HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 18.8s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

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

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
 (*
  (/ (exp (/ (- sinTheta_i) (/ v sinTheta_O))) (* (sinh (/ 1.0 v)) 2.0))
  (* (/ 1.0 v) (* 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 (expf((-sinTheta_i / (v / sinTheta_O))) / (sinhf((1.0f / v)) * 2.0f)) * ((1.0f / v) * (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 = (exp((-sintheta_i / (v / sintheta_o))) / (sinh((1.0e0 / v)) * 2.0e0)) * ((1.0e0 / v) * (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(exp(Float32(Float32(-sinTheta_i) / Float32(v / sinTheta_O))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) * Float32(Float32(Float32(1.0) / v) * 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 = (exp((-sinTheta_i / (v / sinTheta_O))) / (sinh((single(1.0) / v)) * single(2.0))) * ((single(1.0) / v) * (cosTheta_i * (cosTheta_O / v)));
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   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. div-inv 98.7%

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

   2. div-inv 98.7%

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

   3. clear-num 98.8%

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

5. Applied egg-rr 98.8%

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

6. Final simplification 98.8%

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

Alternative 2: 98.5% accurate, 1.8× speedup

Math

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

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_O v)
  (*
   (/ (- cosTheta_i (* (/ cosTheta_i v) (* sinTheta_i sinTheta_O))) 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_O / v) * (((cosTheta_i - ((cosTheta_i / v) * (sinTheta_i * sinTheta_O))) / 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_o / v) * (((costheta_i - ((costheta_i / v) * (sintheta_i * sintheta_o))) / 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_O / v) * Float32(Float32(Float32(cosTheta_i - Float32(Float32(cosTheta_i / v) * Float32(sinTheta_i * sinTheta_O))) / 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_O / v) * (((cosTheta_i - ((cosTheta_i / v) * (sinTheta_i * sinTheta_O))) / v) * (single(0.5) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*r* 98.2%

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

   8. unpow2 98.2%

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

   9. times-frac 98.2%

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

   10. *-commutative 98.2%

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

   11. associate-*l/ 98.2%

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

   12. *-commutative 98.2%

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

   13. associate-/l* 98.2%

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

6. Simplified 98.2%

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

   1. expm1-log1p-u 98.2%

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

   2. expm1-udef 51.9%

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

   3. associate-*l* 51.9%

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

   4. associate-/r/ 51.9%

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

8. Applied egg-rr 51.9%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 98.2%

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

   3. associate-/l/ 98.5%

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

   4. distribute-lft-out-- 98.5%

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

   5. times-frac 98.6%

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

   6. associate-*l/ 98.6%

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

   7. times-frac 98.6%

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

   8. unpow2 98.6%

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

   9. associate-/l* 98.6%

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

   10. unpow2 98.6%

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

   11. *-commutative 98.6%

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

   12. times-frac 98.6%

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

   13. *-commutative 98.6%

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

10. Simplified 98.6%

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

    1. div-sub 98.6%

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

12. Applied egg-rr 98.6%

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

    1. div-sub 98.6%

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

    2. *-rgt-identity 98.6%

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

    3. associate-*r/ 98.5%

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

    4. *-commutative 98.5%

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

    5. associate-/r/ 98.5%

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

    6. associate-/l* 98.5%

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

    7. div-sub 98.5%

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

    8. associate-*r/ 98.5%

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

    9. associate-/l* 98.5%

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

    10. associate-*r* 98.5%

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

    11. associate-/l* 98.5%

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

    12. associate-/r/ 98.5%

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

    13. associate-/r* 98.5%

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

    14. metadata-eval 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 3: 98.5% accurate, 1.8× speedup

