HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 21.0s

Alternatives: 13

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #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) (/ 1.0 v))))
    (* (sinh (/ 1.0 v)) 2.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. times-frac 98.7%

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

   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* 98.9%

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

   9. associate-/l* 98.8%

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

3. Simplified 98.8%

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

   1. div-inv 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. times-frac 98.9%

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

6. Applied egg-rr 98.9%

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

   1. div-inv 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

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 (/ 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(Float32(cosTheta_O_m * 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 98.7%

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

   1. times-frac 98.7%

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

   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* 98.9%

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

   9. associate-/l* 98.8%

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

3. Simplified 98.8%

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

   1. div-inv 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. times-frac 98.9%

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

6. Applied egg-rr 98.9%

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

   1. /-rgt-identity 98.9%

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

   2. associate-*r/ 98.9%

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

8. Applied egg-rr 98.9%

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

Alternative 3: 98.8% accurate, 0.7× speedup

Math

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

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 (/ (/ 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(Float32(cosTheta_O_m * Float32(cosTheta_i * Float32(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 98.7%

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

   1. times-frac 98.7%

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

   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* 98.9%

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

   9. associate-/l* 98.8%

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

3. Simplified 98.8%

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

   1. div-inv 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. times-frac 98.9%

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

6. Applied egg-rr 98.9%

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

   1. /-rgt-identity 98.9%

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

   2. associate-*r/ 98.9%

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

8. Applied egg-rr 98.9%

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

   1. associate-/l* 98.9%

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

10. Simplified 98.9%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #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)))
    (* (/ 1.0 v) (* cosTheta_O_m cosTheta_i)))
   (* v (* (sinh (/ 1.0 v)) 2.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. div-inv 98.8%

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

4. Applied egg-rr 98.8%

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

5. Final simplification 98.8%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

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_i (/ cosTheta_O_m 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_i * (cosTheta_O_m / 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_i * (costheta_o_m / 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(sinTheta_i * Float32(sinTheta_O / Float32(-v)))) * Float32(cosTheta_i * Float32(cosTheta_O_m / 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_i * (cosTheta_O_m / v))) / (v * (sinh((single(1.0) / v)) * single(2.0))));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.7%

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

   3. remove-double-neg 98.7%

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

   4. distribute-rgt-neg-out 98.7%

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

   5. distribute-rgt-neg-out 98.7%

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

   6. distribute-lft-neg-in 98.7%

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

   7. associate-*r/ 98.7%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

Alternative 6: 98.6% accurate, 1.0× speedup

Math

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

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(Float32(cosTheta_O_m * 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 98.7%

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

3. Final simplification 98.7%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

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
  (/
   (* (* cosTheta_i (/ cosTheta_O_m v)) (exp (/ (* sinTheta_i sinTheta_O) 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 * (((cosTheta_i * (cosTheta_O_m / v)) * expf(((sinTheta_i * sinTheta_O) / 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 * (((costheta_i * (costheta_o_m / v)) * exp(((sintheta_i * sintheta_o) / 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(Float32(cosTheta_i * Float32(cosTheta_O_m / v)) * exp(Float32(Float32(sinTheta_i * sinTheta_O) / 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 * (((cosTheta_i * (cosTheta_O_m / v)) * exp(((sinTheta_i * sinTheta_O) / v))) / (v * (sinh((single(1.0) / v)) * single(2.0))));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.7%

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

   3. remove-double-neg 98.7%

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

   4. distribute-rgt-neg-out 98.7%

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

   5. distribute-rgt-neg-out 98.7%

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

   6. distribute-lft-neg-in 98.7%

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

   7. associate-*r/ 98.7%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. associate-*r/ 98.7%

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

   2. add-sqr-sqrt -0.0%

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

   3. sqrt-unprod 98.5%

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

   4. sqr-neg 98.5%

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

   5. sqrt-unprod 98.5%

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

   6. add-sqr-sqrt 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 8: 98.4% accurate, 1.0× speedup

Math

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

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)) (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(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)) - exp((single(-1.0) / v)))) * ((cosTheta_i * (cosTheta_O_m / v)) / v));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.7%

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

   3. remove-double-neg 98.7%

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

   4. distribute-rgt-neg-out 98.7%

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

   5. distribute-rgt-neg-out 98.7%

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

   6. distribute-lft-neg-in 98.7%

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

   7. associate-*r/ 98.7%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. *-commutative 98.7%

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

   2. times-frac 98.8%

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

   3. add-sqr-sqrt 62.9%

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

   4. add-sqr-sqrt 98.8%

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

   5. associate-*r/ 98.8%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 98.6%

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

   8. sqr-neg 98.6%

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

   9. sqrt-unprod 98.6%

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

   10. add-sqr-sqrt 98.6%

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

   11. associate-/l* 98.6%

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

   12. pow-exp 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.5%

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

   1. rec-exp 98.5%

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

   2. distribute-neg-frac 98.5%

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

   3. metadata-eval 98.5%

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

9. Simplified 98.5%

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

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \left(0.5 \cdot \frac{cosTheta\_O\_m}{{v}^{2}}\right)\right) \end{array} \]

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
  (* (/ cosTheta_i (sinh (/ 1.0 v))) (* 0.5 (/ cosTheta_O_m (pow 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 / sinhf((1.0f / v))) * (0.5f * (cosTheta_O_m / powf(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 / sinh((1.0e0 / v))) * (0.5e0 * (costheta_o_m / (v ** 2.0e0))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(cosTheta_i / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(0.5) * Float32(cosTheta_O_m / (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 / sinh((single(1.0) / v))) * (single(0.5) * (cosTheta_O_m / (v ^ single(2.0)))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

