HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%
Time: 13.6s
Alternatives: 8
Speedup: 1.0×

Specification

?
\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]
\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]
(FPCore (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)))
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);
}

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

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

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

\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}

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 8 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 98.5% accurate, 1.0× speedup?

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

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

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

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

\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}

Alternative 1: 98.7% accurate, 1.0× speedup?

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(\left(e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{\frac{1}{v} \cdot cosTheta_O_m}{\frac{v}{cosTheta_i_m}}\right)\right) \end{array} \]
cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* (exp (* sinTheta_O (- (/ sinTheta_i v)))) (/ 0.5 (sinh (/ 1.0 v))))
    (/ (* (/ 1.0 v) cosTheta_O_m) (/ v cosTheta_i_m))))))
cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((expf((sinTheta_O * -(sinTheta_i / v))) * (0.5f / sinhf((1.0f / v)))) * (((1.0f / v) * cosTheta_O_m) / (v / cosTheta_i_m))));
}

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

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

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

\begin{array}{l}
cosTheta_i_m = \left|cosTheta_i\right|
\\
cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right)
\\
cosTheta_O_m = \left|cosTheta_O\right|
\\
cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right)
\\
[cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\
\\
cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(\left(e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{\frac{1}{v} \cdot cosTheta_O_m}{\frac{v}{cosTheta_i_m}}\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup?

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O_m}{v}\right)\right)\right)\right) \end{array} \]
cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ 1.0 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
    (* (/ 1.0 v) (* cosTheta_i_m (/ cosTheta_O_m v)))))))
cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((1.0f / (expf((1.0f / v)) - expf((-1.0f / v)))) * ((1.0f / v) * (cosTheta_i_m * (cosTheta_O_m / v)))));
}

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

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

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

\begin{array}{l}
cosTheta_i_m = \left|cosTheta_i\right|
\\
cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right)
\\
cosTheta_O_m = \left|cosTheta_O\right|
\\
cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right)
\\
[cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\
\\
cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O_m}{v}\right)\right)\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup?

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(\frac{\frac{1}{v} \cdot cosTheta_O_m}{\frac{v}{cosTheta_i_m}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}\right)\right) \end{array} \]
cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ (* (/ 1.0 v) cosTheta_O_m) (/ v cosTheta_i_m))
    (/ 1.0 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))))))
cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((((1.0f / v) * cosTheta_O_m) / (v / cosTheta_i_m)) * (1.0f / (expf((1.0f / v)) - expf((-1.0f / v))))));
}

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

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

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

\begin{array}{l}
cosTheta_i_m = \left|cosTheta_i\right|
\\
cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right)
\\
cosTheta_O_m = \left|cosTheta_O\right|
\\
cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right)
\\
[cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\
\\
cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(\frac{\frac{1}{v} \cdot cosTheta_O_m}{\frac{v}{cosTheta_i_m}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup?

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta_O_m}{v \cdot \frac{v}{cosTheta_i_m}}\right)\right) \end{array} \]
cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ 1.0 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
    (/ cosTheta_O_m (* v (/ v cosTheta_i_m)))))))
cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((1.0f / (expf((1.0f / v)) - expf((-1.0f / v)))) * (cosTheta_O_m / (v * (v / cosTheta_i_m)))));
}

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

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

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

\begin{array}{l}
cosTheta_i_m = \left|cosTheta_i\right|
\\
cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right)
\\
cosTheta_O_m = \left|cosTheta_O\right|
\\
cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right)
\\
[cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\
\\
cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta_O_m}{v \cdot \frac{v}{cosTheta_i_m}}\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 58.4% accurate, 24.4× speedup?

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \frac{1}{\frac{v}{cosTheta_i_m \cdot \left(0.5 \cdot cosTheta_O_m\right)}}\right) \end{array} \]
cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* cosTheta_i_s (/ 1.0 (/ v (* cosTheta_i_m (* 0.5 cosTheta_O_m)))))))
cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (1.0f / (v / (cosTheta_i_m * (0.5f * cosTheta_O_m)))));
}

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

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

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

\begin{array}{l}
cosTheta_i_m = \left|cosTheta_i\right|
\\
cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right)
\\
cosTheta_O_m = \left|cosTheta_O\right|
\\
cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right)
\\
[cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\
\\
cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \frac{1}{\frac{v}{cosTheta_i_m \cdot \left(0.5 \cdot cosTheta_O_m\right)}}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 57.9% accurate, 31.4× speedup?

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(0.5 \cdot \left(cosTheta_O_m \cdot \frac{cosTheta_i_m}{v}\right)\right)\right) \end{array} \]
cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* 0.5 (* cosTheta_O_m (/ cosTheta_i_m v))))))
cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (0.5f * (cosTheta_O_m * (cosTheta_i_m / v))));
}

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

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

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

\begin{array}{l}
cosTheta_i_m = \left|cosTheta_i\right|
\\
cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right)
\\
cosTheta_O_m = \left|cosTheta_O\right|
\\
cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right)
\\
[cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\
\\
cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(0.5 \cdot \left(cosTheta_O_m \cdot \frac{cosTheta_i_m}{v}\right)\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 7: 57.9% accurate, 31.4× speedup?

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(0.5 \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O_m}{v}\right)\right)\right) \end{array} \]
cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* 0.5 (* cosTheta_i_m (/ cosTheta_O_m v))))))
cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (0.5f * (cosTheta_i_m * (cosTheta_O_m / v))));
}

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

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

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

\begin{array}{l}
cosTheta_i_m = \left|cosTheta_i\right|
\\
cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right)
\\
cosTheta_O_m = \left|cosTheta_O\right|
\\
cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right)
\\
[cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\
\\
cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \left(0.5 \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O_m}{v}\right)\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 8: 58.3% accurate, 31.4× speedup?

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \frac{0.5}{\frac{\frac{v}{cosTheta_O_m}}{cosTheta_i_m}}\right) \end{array} \]
cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (/ 0.5 (/ (/ v cosTheta_O_m) cosTheta_i_m)))))
cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (0.5f / ((v / cosTheta_O_m) / cosTheta_i_m)));
}

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

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

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

\begin{array}{l}
cosTheta_i_m = \left|cosTheta_i\right|
\\
cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right)
\\
cosTheta_O_m = \left|cosTheta_O\right|
\\
cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right)
\\
[cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\
\\
cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \frac{0.5}{\frac{\frac{v}{cosTheta_O_m}}{cosTheta_i_m}}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023347 
(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)))