HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%

Time: 15.0s

Alternatives: 12

Speedup: 1.8×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i (* (/ cosTheta_O v) (exp (* sinTheta_O (- (/ sinTheta_i v))))))
  (* (/ 1.0 v) (/ 0.5 (sinh (/ 1.0 v))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * ((cosTheta_O / v) * expf((sinTheta_O * -(sinTheta_i / v))))) * ((1.0f / v) * (0.5f / sinhf((1.0f / v))));
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = (costheta_i * ((costheta_o / v) * exp((sintheta_o * -(sintheta_i / v))))) * ((1.0e0 / v) * (0.5e0 / sinh((1.0e0 / v))))
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * Float32(Float32(cosTheta_O / v) * exp(Float32(sinTheta_O * Float32(-Float32(sinTheta_i / v)))))) * Float32(Float32(Float32(1.0) / v) * Float32(Float32(0.5) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * ((cosTheta_O / v) * exp((sinTheta_O * -(sinTheta_i / v))))) * ((single(1.0) / v) * (single(0.5) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-lft-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\left(-sinTheta_i\right) \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. frac-times 98.5%

      \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. distribute-frac-neg 98.5%

      \[\leadsto \frac{e^{\color{blue}{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/l* 98.5%

      \[\leadsto \frac{e^{-\color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-/l* 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. clear-num 94.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}}} \]

   6. *-commutative 94.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}} \]

   7. associate-/l* 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}} \]

   8. *-commutative 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}} \]

   9. div-inv 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

   10. clear-num 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

   11. associate-/l* 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{-\color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}} \]

   12. distribute-frac-neg 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}} \]

   13. associate-/r/ 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}} \]

5. Applied egg-rr 93.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}} \]
6. Step-by-step derivation

   1. associate-/r/ 98.4%

      \[\leadsto \color{blue}{\frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right)} \]

   2. *-commutative 98.4%

      \[\leadsto \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   3. associate-*l* 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right)\right)} \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   4. *-commutative 98.4%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}}\right)\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   5. distribute-frac-neg 98.4%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \color{blue}{\left(-\frac{sinTheta_i}{v}\right)}}\right)\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   6. associate-/r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   7. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\frac{1}{v}}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \]

7. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\frac{1}{v}}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \]
8. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{1}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)} \]

9. Applied egg-rr 98.6%

   \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{1}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)} \]
10. Step-by-step derivation

    1. *-un-lft-identity 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(1 \cdot \frac{1}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)}\right) \]

    2. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \left(1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \]

    3. metadata-eval 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \left(1 \cdot \frac{\color{blue}{0.5}}{\sinh \left(\frac{1}{v}\right)}\right)\right) \]

11. Applied egg-rr 98.6%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(1 \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right)}\right) \]
12. Step-by-step derivation

    1. *-lft-identity 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \color{blue}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}\right) \]

13. Simplified 98.6%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \color{blue}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}\right) \]

14. Final simplification 98.6%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right) \]

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i (* (/ cosTheta_O v) (exp (* sinTheta_O (- (/ sinTheta_i v))))))
  (/ 0.5 (* v (sinh (/ 1.0 v))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * ((cosTheta_O / v) * expf((sinTheta_O * -(sinTheta_i / v))))) * (0.5f / (v * sinhf((1.0f / v))));
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = (costheta_i * ((costheta_o / v) * exp((sintheta_o * -(sintheta_i / v))))) * (0.5e0 / (v * sinh((1.0e0 / v))))
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * Float32(Float32(cosTheta_O / v) * exp(Float32(sinTheta_O * Float32(-Float32(sinTheta_i / v)))))) * Float32(Float32(0.5) / Float32(v * sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * ((cosTheta_O / v) * exp((sinTheta_O * -(sinTheta_i / v))))) * (single(0.5) / (v * sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-lft-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\left(-sinTheta_i\right) \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. frac-times 98.5%

