HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 16.4s

Alternatives: 13

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (- (/ (* sinTheta_i sinTheta_O) v)))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (* (sinh (/ 1.0 v)) (* 2.0 (/ 1.0 (/ 1.0 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)) * ((1.0f / v) * (cosTheta_i * cosTheta_O))) / (sinhf((1.0f / v)) * (2.0f * (1.0f / (1.0f / 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)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / (sinh((1.0e0 / v)) * (2.0e0 * (1.0e0 / (1.0e0 / 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(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(Float32(1.0) / v)))))
end

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. div-inv 98.8%

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

3. Applied egg-rr 98.8%

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

   1. add-exp-log 98.7%

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

   2. associate-*l* 98.7%

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

5. Applied egg-rr 98.7%

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

   1. add-exp-log 98.8%

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

   2. add-log-exp 93.4%

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

   3. *-commutative 93.4%

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

   4. exp-prod 93.4%

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

   5. *-commutative 93.4%

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

   6. exp-prod 93.3%

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

7. Applied egg-rr 93.3%

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

   1. log-pow 98.6%

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

   2. log-pow 98.7%

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

9. Simplified 98.7%

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

    1. add-log-exp 98.8%

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

    2. /-rgt-identity 98.8%

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

    3. clear-num 98.8%

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

11. Applied egg-rr 98.8%

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

12. Final simplification 98.8%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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)) * ((cosTheta_O * (cosTheta_i * (1.0f / v))) / 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 / (v / sintheta_o))) / (sinh((1.0e0 / v)) * 2.0e0)) * ((costheta_o * (costheta_i * (1.0e0 / v))) / v)
end function

Julia

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(cosTheta_O * Float32(cosTheta_i * Float32(Float32(1.0) / v))) / v))
end

MATLAB

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))) * ((cosTheta_O * (cosTheta_i * (single(1.0) / v))) / v);
end

Derivation

1. Initial program 98.6%

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

   1. 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* 98.5%

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

3. Simplified 98.5%

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

   1. *-un-lft-identity 98.5%

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

   2. div-inv 98.5%

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

   3. clear-num 98.5%

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

5. Applied egg-rr 98.5%

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

   1. *-lft-identity 98.5%

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

   2. associate-*r/ 98.5%

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

   3. associate-*l/ 98.5%

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

   4. *-commutative 98.5%

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

7. Simplified 98.5%

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

   1. div-inv 98.6%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
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(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. div-inv 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((costheta_o / (v * v)) * (costheta_i / exp((sintheta_o / (v / sintheta_i))))) / (sinh((1.0e0 / v)) * 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. 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 cosTheta_i around 0 98.6%

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

   1. times-frac 98.6%

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

   2. unpow2 98.6%

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

   3. associate-/l* 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 5: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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 = ((1.0e0 / v) * (costheta_o / sinh((1.0e0 / v)))) * ((costheta_i / v) / 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. 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 98.4%

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

   1. unpow2 98.4%

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

6. Simplified 98.4%

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

   1. times-frac 98.4%

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

   2. times-frac 98.4%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in cosTheta_O around 0 98.4%

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

    1. *-commutative 98.4%

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

    2. associate-/r* 98.4%

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

    3. associate-/r/ 98.4%

       \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v}}{\frac{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}{2}}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

    4. rec-exp 98.4%

       \[\leadsto \frac{\frac{cosTheta_O}{v}}{\frac{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}}{2}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

    5. sinh-def 98.4%

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

    6. associate-/l/ 98.4%

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

    7. *-commutative 98.4%

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

11. Simplified 98.4%

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

    1. *-un-lft-identity 98.4%

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

    2. times-frac 98.6%

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

13. Applied egg-rr 98.6%

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

14. Final simplification 98.6%

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

Alternative 6: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v \cdot \sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* (/ cosTheta_O v) (/ cosTheta_i (* v (sinh (/ 1.0 v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * ((cosTheta_O / v) * (cosTheta_i / (v * sinhf((1.0f / 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 = 0.5e0 * ((costheta_o / v) * (costheta_i / (v * sinh((1.0e0 / v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(Float32(cosTheta_O / v) * Float32(cosTheta_i / Float32(v * sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((cosTheta_O / v) * (cosTheta_i / (v * sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.6%

