HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 15.0s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (exp (* sinTheta_i (/ sinTheta_O (- v))))
   (*
    (/ cosTheta_i (sinh (/ 1.0 v)))
    (* (/ cosTheta_O_m v) (/ (/ 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 / sinhf((1.0f / v))) * ((cosTheta_O_m / v) * ((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 / sinh((1.0e0 / v))) * ((costheta_o_m / v) * ((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(Float32(sinTheta_i * Float32(sinTheta_O / Float32(-v)))) * Float32(Float32(cosTheta_i / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(cosTheta_O_m / v) * Float32(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 / sinh((single(1.0) / v))) * ((cosTheta_O_m / v) * ((single(1.0) / v) / single(2.0)))));
end

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac2 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-/l* 98.6%

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

   8. distribute-frac-neg2 98.6%

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

   9. distribute-frac-neg 98.6%

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

   10. associate-/l* 98.6%

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

   11. associate-*l* 98.6%

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

   12. times-frac 98.6%

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

   13. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. times-frac 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac2 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-/l* 98.6%

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

   8. distribute-frac-neg2 98.6%

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

   9. distribute-frac-neg 98.6%

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

   10. associate-/l* 98.6%

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

   11. associate-*l* 98.6%

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

   12. times-frac 98.6%

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

   13. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-/l/ 98.6%

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

   3. frac-times 98.6%

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

6. Applied egg-rr 98.6%

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

   1. frac-2neg 98.6%

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

   2. distribute-frac-neg 98.6%

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

   3. associate-*l* 98.7%

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

   4. distribute-lft-neg-in 98.7%

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

   5. distribute-rgt-neg-in 98.7%

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

   6. metadata-eval 98.7%

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

8. Applied egg-rr 98.7%

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

9. Final simplification 98.7%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

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_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(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(cosTheta_i * 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 98.6%

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

3. Final simplification 98.6%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac2 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-/l* 98.6%

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

   8. distribute-frac-neg2 98.6%

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

   9. distribute-frac-neg 98.6%

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

   10. associate-/l* 98.6%

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

   11. associate-*l* 98.6%

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

   12. times-frac 98.6%

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

   13. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac2 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-/l* 98.6%

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

   8. distribute-frac-neg2 98.6%

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

   9. distribute-frac-neg 98.6%

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

   10. associate-/l* 98.6%

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

   11. associate-*l* 98.6%

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

   12. times-frac 98.6%

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

   13. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. times-frac 98.8%

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

6. Applied egg-rr 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/l/ 98.8%

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

   3. *-commutative 98.8%

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

   4. frac-times 98.6%

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

   5. *-un-lft-identity 98.6%

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

8. Applied egg-rr 98.6%

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

9. Final simplification 98.6%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

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
  (*
   (exp (/ (* sinTheta_i sinTheta_O) v))
   (/ cosTheta_i (* (sinh (/ 1.0 v)) (/ (* v 2.0) (/ cosTheta_O_m v)))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac2 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-/l* 98.6%

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

   8. distribute-frac-neg2 98.6%

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

   9. distribute-frac-neg 98.6%

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

   10. associate-/l* 98.6%

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

   11. associate-*l* 98.6%

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

   12. times-frac 98.6%

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

   13. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. *-commutative 98.6%

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

   2. clear-num 98.5%

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

   3. frac-times 98.4%

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

   4. *-un-lft-identity 98.4%

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

   5. *-un-lft-identity 98.4%

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

   6. times-frac 98.4%

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

   7. /-rgt-identity 98.4%

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

6. Applied egg-rr 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*r/ 98.4%

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

8. Simplified 98.4%

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

   1. associate-*r/ 98.4%

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

   2. add-sqr-sqrt 49.1%

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

   3. sqrt-unprod 98.2%

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

   4. sqr-neg 98.2%

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

   5. sqrt-unprod 49.2%

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

   6. add-sqr-sqrt 98.2%

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

   7. *-commutative 98.2%

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

10. Applied egg-rr 98.2%

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

11. Final simplification 98.2%

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

Alternative 7: 58.5% accurate, 1.9× speedup

Math

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

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 (/ (/ 2.0 cosTheta_O_m) cosTheta_i))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac2 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-/l* 98.6%

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

   8. distribute-frac-neg2 98.6%

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

   9. distribute-frac-neg 98.6%

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

   10. associate-/l* 98.6%

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

   11. associate-*l* 98.6%

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

   12. times-frac 98.6%

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

   13. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. *-commutative 98.6%

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

   2. clear-num 98.5%

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

   3. frac-times 98.4%

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

   4. *-un-lft-identity 98.4%

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

   5. *-un-lft-identity 98.4%

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

   6. times-frac 98.4%

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

   7. /-rgt-identity 98.4%

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

6. Applied egg-rr 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*r/ 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in v around inf 56.8%

   \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \frac{cosTheta\_i}{\color{blue}{\frac{1}{v}} \cdot \frac{v \cdot 2}{\frac{cosTheta\_O}{v}}} \]
10. Step-by-step derivation

