HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 18.5s

Alternatives: 15

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: 98.8% accurate, 0.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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 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 / 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 / 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(Float32(cosTheta_i / 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(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 / v) * (single(1.0) / v))) / (sinh((single(1.0) / v)) * single(2.0))));
end

Derivation

1. Initial program 98.8%

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

   1. times-frac 98.8%

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

   2. associate-*l/ 98.7%

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

   3. associate-*r/ 98.7%

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

   4. distribute-frac-neg2 98.7%

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

   5. associate-/l* 98.7%

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

   6. exp-prod 98.7%

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

   7. *-commutative 98.7%

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

   8. associate-/l* 98.8%

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

   9. associate-/l* 98.9%

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

3. Simplified 98.9%

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

   1. div-inv 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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)) 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)) / 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)) / 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(cosTheta_O_m * Float32(Float32(Float32(cosTheta_i * Float32(Float32(1.0) / v)) / v) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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)) / v) / (sinh((single(1.0) / v)) * single(2.0)))));
end

Derivation

1. Initial program 98.8%

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

   1. times-frac 98.8%

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

   2. associate-*l/ 98.7%

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

   3. associate-*r/ 98.7%

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

   4. distribute-frac-neg2 98.7%

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

   5. associate-/l* 98.7%

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

   6. exp-prod 98.7%

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

   7. *-commutative 98.7%

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

   8. associate-/l* 98.8%

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

   9. associate-/l* 98.9%

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

3. Simplified 98.9%

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

   1. associate-/l* 98.9%

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

   2. associate-/l/ 98.9%

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

   3. pow2 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. unpow2 98.9%

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

   3. times-frac 98.9%

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

8. Applied egg-rr 98.9%

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

   1. associate-*r/ 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 3: 98.6% accurate, 0.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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 (sinh (/ 1.0 v))) (/ (/ (/ cosTheta_i v) 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 / sinhf((1.0f / v))) * (((cosTheta_i / v) / 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 / sinh((1.0e0 / v))) * (((costheta_i / v) / 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 / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(Float32(cosTheta_i / v) / v) / Float32(2.0)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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 / sinh((single(1.0) / v))) * (((cosTheta_i / v) / v) / single(2.0))));
end

Derivation

1. Initial program 98.8%

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

   1. times-frac 98.8%

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

   2. associate-*l/ 98.7%

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

   3. associate-*r/ 98.7%

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

   4. distribute-frac-neg2 98.7%

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

   5. associate-/l* 98.7%

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

   6. exp-prod 98.7%

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

   7. *-commutative 98.7%

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

   8. associate-/l* 98.8%

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

   9. associate-/l* 98.9%

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

3. Simplified 98.9%

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

   1. div-inv 99.1%

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

6. Applied egg-rr 99.1%

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

   1. times-frac 99.0%

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

   2. un-div-inv 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 4: 98.6% accurate, 0.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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 (* (sinh (/ 1.0 v)) (* v (* v 2.0))))
   (/ cosTheta_O_m (pow (exp sinTheta_i) (/ sinTheta_O 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_i / (sinhf((1.0f / v)) * (v * (v * 2.0f)))) * (cosTheta_O_m / powf(expf(sinTheta_i), (sinTheta_O / 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 * ((costheta_i / (sinh((1.0e0 / v)) * (v * (v * 2.0e0)))) * (costheta_o_m / (exp(sintheta_i) ** (sintheta_o / 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(cosTheta_i / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0))))) * Float32(cosTheta_O_m / (exp(sinTheta_i) ^ Float32(sinTheta_O / v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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 / (sinh((single(1.0) / v)) * (v * (v * single(2.0))))) * (cosTheta_O_m / (exp(sinTheta_i) ^ (sinTheta_O / v))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. times-frac 98.8%

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

   2. *-commutative 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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)))
    (* cosTheta_O_m (/ cosTheta_i v)))
   (* 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)) * (cosTheta_O_m * (cosTheta_i / v))) / (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 / v))) / (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(cosTheta_O_m * Float32(cosTheta_i / v))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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 / v))) / (v * (sinh((single(1.0) / v)) * single(2.0))));
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in cosTheta_i around 0 98.8%