      \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. distribute-frac-neg 98.5%

      \[\leadsto \frac{e^{\color{blue}{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/l* 98.5%

      \[\leadsto \frac{e^{-\color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-/l* 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. clear-num 94.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}}} \]

   6. *-commutative 94.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}} \]

   7. associate-/l* 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}} \]

   8. *-commutative 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}} \]

   9. div-inv 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

   10. clear-num 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

   11. associate-/l* 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{-\color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}} \]

   12. distribute-frac-neg 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}} \]

   13. associate-/r/ 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}} \]

5. Applied egg-rr 93.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}} \]
6. Step-by-step derivation

   1. associate-/r/ 98.4%

      \[\leadsto \color{blue}{\frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right)} \]

   2. *-commutative 98.4%

      \[\leadsto \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   3. associate-*l* 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right)\right)} \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   4. *-commutative 98.4%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}}\right)\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   5. distribute-frac-neg 98.4%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \color{blue}{\left(-\frac{sinTheta_i}{v}\right)}}\right)\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   6. associate-/r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   7. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\frac{1}{v}}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \]

7. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\frac{1}{v}}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{v}}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)\right)} \]

   2. expm1-udef 96.0%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{1}{v}}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \]

   3. associate-/l/ 95.9%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v}}\right)} - 1\right) \]

   4. *-commutative 95.9%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v}\right)} - 1\right) \]

   5. *-commutative 95.9%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}}\right)} - 1\right) \]

   6. associate-*r* 95.9%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}}\right)} - 1\right) \]

9. Applied egg-rr 95.9%

   \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 98.4%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}\right)\right)} \]

    2. expm1-log1p 98.4%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{1}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

    3. *-commutative 98.4%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{1}{\color{blue}{2 \cdot \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)}} \]

    4. associate-/r* 98.4%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

    5. metadata-eval 98.4%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\color{blue}{0.5}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

11. Simplified 98.4%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

12. Final simplification 98.4%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i (* (/ cosTheta_O v) (exp (* sinTheta_O (- (/ sinTheta_i v))))))
  (/ (/ 0.5 v) (sinh (/ 1.0 v)))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * ((cosTheta_O / v) * expf((sinTheta_O * -(sinTheta_i / v))))) * ((0.5f / v) / sinhf((1.0f / v)));
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = (costheta_i * ((costheta_o / v) * exp((sintheta_o * -(sintheta_i / v))))) * ((0.5e0 / v) / sinh((1.0e0 / v)))
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * Float32(Float32(cosTheta_O / v) * exp(Float32(sinTheta_O * Float32(-Float32(sinTheta_i / v)))))) * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * ((cosTheta_O / v) * exp((sinTheta_O * -(sinTheta_i / v))))) * ((single(0.5) / v) / sinh((single(1.0) / v)));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-lft-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\left(-sinTheta_i\right) \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. frac-times 98.5%

      \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. distribute-frac-neg 98.5%

      \[\leadsto \frac{e^{\color{blue}{-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/l* 98.5%

      \[\leadsto \frac{e^{-\color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-/l* 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. clear-num 94.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}}} \]

   6. *-commutative 94.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}} \]

   7. associate-/l* 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}} \]

   8. *-commutative 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}} \]

   9. div-inv 93.9%

      \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

   10. clear-num 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

   11. associate-/l* 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{-\color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}} \]

   12. distribute-frac-neg 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}} \]

   13. associate-/r/ 93.9%

       \[\leadsto \frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\color{blue}{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}} \]

5. Applied egg-rr 93.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}} \]
6. Step-by-step derivation

   1. associate-/r/ 98.4%

      \[\leadsto \color{blue}{\frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right)} \]

   2. *-commutative 98.4%

      \[\leadsto \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   3. associate-*l* 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}\right)\right)} \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   4. *-commutative 98.4%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}}\right)\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   5. distribute-frac-neg 98.4%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \color{blue}{\left(-\frac{sinTheta_i}{v}\right)}}\right)\right) \cdot \frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

