HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 15.8s

Alternatives: 12

Speedup: 1.9×

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: 98.5% 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: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp(sinTheta_i) ^ (sinTheta_O / -v)) * ((cosTheta_O * ((cosTheta_i_m / 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 98.4%

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

   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* 98.4%

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

   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 98.7%

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

6. Applied egg-rr 98.7%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * ((single(1.0) / v) * (cosTheta_O * cosTheta_i_m))) / (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 98.6%

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

   2. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * ((cosTheta_O * cosTheta_i_m) / v)) / (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. Final simplification 98.6%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((single(1.0) - (sinTheta_i * (sinTheta_O / v))) * ((cosTheta_O * ((cosTheta_i_m * (single(1.0) / v)) / 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 98.4%

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

   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* 98.4%

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

   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. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. mul-1-neg 98.3%

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

   2. associate-*l/ 98.3%

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

   3. *-commutative 98.3%

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

   4. distribute-rgt-neg-in 98.3%

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

   5. distribute-neg-frac2 98.3%

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

7. Simplified 98.3%

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

   1. div-inv 98.5%

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

9. Applied egg-rr 98.5%

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

10. Final simplification 98.5%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((cosTheta_O * (cosTheta_i_m - ((sinTheta_i * (sinTheta_O * cosTheta_i_m)) / v))) / v) / (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. add-cube-cbrt 98.2%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{\left(\sqrt[3]{cosTheta\_i \cdot cosTheta\_O} \cdot \sqrt[3]{cosTheta\_i \cdot cosTheta\_O}\right) \cdot \sqrt[3]{cosTheta\_i \cdot cosTheta\_O}}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. pow3 98.3%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{{\left(\sqrt[3]{cosTheta\_i \cdot cosTheta\_O}\right)}^{3}}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. *-commutative 98.3%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{{\left(\sqrt[3]{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}\right)}^{3}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.3%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{{\left(\sqrt[3]{cosTheta\_O \cdot cosTheta\_i}\right)}^{3}}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Taylor expanded in v around inf 98.4%

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

   1. *-commutative 98.4%

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

   2. remove-double-neg 98.4%

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

   3. distribute-lft-neg-in 98.4%

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

   4. distribute-lft-neg-in 98.4%

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

   5. neg-mul-1 98.4%

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

   6. distribute-lft-in 98.4%

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

   7. neg-mul-1 98.4%

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

   8. *-commutative 98.4%

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

   9. +-commutative 98.4%

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

   10. associate-*r/ 98.4%

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

   11. mul-1-neg 98.4%

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

   12. distribute-neg-frac2 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 6: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((cosTheta_O * ((cosTheta_i_m / 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 98.4%

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

   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* 98.4%

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

   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 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in sinTheta_i around 0 98.3%

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

8. Final simplification 98.3%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((cosTheta_O * ((cosTheta_i_m / v) / 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 98.4%

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

   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* 98.4%

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

   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. Taylor expanded in sinTheta_i around 0 98.1%

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

6. Final simplification 98.1%

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

Alternative 8: 58.6% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 98.4%

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

   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* 98.4%

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

   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. Taylor expanded in v around inf 59.3%

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

   1. associate-*l/ 59.3%

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

   2. *-commutative 59.3%

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

7. Simplified 59.3%

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

   1. associate-*r/ 59.3%

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

   2. clear-num 60.0%

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

   3. *-commutative 60.0%

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

9. Applied egg-rr 60.0%

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

Alternative 9: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   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* 98.4%

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

   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. Taylor expanded in v around inf 59.3%

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

   1. associate-*l/ 59.3%

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

   2. *-commutative 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in cosTheta_i around 0 59.3%

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

   1. associate-*r/ 59.3%

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

   2. associate-*r* 59.3%

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

10. Simplified 59.3%

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

11. Final simplification 59.3%

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

Alternative 10: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   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* 98.4%

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

   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. Taylor expanded in v around inf 59.3%

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

   1. associate-*l/ 59.3%

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

   2. *-commutative 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in cosTheta_i around 0 59.3%

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

   1. associate-*r/ 59.3%

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

10. Simplified 59.3%

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

    1. associate-*r/ 59.3%

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

    2. *-commutative 59.3%

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

12. Applied egg-rr 59.3%

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

    1. associate-*r/ 59.3%

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

    2. *-commutative 59.3%

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

14. Applied egg-rr 59.3%

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

15. Final simplification 59.3%

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

Alternative 11: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   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* 98.4%

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

   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. Taylor expanded in v around inf 59.3%

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

   1. associate-*r/ 59.3%

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

   2. *-commutative 59.3%

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

   3. associate-/l* 59.3%

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

7. Simplified 59.3%

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

Alternative 12: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 98.4%

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

   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* 98.4%

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

   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. Taylor expanded in v around inf 59.3%

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

   1. associate-*l/ 59.3%

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

   2. *-commutative 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in cosTheta_i around 0 59.3%

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

   1. associate-*r/ 59.3%

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

10. Simplified 59.3%

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

Reproduce

herbie shell --seed 2024145 
(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)))