HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 19.2s

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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left({\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O\_m \cdot \left(cosTheta\_i \cdot \left(\frac{1}{v} \cdot \frac{1}{v}\right)\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (pow (exp sinTheta_i) (/ sinTheta_O (- v)))
   (/
    (* cosTheta_O_m (* cosTheta_i (* (/ 1.0 v) (/ 1.0 v))))
    (* (sinh (/ 1.0 v)) 2.0)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * ((exp(sintheta_i) ** (sintheta_o / -v)) * ((costheta_o_m * (costheta_i * ((1.0e0 / v) * (1.0e0 / v)))) / (sinh((1.0e0 / v)) * 2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.5%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}} \]

   2. associate-*l/ 98.5%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-*r/ 98.5%

      \[\leadsto \color{blue}{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   4. distribute-frac-neg2 98.5%

      \[\leadsto e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. associate-/l* 98.5%

      \[\leadsto e^{\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. exp-prod 98.5%

      \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. *-commutative 98.5%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\frac{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. associate-/l* 98.6%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\frac{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. associate-/l* 98.5%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\color{blue}{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{\color{blue}{cosTheta\_i \cdot \frac{1}{v}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. *-un-lft-identity 98.6%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{cosTheta\_i \cdot \frac{1}{v}}{\color{blue}{1 \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. times-frac 98.6%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{1} \cdot \frac{\frac{1}{v}}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Applied egg-rr 98.6%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{1} \cdot \frac{\frac{1}{v}}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
7. Step-by-step derivation

   1. div-inv 98.8%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{1} \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{1}{v}\right)}\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

8. Applied egg-rr 98.8%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{1} \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{1}{v}\right)}\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

9. Final simplification 98.8%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(\frac{1}{v} \cdot \frac{1}{v}\right)\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
10. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O\_m \cdot cosTheta\_i\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (*
    (exp (/ (* sinTheta_i sinTheta_O) (- v)))
    (* (/ 1.0 v) (* cosTheta_O_m cosTheta_i)))
   (* v (* (sinh (/ 1.0 v)) 2.0)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * ((exp(((sintheta_i * sintheta_o) / -v)) * ((1.0e0 / v) * (costheta_o_m * costheta_i))) / (v * (sinh((1.0e0 / v)) * 2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.7%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Final simplification 98.7%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
6. Add Preprocessing

Alternative 3: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(\frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O\_m}{v}}{2}\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (/ (/ 1.0 v) (sinh (/ 1.0 v)))
   (/ (* cosTheta_i (/ cosTheta_O_m v)) 2.0))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (((1.0e0 / v) / sinh((1.0e0 / v))) * ((costheta_i * (costheta_o_m / v)) / 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-/l/ 98.5%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   3. remove-double-neg 98.5%

      \[\leadsto \frac{\color{blue}{-\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   4. distribute-rgt-neg-out 98.5%

      \[\leadsto \frac{-\color{blue}{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(-cosTheta\_i \cdot cosTheta\_O\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   5. distribute-rgt-neg-out 98.5%

      \[\leadsto \frac{-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_i \cdot \left(-cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   6. distribute-lft-neg-in 98.5%

      \[\leadsto \frac{\color{blue}{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \left(cosTheta\_i \cdot \left(-cosTheta\_O\right)\right)}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   7. associate-*r/ 98.5%

      \[\leadsto \color{blue}{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   8. associate-/l/ 98.5%

      \[\leadsto \left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \color{blue}{\frac{\frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   9. associate-*r/ 98.6%

      \[\leadsto \color{blue}{\frac{\left(-e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \frac{cosTheta\_i \cdot \left(-cosTheta\_O\right)}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* 98.5%

      \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

   2. times-frac 98.3%

      \[\leadsto \color{blue}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2}} \]

   3. associate-*r/ 98.3%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   4. add-sqr-sqrt -0.0%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   5. sqrt-unprod 98.2%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   6. sqr-neg 98.2%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\sqrt{\color{blue}{v \cdot v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   7. sqrt-unprod 98.2%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   8. add-sqr-sqrt 98.2%

      \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   9. associate-/l* 98.2%

      \[\leadsto \frac{e^{\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   10. pow-exp 98.2%

       \[\leadsto \frac{\color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

6. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2}} \]

7. Taylor expanded in sinTheta_i around 0 98.2%

   \[\leadsto \color{blue}{\frac{2}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]
8. Step-by-step derivation

   1. metadata-eval 98.2%

      \[\leadsto \frac{\color{blue}{1 \cdot 2}}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   2. associate-*l/ 98.2%

      \[\leadsto \color{blue}{\left(\frac{1}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \cdot 2\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   3. associate-/r* 98.4%

      \[\leadsto \left(\color{blue}{\frac{\frac{1}{v}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot 2\right) \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   4. associate-/r/ 98.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{v}}{\frac{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}{2}}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   5. rec-exp 98.4%

      \[\leadsto \frac{\frac{1}{v}}{\frac{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}}{2}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