   6. associate-/r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   7. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\frac{1}{v}}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \]

7. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\frac{1}{v}}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \]
8. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{1}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)} \]

9. Applied egg-rr 98.6%

   \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{1}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)} \]
10. Step-by-step derivation

    1. *-un-lft-identity 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(1 \cdot \frac{1}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)}\right) \]

    2. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \left(1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \]

    3. metadata-eval 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \left(1 \cdot \frac{\color{blue}{0.5}}{\sinh \left(\frac{1}{v}\right)}\right)\right) \]

11. Applied egg-rr 98.6%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(1 \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right)}\right) \]
12. Step-by-step derivation

    1. *-lft-identity 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \color{blue}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}\right) \]

13. Simplified 98.6%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(\frac{1}{v} \cdot \color{blue}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}\right) \]
14. Step-by-step derivation

    1. expm1-log1p-u 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{v} \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \]

    2. expm1-udef 96.0%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{v} \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \]

    3. associate-*l/ 95.9%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)}}{v}}\right)} - 1\right) \]

    4. *-un-lft-identity 95.9%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}}{v}\right)} - 1\right) \]

15. Applied egg-rr 95.9%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}{v}\right)} - 1\right)} \]
16. Step-by-step derivation

    1. expm1-def 98.5%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}{v}\right)\right)} \]

    2. expm1-log1p 98.5%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}{v}} \]

    3. associate-/l/ 98.4%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

    4. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

17. Simplified 98.6%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \color{blue}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

18. Final simplification 98.6%

    \[\leadsto \left(cosTheta_i \cdot \left(\frac{cosTheta_O}{v} \cdot e^{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}\right)\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \]

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/
   (/ (* cosTheta_i cosTheta_O) (exp (/ (* sinTheta_O sinTheta_i) v)))
   (* v v))
  (* (sinh (/ 1.0 v)) 2.0)))

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = (((costheta_i * costheta_o) / exp(((sintheta_o * sintheta_i) / v))) / (v * v)) / (sinh((1.0e0 / v)) * 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-/l/ 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. *-commutative 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. /-rgt-identity 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{1}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-/r/ 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. exp-neg 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. remove-double-div 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   10. *-commutative 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   11. associate-*l/ 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   12. exp-prod 98.5%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in cosTheta_i around 0 98.4%

   \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

5. Final simplification 98.4%

   \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

Alternative 5: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{\frac{\left(cosTheta_i \cdot cosTheta_O - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right) - \left(sinTheta_i \cdot sinTheta_i\right) \cdot \left(\frac{cosTheta_O}{\frac{v \cdot v}{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_O\right)}} \cdot -0.5\right)}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/
   (-
    (-
     (* cosTheta_i cosTheta_O)
     (/ (* cosTheta_O (* cosTheta_i (* sinTheta_O sinTheta_i))) v))
    (*
     (* sinTheta_i sinTheta_i)
     (*
      (/ cosTheta_O (/ (* v v) (* cosTheta_i (* sinTheta_O sinTheta_O))))
      -0.5)))
   (* v v))
  (* (sinh (/ 1.0 v)) 2.0)))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((((cosTheta_i * cosTheta_O) - ((cosTheta_O * (cosTheta_i * (sinTheta_O * sinTheta_i))) / v)) - ((sinTheta_i * sinTheta_i) * ((cosTheta_O / ((v * v) / (cosTheta_i * (sinTheta_O * sinTheta_O)))) * -0.5f))) / (v * v)) / (sinhf((1.0f / v)) * 2.0f);
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = ((((costheta_i * costheta_o) - ((costheta_o * (costheta_i * (sintheta_o * sintheta_i))) / v)) - ((sintheta_i * sintheta_i) * ((costheta_o / ((v * v) / (costheta_i * (sintheta_o * sintheta_o)))) * (-0.5e0)))) / (v * v)) / (sinh((1.0e0 / v)) * 2.0e0)
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(Float32(cosTheta_O * Float32(cosTheta_i * Float32(sinTheta_O * sinTheta_i))) / v)) - Float32(Float32(sinTheta_i * sinTheta_i) * Float32(Float32(cosTheta_O / Float32(Float32(v * v) / Float32(cosTheta_i * Float32(sinTheta_O * sinTheta_O)))) * Float32(-0.5)))) / Float32(v * v)) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((((cosTheta_i * cosTheta_O) - ((cosTheta_O * (cosTheta_i * (sinTheta_O * sinTheta_i))) / v)) - ((sinTheta_i * sinTheta_i) * ((cosTheta_O / ((v * v) / (cosTheta_i * (sinTheta_O * sinTheta_O)))) * single(-0.5)))) / (v * v)) / (sinh((single(1.0) / v)) * single(2.0));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-/l/ 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. *-commutative 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. /-rgt-identity 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{1}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-/r/ 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. exp-neg 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. remove-double-div 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   10. *-commutative 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   11. associate-*l/ 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   12. exp-prod 98.5%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in sinTheta_i around 0 98.4%