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

   1. associate-*r/ 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
   (* (/ 1.0 v) (* 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 / (expf((1.0f / v)) - expf((-1.0f / v)))) * ((1.0f / v) * (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 / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * ((1.0e0 / v) * (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(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * Float32(cosTheta_O_m / v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * ((single(1.0) / v) * (cosTheta_i * (cosTheta_O_m / v))));
end

Derivation

1. Initial program 98.8%

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

   1. associate-*r/ 98.8%

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

   2. frac-times 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 98.6%

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

   6. sqr-neg 98.6%

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

   7. sqrt-unprod 98.6%

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

   8. add-sqr-sqrt 98.6%

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

   9. associate-/l* 98.6%

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

   10. pow-exp 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]
6. Step-by-step derivation

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

7. Simplified 98.6%

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

   1. div-inv 98.8%

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 7: 98.3% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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 (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
   (* cosTheta_i (/ (/ cosTheta_O_m v) 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 / (expf((1.0f / v)) - expf((-1.0f / v)))) * (cosTheta_i * ((cosTheta_O_m / v) / 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 / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * (costheta_i * ((costheta_o_m / v) / 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(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) * Float32(cosTheta_i * Float32(Float32(cosTheta_O_m / v) / v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * (cosTheta_i * ((cosTheta_O_m / v) / v)));
end

Derivation

1. Initial program 98.8%

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

   1. associate-*r/ 98.8%

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

   2. frac-times 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 98.6%

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

   6. sqr-neg 98.6%

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

   7. sqrt-unprod 98.6%

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

   8. add-sqr-sqrt 98.6%

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

   9. associate-/l* 98.6%

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

   10. pow-exp 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]
6. Step-by-step derivation

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

7. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 8: 68.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(\frac{1}{e^{\frac{1}{v}} + \left(-1 + \frac{1 - \frac{0.5 + \frac{-0.16666666666666666}{v}}{v}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O\_m}{v}}{v}\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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
    (+
     (exp (/ 1.0 v))
     (+ -1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.16666666666666666 v)) v)) v))))
   (/ (* cosTheta_i (/ cosTheta_O_m v)) 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 / (expf((1.0f / v)) + (-1.0f + ((1.0f - ((0.5f + (-0.16666666666666666f / v)) / v)) / v)))) * ((cosTheta_i * (cosTheta_O_m / v)) / 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 / (exp((1.0e0 / v)) + ((-1.0e0) + ((1.0e0 - ((0.5e0 + ((-0.16666666666666666e0) / v)) / v)) / v)))) * ((costheta_i * (costheta_o_m / v)) / 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(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(Float32(-1.0) + Float32(Float32(Float32(1.0) - Float32(Float32(Float32(0.5) + Float32(Float32(-0.16666666666666666) / v)) / v)) / v)))) * Float32(Float32(cosTheta_i * Float32(cosTheta_O_m / v)) / v)))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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) / (exp((single(1.0) / v)) + (single(-1.0) + ((single(1.0) - ((single(0.5) + (single(-0.16666666666666666) / v)) / v)) / v)))) * ((cosTheta_i * (cosTheta_O_m / v)) / v));
end

Derivation

1. Initial program 98.8%

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

   1. associate-*r/ 98.8%

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

   2. frac-times 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 98.6%

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

   6. sqr-neg 98.6%

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

   7. sqrt-unprod 98.6%

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

   8. add-sqr-sqrt 98.6%

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

   9. associate-/l* 98.6%

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

   10. pow-exp 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]
6. Step-by-step derivation

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in v around -inf 69.9%

   \[\leadsto \frac{1}{e^{\frac{1}{v}} - \color{blue}{\left(1 + -1 \cdot \frac{1 + -1 \cdot \frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]
9. Step-by-step derivation