Math

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

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_O v)
  (/
   (- cosTheta_i (/ (* sinTheta_O cosTheta_i) (/ v sinTheta_i)))
   (* (sinh (/ 1.0 v)) (* 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 - ((sinTheta_O * cosTheta_i) / (v / sinTheta_i))) / (sinhf((1.0f / v)) * (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 - ((sintheta_o * costheta_i) / (v / sintheta_i))) / (sinh((1.0e0 / v)) * (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_O / v) * Float32(Float32(cosTheta_i - Float32(Float32(sinTheta_O * cosTheta_i) / Float32(v / sinTheta_i))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(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 - ((sinTheta_O * cosTheta_i) / (v / sinTheta_i))) / (sinh((single(1.0) / v)) * (v * single(2.0))));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*r* 98.2%

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

   8. unpow2 98.2%

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

   9. times-frac 98.2%

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

   10. *-commutative 98.2%

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

   11. associate-*l/ 98.2%

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

   12. *-commutative 98.2%

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

   13. associate-/l* 98.2%

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

6. Simplified 98.2%

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

   1. expm1-log1p-u 98.2%

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

   2. expm1-udef 51.9%

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

   3. associate-*l* 51.9%

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

   4. associate-/r/ 51.9%

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

8. Applied egg-rr 51.9%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 98.2%

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

   3. associate-/l/ 98.5%

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

   4. distribute-lft-out-- 98.5%

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

   5. associate-*l/ 98.5%

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

   6. associate-/r/ 98.5%

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

   7. *-commutative 98.5%

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

10. Simplified 98.5%

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

    1. *-un-lft-identity 98.5%

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

    2. sub-div 98.5%

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

12. Applied egg-rr 98.5%

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

    1. *-lft-identity 98.5%

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

    2. times-frac 98.6%

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

    3. associate-/r* 98.6%

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

    4. associate-*r/ 98.6%

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

    5. associate-*r* 98.6%

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

14. Simplified 98.6%

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

15. Final simplification 98.6%

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

Alternative 4: 98.4% 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 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-/l/ 98.6%

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

   4. associate-*r/ 98.6%

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

   5. *-commutative 98.6%

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

   6. /-rgt-identity 98.6%

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

   7. associate-/r/ 98.6%

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

   8. exp-neg 98.6%

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

   9. remove-double-div 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

   12. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in sinTheta_O around 0 98.5%

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

   1. unpow2 98.5%

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

   2. times-frac 98.5%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

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

Alternative 5: 63.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{cosTheta_O \cdot \left(\frac{cosTheta_i}{v} - \frac{cosTheta_i \cdot \frac{sinTheta_O}{\frac{v}{sinTheta_i}}}{v}\right)}{2 + \frac{0.3333333333333333}{v \cdot v}} \end{array} \]

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
 (/
  (*
   cosTheta_O
   (- (/ cosTheta_i v) (/ (* cosTheta_i (/ sinTheta_O (/ v sinTheta_i))) v)))
  (+ 2.0 (/ 0.3333333333333333 (* v 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_O * ((cosTheta_i / v) - ((cosTheta_i * (sinTheta_O / (v / sinTheta_i))) / v))) / (2.0f + (0.3333333333333333f / (v * 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_o * ((costheta_i / v) - ((costheta_i * (sintheta_o / (v / sintheta_i))) / v))) / (2.0e0 + (0.3333333333333333e0 / (v * 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_O * Float32(Float32(cosTheta_i / v) - Float32(Float32(cosTheta_i * Float32(sinTheta_O / Float32(v / sinTheta_i))) / v))) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * 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_O * ((cosTheta_i / v) - ((cosTheta_i * (sinTheta_O / (v / sinTheta_i))) / v))) / (single(2.0) + (single(0.3333333333333333) / (v * v)));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*r* 98.2%