   \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left({sinTheta_i}^{2} \cdot \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}}\right)\right) + \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v} + cosTheta_O \cdot cosTheta_i\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
5. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v} + cosTheta_O \cdot cosTheta_i\right) + -1 \cdot \left({sinTheta_i}^{2} \cdot \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}}\right)\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. mul-1-neg 98.4%

      \[\leadsto \frac{\frac{\left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v} + cosTheta_O \cdot cosTheta_i\right) + \color{blue}{\left(-{sinTheta_i}^{2} \cdot \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}}\right)\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. unsub-neg 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v} + cosTheta_O \cdot cosTheta_i\right) - {sinTheta_i}^{2} \cdot \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}}\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. +-commutative 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_O \cdot cosTheta_i + -1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right)} - {sinTheta_i}^{2} \cdot \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}}\right)}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. mul-1-neg 98.4%

      \[\leadsto \frac{\frac{\left(cosTheta_O \cdot cosTheta_i + \color{blue}{\left(-\frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right)}\right) - {sinTheta_i}^{2} \cdot \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}}\right)}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. unsub-neg 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right)} - {sinTheta_i}^{2} \cdot \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}}\right)}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. unpow2 98.4%

      \[\leadsto \frac{\frac{\left(cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right) - \color{blue}{\left(sinTheta_i \cdot sinTheta_i\right)} \cdot \left(-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} + 0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}}\right)}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. distribute-rgt-out 98.4%

      \[\leadsto \frac{\frac{\left(cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right) - \left(sinTheta_i \cdot sinTheta_i\right) \cdot \color{blue}{\left(\frac{cosTheta_O \cdot \left(cosTheta_i \cdot {sinTheta_O}^{2}\right)}{{v}^{2}} \cdot \left(-1 + 0.5\right)\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Simplified 98.4%

   \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right) - \left(sinTheta_i \cdot sinTheta_i\right) \cdot \left(\frac{cosTheta_O}{\frac{v \cdot v}{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_O\right)}} \cdot -0.5\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

7. Final simplification 98.4%

   \[\leadsto \frac{\frac{\left(cosTheta_i \cdot cosTheta_O - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right) - \left(sinTheta_i \cdot sinTheta_i\right) \cdot \left(\frac{cosTheta_O}{\frac{v \cdot v}{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_O\right)}} \cdot -0.5\right)}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