   1. associate-*l* 98.7%

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

   2. times-frac 98.7%

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

   3. *-commutative 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in v around inf 98.5%

   \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\left(0.5 \cdot \frac{cosTheta\_O}{{v}^{2}}\right)} \]
8. Add Preprocessing

Alternative 10: 59.6% accurate, 1.9× speedup

Math

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

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 cosTheta_i)
    (/ (fma 2.0 (* sinTheta_O (/ sinTheta_i v)) 2.0) cosTheta_O_m)))))

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 / cosTheta_i) * (fmaf(2.0f, (sinTheta_O * (sinTheta_i / v)), 2.0f) / cosTheta_O_m)));
}

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(v / cosTheta_i) * Float32(fma(Float32(2.0), Float32(sinTheta_O * Float32(sinTheta_i / v)), Float32(2.0)) / cosTheta_O_m))))
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 62.6%

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

   1. clear-num 63.2%

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

   2. inv-pow 63.2%

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

   3. +-commutative 63.2%

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

   4. fma-define 63.2%

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

   5. associate-/l* 63.2%

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

7. Applied egg-rr 63.2%

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

   1. unpow-1 63.2%

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

   2. times-frac 63.2%

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

9. Simplified 63.2%

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

Alternative 11: 59.1% accurate, 14.7× speedup

Math

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

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
  (*
   0.5
   (/
    (*
     cosTheta_O_m
     (+ cosTheta_i (* cosTheta_i (* sinTheta_O (/ sinTheta_i 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 * (0.5f * ((cosTheta_O_m * (cosTheta_i + (cosTheta_i * (sinTheta_O * (sinTheta_i / 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 * (0.5e0 * ((costheta_o_m * (costheta_i + (costheta_i * (sintheta_o * (sintheta_i / 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(0.5) * Float32(Float32(cosTheta_O_m * Float32(cosTheta_i + Float32(cosTheta_i * Float32(sinTheta_O * Float32(sinTheta_i / 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(0.5) * ((cosTheta_O_m * (cosTheta_i + (cosTheta_i * (sinTheta_O * (sinTheta_i / v))))) / v));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.7%

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

   3. remove-double-neg 98.7%

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

   4. distribute-rgt-neg-out 98.7%

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

   5. distribute-rgt-neg-out 98.7%

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

   6. distribute-lft-neg-in 98.7%

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

   7. associate-*r/ 98.7%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. associate-*r/ 98.7%

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

   2. add-sqr-sqrt -0.0%

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

   3. sqrt-unprod 98.5%

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

   4. sqr-neg 98.5%

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

   5. sqrt-unprod 98.5%

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

   6. add-sqr-sqrt 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in v around inf 62.7%

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

   1. distribute-lft-out 62.7%

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

   2. *-lft-identity 62.7%

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

   3. times-frac 62.7%

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

   4. metadata-eval 62.7%

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

   5. associate-/l* 62.7%

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

   6. distribute-lft-out 62.7%

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

   7. *-lft-identity 62.7%

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

   8. times-frac 62.7%

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

   9. /-rgt-identity 62.7%

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

   10. associate-/l* 62.7%

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

9. Simplified 62.7%

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

Alternative 12: 59.1% accurate, 31.4× speedup

Math

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

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_O_m cosTheta_i) v) 0.5)))

C

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

Fortran

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.7%

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

   3. remove-double-neg 98.7%

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

   4. distribute-rgt-neg-out 98.7%

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

   5. distribute-rgt-neg-out 98.7%

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

   6. distribute-lft-neg-in 98.7%

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

   7. associate-*r/ 98.7%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. *-commutative 98.7%

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

   2. times-frac 98.8%

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

   3. add-sqr-sqrt 62.9%

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

   4. add-sqr-sqrt 98.8%

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

   5. associate-*r/ 98.8%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 98.6%

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

   8. sqr-neg 98.6%

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

   9. sqrt-unprod 98.6%

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

   10. add-sqr-sqrt 98.6%

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

   11. associate-/l* 98.6%

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

   12. pow-exp 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in v around inf 62.7%

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

8. Final simplification 62.7%

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

Alternative 13: 59.1% accurate, 31.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #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)) 0.5)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.7%

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

   3. remove-double-neg 98.7%

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

   4. distribute-rgt-neg-out 98.7%

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

   5. distribute-rgt-neg-out 98.7%

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

   6. distribute-lft-neg-in 98.7%

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

   7. associate-*r/ 98.7%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. *-commutative 98.7%

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

   2. times-frac 98.8%

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

   3. add-sqr-sqrt 62.9%

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

   4. add-sqr-sqrt 98.8%

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

   5. associate-*r/ 98.8%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 98.6%

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

   8. sqr-neg 98.6%

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

   9. sqrt-unprod 98.6%

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

   10. add-sqr-sqrt 98.6%

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

   11. associate-/l* 98.6%

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

   12. pow-exp 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in v around inf 62.7%

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

   1. *-commutative 62.7%

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

   2. associate-*r/ 62.6%

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

9. Simplified 62.6%

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

10. Final simplification 62.6%

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

Reproduce

herbie shell --seed 2024090 
(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)))