    1. clear-num 57.6%

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

    2. inv-pow 57.6%

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

    3. frac-times 57.6%

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

    4. *-un-lft-identity 57.6%

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

    5. times-frac 57.6%

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

11. Applied egg-rr 57.6%

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

    1. unpow-1 57.6%

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

    2. times-frac 57.6%

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

    3. associate-/l* 57.6%

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

    4. *-rgt-identity 57.6%

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

    5. *-commutative 57.6%

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

    6. times-frac 58.0%

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

    7. /-rgt-identity 58.0%

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

    8. *-commutative 58.0%

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

    9. associate-*l/ 58.0%

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

    10. *-lft-identity 58.0%

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

    11. times-frac 58.0%

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

    12. /-rgt-identity 58.0%

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

    13. *-inverses 58.0%

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

    14. *-rgt-identity 58.0%

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

13. Simplified 58.0%

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

14. Final simplification 58.0%

    \[\leadsto e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \frac{1}{v \cdot \frac{\frac{2}{cosTheta\_O}}{cosTheta\_i}} \]
15. Add Preprocessing

Alternative 8: 58.5% accurate, 1.9× speedup

Math

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

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 (/ (/ 2.0 cosTheta_O_m) cosTheta_i))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac2 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-/l* 98.6%

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

   8. distribute-frac-neg2 98.6%

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

   9. distribute-frac-neg 98.6%

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

   10. associate-/l* 98.6%

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

   11. associate-*l* 98.6%

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

   12. times-frac 98.6%

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

   13. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. *-commutative 98.6%

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

   2. clear-num 98.5%

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

   3. frac-times 98.4%

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

   4. *-un-lft-identity 98.4%

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

   5. *-un-lft-identity 98.4%

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

   6. times-frac 98.4%

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

   7. /-rgt-identity 98.4%

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

6. Applied egg-rr 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*r/ 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in v around inf 56.8%

   \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \frac{cosTheta\_i}{\color{blue}{\frac{1}{v}} \cdot \frac{v \cdot 2}{\frac{cosTheta\_O}{v}}} \]
10. Step-by-step derivation

    1. associate-*r/ 98.4%

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

    2. add-sqr-sqrt 49.1%

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

    3. sqrt-unprod 98.2%

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

    4. sqr-neg 98.2%

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

    5. sqrt-unprod 49.2%

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

    6. add-sqr-sqrt 98.2%

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

    7. *-commutative 98.2%

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

11. Applied egg-rr 56.8%

    \[\leadsto e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{\frac{1}{v} \cdot \frac{v \cdot 2}{\frac{cosTheta\_O}{v}}} \]
12. Step-by-step derivation

    1. clear-num 57.6%

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

    2. inv-pow 57.6%

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

    3. frac-times 57.6%

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

    4. *-un-lft-identity 57.6%

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

    5. times-frac 57.6%

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

13. Applied egg-rr 57.6%

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

    1. unpow-1 57.6%

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

    2. times-frac 57.6%

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

    3. associate-/l* 57.6%

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

    4. *-rgt-identity 57.6%

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

    5. *-commutative 57.6%

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

    6. times-frac 58.0%

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

    7. /-rgt-identity 58.0%

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

    8. *-commutative 58.0%

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

    9. associate-*l/ 58.0%

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

    10. *-lft-identity 58.0%

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

    11. times-frac 58.0%

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

    12. /-rgt-identity 58.0%

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

    13. *-inverses 58.0%

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

    14. *-rgt-identity 58.0%

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

15. Simplified 58.0%

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

16. Final simplification 58.0%

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

Alternative 9: 58.3% accurate, 24.4× speedup

Math

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

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 (/ 1.0 (/ v (* cosTheta_i 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 * (0.5f * (1.0f / (v / (cosTheta_i * cosTheta_O_m))));
}

Fortran

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r/ 98.6%

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

   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/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 56.9%

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

6. Taylor expanded in cosTheta_i around 0 56.9%

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

7. Taylor expanded in sinTheta_O around 0 56.8%

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

   1. associate-/l* 56.8%

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

9. Simplified 56.8%

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

    1. associate-*r/ 56.8%

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

    2. clear-num 57.7%

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

11. Applied egg-rr 57.7%

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

12. Final simplification 57.7%

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

Alternative 10: 57.9% accurate, 31.4× speedup

Math

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

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 0.5)) v)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r/ 98.6%

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

   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/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 56.9%

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

6. Taylor expanded in cosTheta_i around 0 56.9%

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

7. Taylor expanded in sinTheta_O around 0 56.8%

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

   1. associate-/l* 56.8%

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

9. Simplified 56.8%

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

10. Taylor expanded in cosTheta_O around 0 56.8%

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

    1. associate-/l* 56.8%

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

    2. associate-*r* 56.8%

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

    3. *-commutative 56.8%

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

    4. *-commutative 56.8%

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

    5. associate-*l/ 56.9%

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

    6. *-commutative 56.9%

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

12. Simplified 56.9%

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

13. Final simplification 56.9%

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

Alternative 11: 57.8% accurate, 31.4× speedup

Math

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

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_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(Float32(cosTheta_i * 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 98.6%

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

   1. associate-*r/ 98.6%

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

   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/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 56.9%

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

6. Taylor expanded in cosTheta_i around 0 56.9%

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

7. Taylor expanded in sinTheta_O around 0 56.8%

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

   1. *-commutative 56.8%

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

9. Simplified 56.8%

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

10. Final simplification 56.8%

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

Alternative 12: 57.8% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r/ 98.6%

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

   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/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 56.9%

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

6. Taylor expanded in cosTheta_i around 0 56.9%

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

7. Taylor expanded in sinTheta_O around 0 56.8%

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

   1. associate-/l* 56.8%

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

9. Simplified 56.8%

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

Reproduce

herbie shell --seed 2024107 
(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)))