Alternative 6: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_O}{v \cdot v} \cdot \left(cosTheta_i - \frac{sinTheta_O \cdot \left(cosTheta_i \cdot sinTheta_i\right)}{v}\right)\right) \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 (sinh (/ 1.0 v)))
  (*
   (/ cosTheta_O (* v v))
   (- cosTheta_i (/ (* sinTheta_O (* cosTheta_i sinTheta_i)) v)))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / sinhf((1.0f / v))) * ((cosTheta_O / (v * v)) * (cosTheta_i - ((sinTheta_O * (cosTheta_i * sinTheta_i)) / v)));
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = (0.5e0 / sinh((1.0e0 / v))) * ((costheta_o / (v * v)) * (costheta_i - ((sintheta_o * (costheta_i * sintheta_i)) / v)))
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(cosTheta_O / Float32(v * v)) * Float32(cosTheta_i - Float32(Float32(sinTheta_O * Float32(cosTheta_i * sinTheta_i)) / v))))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / sinh((single(1.0) / v))) * ((cosTheta_O / (v * v)) * (cosTheta_i - ((sinTheta_O * (cosTheta_i * sinTheta_i)) / v)));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-/l/ 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. *-commutative 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. /-rgt-identity 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{1}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-/r/ 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. exp-neg 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. remove-double-div 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   10. *-commutative 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   11. associate-*l/ 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   12. exp-prod 98.5%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in sinTheta_O around 0 98.4%

   \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v} + cosTheta_O \cdot cosTheta_i}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
5. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i + -1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. mul-1-neg 98.4%

      \[\leadsto \frac{\frac{cosTheta_O \cdot cosTheta_i + \color{blue}{\left(-\frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. unsub-neg 98.4%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Simplified 98.4%

   \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

7. Taylor expanded in cosTheta_O around 0 98.4%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta_O \cdot \left(cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}\right)}{{v}^{2}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
8. Step-by-step derivation

   1. unpow2 98.4%

      \[\leadsto \frac{\frac{cosTheta_O \cdot \left(cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}\right)}{\color{blue}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. times-frac 98.4%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{\color{blue}{\left(sinTheta_O \cdot sinTheta_i\right) \cdot cosTheta_i}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*l* 98.4%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{\color{blue}{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

9. Simplified 98.4%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
10. Step-by-step derivation

    1. expm1-log1p-u 98.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right)\right)} \]

    2. expm1-udef 49.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right)} - 1} \]

    3. times-frac 49.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}{v}}{v}}{2}}\right)} - 1 \]

    4. associate-/l* 49.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i \cdot cosTheta_i}}}}{v}}{2}\right)} - 1 \]

    5. *-commutative 49.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{\color{blue}{cosTheta_i \cdot sinTheta_i}}}}{v}}{2}\right)} - 1 \]

11. Applied egg-rr 49.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{2}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 98.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{2}\right)\right)} \]

    2. expm1-log1p 98.4%

       \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{2}} \]

    3. associate-*r/ 98.3%

       \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{2}} \]

    4. associate-*l/ 98.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{\sinh \left(\frac{1}{v}\right)}}}{2} \]

    5. associate-/r* 98.4%

       \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

    6. *-rgt-identity 98.4%

       \[\leadsto \frac{\color{blue}{\left(\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}\right) \cdot 1}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

    7. associate-*r/ 98.3%

       \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

    8. *-commutative 98.3%

       \[\leadsto \color{blue}{\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}\right)} \]

    9. *-commutative 98.3%

       \[\leadsto \frac{1}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \cdot \left(\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}\right) \]

    10. associate-/r* 98.3%

        \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sinh \left(\frac{1}{v}\right)}} \cdot \left(\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}\right) \]

    11. metadata-eval 98.3%

        \[\leadsto \frac{\color{blue}{0.5}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}\right) \]

    12. associate-*r/ 98.3%

        \[\leadsto \frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot \left(cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}\right)}{v}} \]

    13. associate-/l* 98.3%

        \[\leadsto \frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{\frac{cosTheta_O}{v}}{\frac{v}{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}}} \]

    14. associate-/r/ 98.3%

        \[\leadsto \frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\left(\frac{\frac{cosTheta_O}{v}}{v} \cdot \left(cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}\right)\right)} \]