   1. mul-1-neg 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \left(1 + \color{blue}{\left(-\frac{1 + -1 \cdot \frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

   2. unsub-neg 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \color{blue}{\left(1 - \frac{1 + -1 \cdot \frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

   3. mul-1-neg 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \left(1 - \frac{1 + \color{blue}{\left(-\frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}\right)}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

   4. unsub-neg 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \left(1 - \frac{\color{blue}{1 - \frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

   5. sub-neg 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{\color{blue}{0.5 + \left(-0.16666666666666666 \cdot \frac{1}{v}\right)}}{v}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

   6. associate-*r/ 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{0.5 + \left(-\color{blue}{\frac{0.16666666666666666 \cdot 1}{v}}\right)}{v}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

   7. metadata-eval 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{0.5 + \left(-\frac{\color{blue}{0.16666666666666666}}{v}\right)}{v}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

   8. distribute-neg-frac 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{0.5 + \color{blue}{\frac{-0.16666666666666666}{v}}}{v}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

   9. metadata-eval 69.9%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{0.5 + \frac{\color{blue}{-0.16666666666666666}}{v}}{v}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

10. Simplified 69.9%

    \[\leadsto \frac{1}{e^{\frac{1}{v}} - \color{blue}{\left(1 - \frac{1 - \frac{0.5 + \frac{-0.16666666666666666}{v}}{v}}{v}\right)}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]

11. Final simplification 69.9%

    \[\leadsto \frac{1}{e^{\frac{1}{v}} + \left(-1 + \frac{1 - \frac{0.5 + \frac{-0.16666666666666666}{v}}{v}}{v}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]
12. Add Preprocessing

Alternative 9: 67.8% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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)) v)
   (/ 1.0 (+ (exp (/ 1.0 v)) (+ (/ 1.0 v) -1.0))))))

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_i * (cosTheta_O_m / v)) / v) * (1.0f / (expf((1.0f / v)) + ((1.0f / v) + -1.0f))));
}

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_i * (costheta_o_m / v)) / v) * (1.0e0 / (exp((1.0e0 / v)) + ((1.0e0 / v) + (-1.0e0)))))
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(cosTheta_i * Float32(cosTheta_O_m / v)) / v) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(Float32(Float32(1.0) / v) + Float32(-1.0))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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)) / v) * (single(1.0) / (exp((single(1.0) / v)) + ((single(1.0) / v) + single(-1.0)))));
end

Derivation

1. Initial program 98.8%

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

   1. associate-*r/ 98.8%

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

   2. frac-times 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 98.6%

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

   6. sqr-neg 98.6%

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

   7. sqrt-unprod 98.6%

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

   8. add-sqr-sqrt 98.6%

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

   9. associate-/l* 98.6%

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

   10. pow-exp 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]
6. Step-by-step derivation

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in v around inf 68.6%

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

9. Final simplification 68.6%

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

Alternative 10: 64.4% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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)) v)
   (/ 1.0 (/ (+ 2.0 (/ 0.3333333333333333 (pow 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) {
	return cosTheta_O_s * (((cosTheta_i * (cosTheta_O_m / v)) / v) * (1.0f / ((2.0f + (0.3333333333333333f / powf(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
    code = costheta_o_s * (((costheta_i * (costheta_o_m / v)) / v) * (1.0e0 / ((2.0e0 + (0.3333333333333333e0 / (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)
	return Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_i * Float32(cosTheta_O_m / v)) / v) * Float32(Float32(1.0) / Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / (v ^ Float32(2.0)))) / v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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)) / v) * (single(1.0) / ((single(2.0) + (single(0.3333333333333333) / (v ^ single(2.0)))) / v)));
end

Derivation

1. Initial program 98.8%

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

   1. associate-*r/ 98.8%

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

   2. frac-times 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 98.6%

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

   6. sqr-neg 98.6%

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

   7. sqrt-unprod 98.6%

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

   8. add-sqr-sqrt 98.6%

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

   9. associate-/l* 98.6%

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

   10. pow-exp 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v} \]
6. Step-by-step derivation