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

   8. unpow2 98.2%

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

   9. times-frac 98.2%

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

   10. *-commutative 98.2%

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

   11. associate-*l/ 98.2%

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

   12. *-commutative 98.2%

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

   13. associate-/l* 98.2%

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

6. Simplified 98.2%

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

   1. expm1-log1p-u 98.2%

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

   2. expm1-udef 51.9%

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

   3. associate-*l* 51.9%

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

   4. associate-/r/ 51.9%

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

8. Applied egg-rr 51.9%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 98.2%

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

   3. associate-/l/ 98.5%

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

   4. distribute-lft-out-- 98.5%

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

   5. associate-*l/ 98.5%

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

   6. associate-/r/ 98.5%

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

   7. *-commutative 98.5%

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

10. Simplified 98.5%

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

11. Taylor expanded in v around inf 63.6%

    \[\leadsto \frac{cosTheta_O \cdot \left(\frac{cosTheta_i}{v} - \frac{cosTheta_i \cdot \frac{sinTheta_O}{\frac{v}{sinTheta_i}}}{v}\right)}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
12. Step-by-step derivation

    1. associate-*r/ 63.6%

       \[\leadsto \frac{cosTheta_O \cdot \left(\frac{cosTheta_i}{v} - \frac{cosTheta_i \cdot \frac{sinTheta_O}{\frac{v}{sinTheta_i}}}{v}\right)}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

    2. metadata-eval 63.6%

       \[\leadsto \frac{cosTheta_O \cdot \left(\frac{cosTheta_i}{v} - \frac{cosTheta_i \cdot \frac{sinTheta_O}{\frac{v}{sinTheta_i}}}{v}\right)}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

    3. unpow2 63.6%

       \[\leadsto \frac{cosTheta_O \cdot \left(\frac{cosTheta_i}{v} - \frac{cosTheta_i \cdot \frac{sinTheta_O}{\frac{v}{sinTheta_i}}}{v}\right)}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

13. Simplified 63.6%

    \[\leadsto \frac{cosTheta_O \cdot \left(\frac{cosTheta_i}{v} - \frac{cosTheta_i \cdot \frac{sinTheta_O}{\frac{v}{sinTheta_i}}}{v}\right)}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

14. Final simplification 63.6%

    \[\leadsto \frac{cosTheta_O \cdot \left(\frac{cosTheta_i}{v} - \frac{cosTheta_i \cdot \frac{sinTheta_O}{\frac{v}{sinTheta_i}}}{v}\right)}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

Alternative 6: 58.3% accurate, 12.9× speedup

Math

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

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
 (/
  (+ (* (* sinTheta_i sinTheta_O) -0.5) (* v 0.5))
  (/ v (* cosTheta_O (/ cosTheta_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 (((sinTheta_i * sinTheta_O) * -0.5f) + (v * 0.5f)) / (v / (cosTheta_O * (cosTheta_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 = (((sintheta_i * sintheta_o) * (-0.5e0)) + (v * 0.5e0)) / (v / (costheta_o * (costheta_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(Float32(sinTheta_i * sinTheta_O) * Float32(-0.5)) + Float32(v * Float32(0.5))) / Float32(v / Float32(cosTheta_O * Float32(cosTheta_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 = (((sinTheta_i * sinTheta_O) * single(-0.5)) + (v * single(0.5))) / (v / (cosTheta_O * (cosTheta_i / v)));
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   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. associate-*r/ 98.2%

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

   2. div-inv 98.2%

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

   3. clear-num 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-/l* 95.4%

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

   2. associate-/r/ 95.4%

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

   3. *-commutative 95.4%

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

   4. associate-*r/ 95.3%

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

   5. associate-*l/ 95.3%

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

   6. *-commutative 95.3%

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

7. Simplified 95.3%

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

8. Taylor expanded in v around inf 58.1%

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

9. Final simplification 58.1%

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

Alternative 7: 58.3% accurate, 20.0× speedup

Math

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

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
 (/ (* v 0.5) (/ v (* cosTheta_O (/ cosTheta_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 (v * 0.5f) / (v / (cosTheta_O * (cosTheta_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 = (v * 0.5e0) / (v / (costheta_o * (costheta_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(v * Float32(0.5)) / Float32(v / Float32(cosTheta_O * Float32(cosTheta_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 = (v * single(0.5)) / (v / (cosTheta_O * (cosTheta_i / v)));
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   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. associate-*r/ 98.2%