13. Simplified 98.2%

    \[\leadsto \color{blue}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_O}{v \cdot v} \cdot \left(cosTheta_i - \frac{sinTheta_O \cdot \left(cosTheta_i \cdot sinTheta_i\right)}{v}\right)\right)} \]

14. Final simplification 98.2%

    \[\leadsto \frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_O}{v \cdot v} \cdot \left(cosTheta_i - \frac{sinTheta_O \cdot \left(cosTheta_i \cdot sinTheta_i\right)}{v}\right)\right) \]

Alternative 7: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{2} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (/ cosTheta_O v) (sinh (/ 1.0 v)))
  (/ (/ (- cosTheta_i (/ sinTheta_O (/ v (* cosTheta_i sinTheta_i)))) v) 2.0)))

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = ((costheta_o / v) / sinh((1.0e0 / v))) * (((costheta_i - (sintheta_o / (v / (costheta_i * sintheta_i)))) / v) / 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-/l/ 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. *-commutative 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. /-rgt-identity 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{1}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-/r/ 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. exp-neg 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. remove-double-div 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   10. *-commutative 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   11. associate-*l/ 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   12. exp-prod 98.5%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in sinTheta_O around 0 98.4%

   \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v} + cosTheta_O \cdot cosTheta_i}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
5. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i + -1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. mul-1-neg 98.4%

      \[\leadsto \frac{\frac{cosTheta_O \cdot cosTheta_i + \color{blue}{\left(-\frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. unsub-neg 98.4%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Simplified 98.4%

   \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

7. Taylor expanded in cosTheta_O around 0 98.4%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta_O \cdot \left(cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}\right)}{{v}^{2}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
8. Step-by-step derivation

   1. unpow2 98.4%

      \[\leadsto \frac{\frac{cosTheta_O \cdot \left(cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}\right)}{\color{blue}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. times-frac 98.4%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{\color{blue}{\left(sinTheta_O \cdot sinTheta_i\right) \cdot cosTheta_i}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*l* 98.4%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{\color{blue}{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

9. Simplified 98.4%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
10. Step-by-step derivation

    1. times-frac 98.4%

       \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}{v}}{v}}{2}} \]

    2. associate-/l* 98.4%

       \[\leadsto \frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i \cdot cosTheta_i}}}}{v}}{2} \]

    3. *-commutative 98.4%

       \[\leadsto \frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{\color{blue}{cosTheta_i \cdot sinTheta_i}}}}{v}}{2} \]

11. Applied egg-rr 98.4%

    \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{2}} \]

12. Final simplification 98.4%

    \[\leadsto \frac{\frac{cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_i - \frac{sinTheta_O}{\frac{v}{cosTheta_i \cdot sinTheta_i}}}{v}}{2} \]

Alternative 8: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{cosTheta_i \cdot \frac{cosTheta_O}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_i (/ cosTheta_O (* v v))) (* (sinh (/ 1.0 v)) 2.0)))

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = (costheta_i * (costheta_o / (v * v))) / (sinh((1.0e0 / v)) * 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-/l/ 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. *-commutative 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. /-rgt-identity 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{1}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-/r/ 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. exp-neg 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. remove-double-div 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   10. *-commutative 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   11. associate-*l/ 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   12. exp-prod 98.5%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in sinTheta_O around 0 98.4%

   \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v} + cosTheta_O \cdot cosTheta_i}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
5. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i + -1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. mul-1-neg 98.4%

      \[\leadsto \frac{\frac{cosTheta_O \cdot cosTheta_i + \color{blue}{\left(-\frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. unsub-neg 98.4%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Simplified 98.4%

   \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

7. Taylor expanded in cosTheta_O around 0 98.4%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta_O \cdot \left(cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}\right)}{{v}^{2}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
8. Step-by-step derivation

   1. unpow2 98.4%

      \[\leadsto \frac{\frac{cosTheta_O \cdot \left(cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}\right)}{\color{blue}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. times-frac 98.4%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{\color{blue}{\left(sinTheta_O \cdot sinTheta_i\right) \cdot cosTheta_i}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*l* 98.4%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{\color{blue}{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