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in v around inf 66.4%

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

   1. associate-*r/ 66.4%

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

   2. metadata-eval 66.4%

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

10. Simplified 66.4%

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

11. Final simplification 66.4%

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

Alternative 11: 59.4% accurate, 24.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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 (/ 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 * (1.0f / (2.0f * (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 * (1.0e0 / (2.0e0 * (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(1.0) / Float32(Float32(2.0) * Float32(v / Float32(cosTheta_O_m * cosTheta_i)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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) * (v / (cosTheta_O_m * cosTheta_i))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around inf 60.5%

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

6. Taylor expanded in v around inf 60.5%

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

   1. *-commutative 60.5%

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

8. Simplified 60.5%

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

   1. clear-num 60.9%

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

   2. inv-pow 60.9%

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

   3. *-commutative 60.9%

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

10. Applied egg-rr 60.9%

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

    1. unpow-1 60.9%

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

    2. *-commutative 60.9%

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

    3. associate-*r/ 60.9%

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

12. Simplified 60.9%

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

13. Final simplification 60.9%

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

Alternative 12: 59.5% accurate, 24.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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 (/ 2.0 (* 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 * (1.0f / (v * (2.0f / (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 * (1.0e0 / (v * (2.0e0 / (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(1.0) / Float32(v * Float32(Float32(2.0) / Float32(cosTheta_O_m * cosTheta_i)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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 * (single(2.0) / (cosTheta_O_m * cosTheta_i))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around inf 60.5%

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

   1. clear-num 60.9%

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

   2. inv-pow 60.9%

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

   3. +-commutative 60.9%

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

   4. *-commutative 60.9%

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

   5. fma-define 60.9%

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

   6. *-commutative 60.9%

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

   7. associate-/l* 60.9%

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

7. Applied egg-rr 60.9%

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

   1. unpow-1 60.9%

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

   2. associate-/l* 60.9%

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

9. Simplified 60.9%

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

10. Taylor expanded in sinTheta_O around 0 60.9%

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

11. Final simplification 60.9%

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

Alternative 13: 59.5% accurate, 24.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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) (/ 2.0 (* 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 * ((1.0f / v) / (2.0f / (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 * ((1.0e0 / v) / (2.0e0 / (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(Float32(1.0) / v) / Float32(Float32(2.0) / Float32(cosTheta_O_m * cosTheta_i))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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) / (single(2.0) / (cosTheta_O_m * cosTheta_i)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around inf 60.5%

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

   1. clear-num 60.9%

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

   2. inv-pow 60.9%

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

   3. +-commutative 60.9%

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

   4. *-commutative 60.9%

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

   5. fma-define 60.9%

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

   6. *-commutative 60.9%

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

   7. associate-/l* 60.9%

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

7. Applied egg-rr 60.9%

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

   1. unpow-1 60.9%

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

   2. associate-/l* 60.9%

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

   3. associate-/r* 60.9%

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

9. Simplified 60.9%

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

10. Taylor expanded in sinTheta_O around 0 60.9%

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

11. Final simplification 60.9%

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

Alternative 14: 58.8% accurate, 31.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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 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_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 * (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(cosTheta_i * Float32(cosTheta_O_m / Float32(v * Float32(2.0)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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(2.0))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around inf 60.5%

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

6. Taylor expanded in v around inf 60.5%

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

   1. *-commutative 60.5%

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

8. Simplified 60.5%

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

   1. associate-/l* 60.5%

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

10. Applied egg-rr 60.5%

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

11. Final simplification 60.5%

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

Alternative 15: 58.8% accurate, 31.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) 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) (* 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) / (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) / (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(v * Float32(2.0))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(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) / (v * single(2.0)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around inf 60.5%

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

6. Taylor expanded in v around inf 60.5%

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

   1. *-commutative 60.5%

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

8. Simplified 60.5%

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

9. Final simplification 60.5%

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

Reproduce

herbie shell --seed 2024130 
(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)))