   6. sinh-def 98.5%

      \[\leadsto \frac{\frac{1}{v}}{\color{blue}{\sinh \left(\frac{1}{v}\right)}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

9. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]

10. Final simplification 98.5%

    \[\leadsto \frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]
11. Add Preprocessing

Alternative 4: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_O\_m \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* cosTheta_O_m (/ cosTheta_i (fma v 2.0 (/ 0.3333333333333333 v))))))

C

cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_O_m * (cosTheta_i / fmaf(v, 2.0f, (0.3333333333333333f / v))));
}

Julia

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_O_m * Float32(cosTheta_i / fma(v, Float32(2.0), Float32(Float32(0.3333333333333333) / v)))))
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 63.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Taylor expanded in sinTheta_i around 0 63.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]
7. Step-by-step derivation

   1. associate-/l* 63.3%

      \[\leadsto \color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]

   2. associate-*r/ 63.3%

      \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{2 \cdot v + \color{blue}{\frac{0.3333333333333333 \cdot 1}{v}}} \]

   3. metadata-eval 63.3%

      \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{2 \cdot v + \frac{\color{blue}{0.3333333333333333}}{v}} \]

   4. *-commutative 63.3%

      \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{v \cdot 2} + \frac{0.3333333333333333}{v}} \]

   5. fma-undefine 63.3%

      \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}} \]

8. Simplified 63.3%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}} \]

9. Final simplification 63.3%

   \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \]
10. Add Preprocessing

Alternative 5: 63.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := \frac{1}{v} \cdot 0.3333333333333333\\ cosTheta\_O\_s \cdot \frac{cosTheta\_O\_m \cdot cosTheta\_i}{t\_0 + \left(v \cdot 2 + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(t\_0 + v \cdot 2\right)\right)}{v}\right)} \end{array} \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* (/ 1.0 v) 0.3333333333333333)))
   (*
    cosTheta_O_s
    (/
     (* cosTheta_O_m cosTheta_i)
     (+
      t_0
      (+ (* v 2.0) (/ (* sinTheta_O (* sinTheta_i (+ t_0 (* v 2.0)))) v)))))))

C

cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (1.0f / v) * 0.3333333333333333f;
	return cosTheta_O_s * ((cosTheta_O_m * cosTheta_i) / (t_0 + ((v * 2.0f) + ((sinTheta_O * (sinTheta_i * (t_0 + (v * 2.0f)))) / v))));
}

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = (1.0e0 / v) * 0.3333333333333333e0
    code = costheta_o_s * ((costheta_o_m * costheta_i) / (t_0 + ((v * 2.0e0) + ((sintheta_o * (sintheta_i * (t_0 + (v * 2.0e0)))) / v))))
end function

Julia

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333))
	return Float32(cosTheta_O_s * Float32(Float32(cosTheta_O_m * cosTheta_i) / Float32(t_0 + Float32(Float32(v * Float32(2.0)) + Float32(Float32(sinTheta_O * Float32(sinTheta_i * Float32(t_0 + Float32(v * Float32(2.0))))) / v)))))
end

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	t_0 = (single(1.0) / v) * single(0.3333333333333333);
	tmp = cosTheta_O_s * ((cosTheta_O_m * cosTheta_i) / (t_0 + ((v * single(2.0)) + ((sinTheta_O * (sinTheta_i * (t_0 + (v * single(2.0))))) / v))));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 63.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Taylor expanded in sinTheta_i around 0 63.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{0.3333333333333333 \cdot \frac{1}{v} + \left(2 \cdot v + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)\right)}{v}\right)}} \]

7. Final simplification 63.3%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\frac{1}{v} \cdot 0.3333333333333333 + \left(v \cdot 2 + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)\right)}{v}\right)} \]
8. Add Preprocessing

Alternative 6: 63.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{cosTheta\_O\_m \cdot cosTheta\_i}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (* cosTheta_O_m cosTheta_i)
   (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0)))))

C

cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * ((cosTheta_O_m * cosTheta_i) / (((1.0f / v) * 0.3333333333333333f) + (v * 2.0f)));
}

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * costheta_i) / (((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0)))
end function

Julia

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(cosTheta_O_m * cosTheta_i) / Float32(Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0)))))
end

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((cosTheta_O_m * cosTheta_i) / (((single(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0))));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 63.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Taylor expanded in sinTheta_i around 0 63.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]