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

   2. div-inv 98.2%

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

   3. clear-num 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-/l* 95.4%

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

   2. associate-/r/ 95.4%

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

   3. *-commutative 95.4%

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

   4. associate-*r/ 95.3%

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

   5. associate-*l/ 95.3%

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

   6. *-commutative 95.3%

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

7. Simplified 95.3%

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

8. Taylor expanded in v around inf 58.1%

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

   1. *-commutative 58.1%

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

10. Simplified 58.1%

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

11. Final simplification 58.1%

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

Alternative 8: 58.0% accurate, 31.4× speedup

Math

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

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_O (/ cosTheta_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 * (cosTheta_O * (cosTheta_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 * (costheta_o * (costheta_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(0.5) * Float32(cosTheta_O * Float32(cosTheta_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) * (cosTheta_O * (cosTheta_i / v));
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-/l/ 98.6%

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

   4. associate-*r/ 98.6%

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

   5. *-commutative 98.6%

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

   6. /-rgt-identity 98.6%

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

   7. associate-/r/ 98.6%

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

   8. exp-neg 98.6%

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

   9. remove-double-div 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

   12. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.8%

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

   1. *-commutative 57.8%

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

   2. associate-*l/ 57.8%

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

   3. *-commutative 57.8%

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

6. Simplified 57.8%

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

7. Final simplification 57.8%

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

Alternative 9: 58.0% accurate, 31.4× speedup

Math

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

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)) 0.5))

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)) * 0.5f;
}

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)) * 0.5e0
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 / v)) * Float32(0.5))
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)) * single(0.5);
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-/l/ 98.6%

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

   4. associate-*r/ 98.6%

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

   5. *-commutative 98.6%

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

   6. /-rgt-identity 98.6%

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

   7. associate-/r/ 98.6%

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

   8. exp-neg 98.6%

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

   9. remove-double-div 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

   12. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.8%

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

   1. *-commutative 57.8%

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

   2. associate-*l/ 57.8%

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

   3. *-commutative 57.8%

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

6. Simplified 57.8%

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

7. Taylor expanded in cosTheta_O around 0 57.8%

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

   1. *-commutative 57.8%

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

   2. associate-*r/ 57.8%

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

9. Simplified 57.8%

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

10. Final simplification 57.8%

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

Alternative 10: 58.0% 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 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-/l/ 98.6%

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

   4. associate-*r/ 98.6%

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

   5. *-commutative 98.6%

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

   6. /-rgt-identity 98.6%

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

   7. associate-/r/ 98.6%

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

   8. exp-neg 98.6%

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

   9. remove-double-div 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

   12. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.8%

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

5. Final simplification 57.8%

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

Alternative 11: 58.0% 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 98.5%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 57.8%

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

5. Final simplification 57.8%

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

Alternative 12: 58.0% accurate, 31.4× speedup

Math

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

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
 (/ (* cosTheta_i (* cosTheta_O 0.5)) 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 * 0.5f)) / 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 * 0.5e0)) / 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(cosTheta_O * Float32(0.5))) / 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 * single(0.5))) / v;
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-/l/ 98.6%

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

   4. associate-*r/ 98.6%

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

   5. *-commutative 98.6%

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

   6. /-rgt-identity 98.6%

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

   7. associate-/r/ 98.6%

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

   8. exp-neg 98.6%

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

   9. remove-double-div 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

   12. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.8%

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

   1. *-commutative 57.8%

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

   2. associate-*l/ 57.8%

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

   3. *-commutative 57.8%

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

6. Simplified 57.8%

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

7. Taylor expanded in cosTheta_O around 0 57.8%

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

   1. *-commutative 57.8%

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

   2. associate-*r/ 57.8%

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

9. Simplified 57.8%

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

10. Taylor expanded in cosTheta_i around 0 57.8%

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

    1. associate-*r/ 57.8%

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

    2. *-commutative 57.8%

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

    3. *-commutative 57.8%

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

    4. associate-*l* 57.8%

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

12. Simplified 57.8%

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

13. Final simplification 57.8%

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

Reproduce

herbie shell --seed 2023279 
(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)))