m * Float32(cosTheta_O / v)))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp((sinTheta_i / (v / -sinTheta_O))) / ((single(0.5) * (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * single(2.0))) * ((single(1.0) / v) * (cosTheta_i_m * (cosTheta_O / v))));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(\left(\frac{1}{v} \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (* (/ 1.0 v) (* cosTheta_i_m (/ cosTheta_O v)))
   (/ (exp (/ sinTheta_i (/ v (- sinTheta_O)))) (* 2.0 (sinh (/ 1.0 v)))))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((1.0f / v) * (cosTheta_i_m * (cosTheta_O / v))) * (expf((sinTheta_i / (v / -sinTheta_O))) / (2.0f * sinhf((1.0f / v)))));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (((1.0e0 / v) * (costheta_i_m * (costheta_o / v))) * (exp((sintheta_i / (v / -sintheta_o))) / (2.0e0 * sinh((1.0e0 / v)))))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i_m * Float32(cosTheta_O / v))) * Float32(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((single(1.0) / v) * (cosTheta_i_m * (cosTheta_O / v))) * (exp((sinTheta_i / (v / -sinTheta_O))) / (single(2.0) * sinh((single(1.0) / v)))));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(\left(\frac{1}{v} \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (* (/ 1.0 v) (* cosTheta_i_m (/ cosTheta_O v)))
   (/ 1.0 (- (exp (/ 1.0 v)) (exp (/ -1.0 v)))))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((1.0f / v) * (cosTheta_i_m * (cosTheta_O / v))) * (1.0f / (expf((1.0f / v)) - expf((-1.0f / v)))));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (((1.0e0 / v) * (costheta_i_m * (costheta_o / v))) * (1.0e0 / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i_m * Float32(cosTheta_O / v))) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v))))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((single(1.0) / v) * (cosTheta_i_m * (cosTheta_O / v))) * (single(1.0) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta_i_m}{v \cdot \frac{v}{cosTheta_O}}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (/ 1.0 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
   (/ cosTheta_i_m (* v (/ v cosTheta_O))))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((1.0f / (expf((1.0f / v)) - expf((-1.0f / v)))) * (cosTheta_i_m / (v * (v / cosTheta_O))));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * ((1.0e0 / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * (costheta_i_m / (v * (v / costheta_o))))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) * Float32(cosTheta_i_m / Float32(v * Float32(v / cosTheta_O)))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((single(1.0) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * (cosTheta_i_m / (v * (v / cosTheta_O))));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{\frac{cosTheta_i_m}{\frac{v}{cosTheta_O}}}{v}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (/ 1.0 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
   (/ (/ cosTheta_i_m (/ v cosTheta_O)) v))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((1.0f / (expf((1.0f / v)) - expf((-1.0f / v)))) * ((cosTheta_i_m / (v / cosTheta_O)) / v));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * ((1.0e0 / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * ((costheta_i_m / (v / costheta_o)) / v))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) * Float32(Float32(cosTheta_i_m / Float32(v / cosTheta_O)) / v)))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((single(1.0) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * ((cosTheta_i_m / (v / cosTheta_O)) / v));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 59.0% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(0.5 \cdot \frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i_m}}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* 0.5 (/ 1.0 (/ (/ v cosTheta_O) cosTheta_i_m)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (0.5f * (1.0f / ((v / cosTheta_O) / cosTheta_i_m)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 * (1.0e0 / ((v / costheta_o) / costheta_i_m)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(0.5) * Float32(Float32(1.0) / Float32(Float32(v / cosTheta_O) / cosTheta_i_m))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(0.5) * (single(1.0) / ((v / cosTheta_O) / cosTheta_i_m)));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 59.0% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \frac{1}{\frac{\frac{v}{cosTheta_O}}{0.5 \cdot cosTheta_i_m}} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (/ 1.0 (/ (/ v cosTheta_O) (* 0.5 cosTheta_i_m)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (1.0f / ((v / cosTheta_O) / (0.5f * cosTheta_i_m)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (1.0e0 / ((v / costheta_o) / (0.5e0 * costheta_i_m)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(1.0) / Float32(Float32(v / cosTheta_O) / Float32(Float32(0.5) * cosTheta_i_m))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(1.0) / ((v / cosTheta_O) / (single(0.5) * cosTheta_i_m)));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 58.5% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(0.5 \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O}{v}\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* 0.5 (* cosTheta_i_m (/ cosTheta_O v)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (0.5f * (cosTheta_i_m * (cosTheta_O / v)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 * (costheta_i_m * (costheta_o / v)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(0.5) * Float32(cosTheta_i_m * Float32(cosTheta_O / v))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(0.5) * (cosTheta_i_m * (cosTheta_O / v)));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 58.5% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i_m}{v}\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* 0.5 (* cosTheta_O (/ cosTheta_i_m v)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (0.5f * (cosTheta_O * (cosTheta_i_m / v)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 * (costheta_o * (costheta_i_m / v)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(0.5) * Float32(cosTheta_O * Float32(cosTheta_i_m / v))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(0.5) * (cosTheta_O * (cosTheta_i_m / v)));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 59.0% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \frac{0.5}{\frac{v}{cosTheta_i_m \cdot cosTheta_O}} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (/ 0.5 (/ v (* cosTheta_i_m cosTheta_O)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (0.5f / (v / (cosTheta_i_m * cosTheta_O)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 / (v / (costheta_i_m * costheta_o)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(0.5) / Float32(v / Float32(cosTheta_i_m * cosTheta_O))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(0.5) / (v / (cosTheta_i_m * cosTheta_O)));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2024008 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))