7. Final simplification 63.3%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} \]
8. Add Preprocessing

Alternative 7: 58.7% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(0.5 \cdot \frac{1}{\frac{v}{cosTheta\_O\_m \cdot cosTheta\_i}}\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (/ 1.0 (/ v (* cosTheta_O_m cosTheta_i))))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (0.5e0 * (1.0e0 / (v / (costheta_o_m * costheta_i))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 57.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot v + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}} \]
6. Step-by-step derivation

   1. distribute-lft-out 57.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot \left(v + sinTheta\_O \cdot sinTheta\_i\right)}} \]

7. Simplified 57.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot \left(v + sinTheta\_O \cdot sinTheta\_i\right)}} \]

8. Taylor expanded in v around inf 57.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

   1. associate-/l* 57.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

10. Simplified 57.3%

    \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

11. Taylor expanded in cosTheta_O around 0 57.3%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
12. Step-by-step derivation

    1. *-commutative 57.3%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

    2. associate-*r/ 57.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]

13. Simplified 57.3%

    \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]
14. Step-by-step derivation

    1. associate-*r/ 57.3%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}} \]

    2. clear-num 57.7%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]

15. Applied egg-rr 57.7%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]

16. Final simplification 57.7%

    \[\leadsto 0.5 \cdot \frac{1}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}} \]
17. Add Preprocessing

Alternative 8: 58.7% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{1}{\frac{2}{cosTheta\_i \cdot \frac{cosTheta\_O\_m}{v}}} \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (/ 1.0 (/ 2.0 (* cosTheta_i (/ cosTheta_O_m v))))))

C

cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (1.0f / (2.0f / (cosTheta_i * (cosTheta_O_m / v))));
}

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (1.0e0 / (2.0e0 / (costheta_i * (costheta_o_m / v))))
end function

Julia

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(1.0) / Float32(Float32(2.0) / Float32(cosTheta_i * Float32(cosTheta_O_m / v)))))
end

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 57.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot v + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}} \]
6. Step-by-step derivation

   1. distribute-lft-out 57.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot \left(v + sinTheta\_O \cdot sinTheta\_i\right)}} \]

7. Simplified 57.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot \left(v + sinTheta\_O \cdot sinTheta\_i\right)}} \]

8. Taylor expanded in v around inf 57.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

   1. associate-/l* 57.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

10. Simplified 57.3%

    \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

11. Taylor expanded in cosTheta_O around 0 57.3%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
12. Step-by-step derivation

    1. *-commutative 57.3%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

    2. associate-*r/ 57.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]

13. Simplified 57.3%

    \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]
14. Step-by-step derivation

    1. *-commutative 57.3%

       \[\leadsto \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \cdot 0.5} \]

    2. metadata-eval 57.3%

       \[\leadsto \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \cdot \color{blue}{\frac{1}{2}} \]

    3. div-inv 57.3%

       \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2}} \]

    4. clear-num 57.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{2}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}} \]

15. Applied egg-rr 57.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{2}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}} \]

16. Final simplification 57.8%

    \[\leadsto \frac{1}{\frac{2}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}} \]
17. Add Preprocessing

Alternative 9: 58.2% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(0.5 \cdot \left(cosTheta\_O\_m \cdot \frac{cosTheta\_i}{v}\right)\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (* cosTheta_O_m (/ cosTheta_i v)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (0.5e0 * (costheta_o_m * (costheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 57.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot v + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}} \]
6. Step-by-step derivation

   1. distribute-lft-out 57.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot \left(v + sinTheta\_O \cdot sinTheta\_i\right)}} \]

7. Simplified 57.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot \left(v + sinTheta\_O \cdot sinTheta\_i\right)}} \]

8. Taylor expanded in v around inf 57.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

   1. associate-/l* 57.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

10. Simplified 57.3%

    \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

11. Final simplification 57.3%

    \[\leadsto 0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \]
12. Add Preprocessing

Alternative 10: 58.2% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta\_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(\left(cosTheta\_i \cdot \frac{cosTheta\_O\_m}{v}\right) \cdot 0.5\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* (* cosTheta_i (/ cosTheta_O_m v)) 0.5)))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (costheta_o_m / v)) * 0.5e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 57.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot v + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}} \]
6. Step-by-step derivation

   1. distribute-lft-out 57.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot \left(v + sinTheta\_O \cdot sinTheta\_i\right)}} \]

7. Simplified 57.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot \left(v + sinTheta\_O \cdot sinTheta\_i\right)}} \]

8. Taylor expanded in v around inf 57.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

   1. associate-/l* 57.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

10. Simplified 57.3%

    \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

11. Taylor expanded in cosTheta_O around 0 57.3%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
12. Step-by-step derivation

    1. *-commutative 57.3%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

    2. associate-*r/ 57.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]

13. Simplified 57.3%

    \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]

14. Final simplification 57.3%

    \[\leadsto \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \cdot 0.5 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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)))