9. Simplified 98.4%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i - \frac{sinTheta_O \cdot \left(sinTheta_i \cdot cosTheta_i\right)}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

10. Taylor expanded in v around inf 97.8%

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

    1. associate-/l* 97.8%

       \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{\frac{{v}^{2}}{cosTheta_i}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

    2. associate-/r/ 97.8%

       \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{{v}^{2}} \cdot cosTheta_i}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

    3. unpow2 97.8%

       \[\leadsto \frac{\frac{cosTheta_O}{\color{blue}{v \cdot v}} \cdot cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

12. Simplified 97.8%

    \[\leadsto \frac{\color{blue}{\frac{cosTheta_O}{v \cdot v} \cdot cosTheta_i}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

13. Final simplification 97.8%

    \[\leadsto \frac{cosTheta_i \cdot \frac{cosTheta_O}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

Alternative 9: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* (/ cosTheta_O v) (/ cosTheta_i v)) (* (sinh (/ 1.0 v)) 2.0)))

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = ((costheta_o / v) * (costheta_i / v)) / (sinh((1.0e0 / v)) * 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-/l/ 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. *-commutative 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. /-rgt-identity 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{1}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-/r/ 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. exp-neg 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. remove-double-div 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   10. *-commutative 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   11. associate-*l/ 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   12. exp-prod 98.5%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in sinTheta_O around 0 98.4%

   \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v} + cosTheta_O \cdot cosTheta_i}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
5. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i + -1 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. mul-1-neg 98.4%

      \[\leadsto \frac{\frac{cosTheta_O \cdot cosTheta_i + \color{blue}{\left(-\frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}\right)}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. unsub-neg 98.4%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Simplified 98.4%

   \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i - \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)\right)}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

7. Taylor expanded in sinTheta_O around 0 97.8%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
8. Step-by-step derivation

   1. *-commutative 97.8%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{{v}^{2}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. unpow2 97.8%

      \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. times-frac 97.9%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

9. Simplified 97.9%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

10. Final simplification 97.9%

    \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

Alternative 10: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ 0.5 \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* cosTheta_i (/ cosTheta_O v))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_i * (cosTheta_O / v));
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = 0.5e0 * (costheta_i * (costheta_o / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-/l/ 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. *-commutative 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. /-rgt-identity 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{1}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-/r/ 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. exp-neg 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. remove-double-div 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   10. *-commutative 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   11. associate-*l/ 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   12. exp-prod 98.5%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in v around inf 55.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. associate-/l* 55.3%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]

6. Simplified 55.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]
7. Step-by-step derivation

   1. associate-/r/ 55.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \]

8. Applied egg-rr 55.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \]

9. Final simplification 55.3%

   \[\leadsto 0.5 \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \]

Alternative 11: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ 0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (* cosTheta_i cosTheta_O) v)))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * ((cosTheta_i * cosTheta_O) / v);
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = 0.5e0 * ((costheta_i * costheta_o) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-/l/ 98.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. *-commutative 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. /-rgt-identity 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{1}} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-/r/ 98.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. exp-neg 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. remove-double-div 98.4%

      \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   10. *-commutative 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   11. associate-*l/ 98.4%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   12. exp-prod 98.5%

       \[\leadsto \frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}{v \cdot v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in v around inf 55.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]

5. Final simplification 55.3%

   \[\leadsto 0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]

Alternative 12: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{0.5 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (* cosTheta_i cosTheta_O)) v))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * (cosTheta_i * cosTheta_O)) / v;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
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 = (0.5e0 * (costheta_i * costheta_o)) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{v}} \]

4. Taylor expanded in v around inf 55.4%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \left(cosTheta_O \cdot cosTheta_i\right)}}{v} \]

5. Final simplification 55.4%

   \[\leadsto \frac{0.5 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v} \]

Reproduce

herbie shell --seed 2023271 
(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)))