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

   1. 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 98.4%

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

   1. unpow2 98.4%

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

6. Simplified 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. *-commutative 98.4%

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

   3. times-frac 98.3%

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

   4. metadata-eval 98.3%

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

   5. div-inv 98.3%

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

   6. *-commutative 98.3%

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

   7. pow2 98.3%

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

   8. pow-flip 98.3%

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

   9. metadata-eval 98.3%

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

8. Applied egg-rr 98.3%

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

   1. metadata-eval 98.3%

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

   2. pow-sqr 98.5%

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

   3. unpow-1 98.5%

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

   4. unpow-1 98.5%

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

   5. associate-*r/ 98.4%

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

   6. associate-/r/ 98.4%

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

   7. /-rgt-identity 98.4%

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

   8. associate-*r/ 98.4%

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

   9. associate-*l/ 98.4%

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

   10. associate-*r/ 98.3%

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

   11. *-rgt-identity 98.3%

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

   12. associate-*l/ 98.2%

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

   13. associate-*r/ 98.3%

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

   14. *-commutative 98.3%

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

   15. associate-*r/ 98.3%

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

   16. associate-/l/ 98.3%

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

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 7: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ cosTheta_O \cdot \frac{\frac{cosTheta_i}{v} \cdot \frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (/ (* (/ cosTheta_i v) (/ 0.5 v)) (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * (((cosTheta_i / v) * (0.5f / v)) / sinhf((1.0f / 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 = costheta_o * (((costheta_i / v) * (0.5e0 / v)) / sinh((1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(Float32(cosTheta_i / v) * Float32(Float32(0.5) / v)) / sinh(Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * (((cosTheta_i / v) * (single(0.5) / v)) / sinh((single(1.0) / v)));
end

Derivation

1. Initial program 98.6%

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

   1. 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 98.4%

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

   1. unpow2 98.4%

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

6. Simplified 98.4%

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

   1. times-frac 98.4%

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

   2. times-frac 98.4%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in cosTheta_O around 0 98.4%

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

    1. *-commutative 98.4%

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

    2. associate-/r* 98.4%

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

    3. associate-/r/ 98.4%

       \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v}}{\frac{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}{2}}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

    4. rec-exp 98.4%

       \[\leadsto \frac{\frac{cosTheta_O}{v}}{\frac{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}}{2}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

    5. sinh-def 98.4%

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

    6. associate-/l/ 98.4%

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

    7. *-commutative 98.4%

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

11. Simplified 98.4%

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

    1. expm1-log1p-u 98.4%

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

    2. expm1-udef 54.7%

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

    3. associate-*l/ 54.7%

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

    4. times-frac 54.7%

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

    5. associate-/l/ 54.7%

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

    6. *-un-lft-identity 54.7%

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

    7. times-frac 54.7%

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

    8. metadata-eval 54.7%

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

13. Applied egg-rr 54.7%

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

    1. expm1-def 98.4%

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

    2. expm1-log1p 98.4%

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

    3. *-commutative 98.4%

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

    4. associate-*l/ 98.3%

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

    5. associate-*r/ 98.3%

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

    6. associate-/r* 98.3%

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

    7. associate-*l/ 98.4%

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

    8. *-commutative 98.4%

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

    9. times-frac 98.5%

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

    10. associate-*r/ 98.5%

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

15. Simplified 98.5%

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

16. Final simplification 98.5%

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

Alternative 8: 64.7% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta_i}{v}}{2} \cdot \frac{cosTheta_O}{1 + \frac{0.16666666666666666}{v \cdot v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (/ cosTheta_i v) 2.0)
  (/ cosTheta_O (+ 1.0 (/ 0.16666666666666666 (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i / v) / 2.0f) * (cosTheta_O / (1.0f + (0.16666666666666666f / (v * 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 = ((costheta_i / v) / 2.0e0) * (costheta_o / (1.0e0 + (0.16666666666666666e0 / (v * v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i / v) / Float32(2.0)) * Float32(cosTheta_O / Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i / v) / single(2.0)) * (cosTheta_O / (single(1.0) + (single(0.16666666666666666) / (v * v))));
end

Derivation

1. Initial program 98.6%

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

   1. 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 98.4%

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

   1. unpow2 98.4%

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

6. Simplified 98.4%

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

   1. times-frac 98.4%

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

   2. times-frac 98.4%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in cosTheta_O around 0 98.4%

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

    1. *-commutative 98.4%

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

    2. associate-/r* 98.4%

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

    3. associate-/r/ 98.4%

       \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v}}{\frac{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}{2}}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

    4. rec-exp 98.4%

       \[\leadsto \frac{\frac{cosTheta_O}{v}}{\frac{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}}{2}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

    5. sinh-def 98.4%

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

    6. associate-/l/ 98.4%

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

    7. *-commutative 98.4%

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

11. Simplified 98.4%

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

12. Taylor expanded in v around inf 65.0%

    \[\leadsto \frac{cosTheta_O}{\color{blue}{1 + 0.16666666666666666 \cdot \frac{1}{{v}^{2}}}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]
13. Step-by-step derivation

    1. unpow2 65.0%

       \[\leadsto \frac{cosTheta_O}{1 + 0.16666666666666666 \cdot \frac{1}{\color{blue}{v \cdot v}}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

    2. associate-*r/ 65.0%

       \[\leadsto \frac{cosTheta_O}{1 + \color{blue}{\frac{0.16666666666666666 \cdot 1}{v \cdot v}}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

    3. metadata-eval 65.0%

       \[\leadsto \frac{cosTheta_O}{1 + \frac{\color{blue}{0.16666666666666666}}{v \cdot v}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

14. Simplified 65.0%

    \[\leadsto \frac{cosTheta_O}{\color{blue}{1 + \frac{0.16666666666666666}{v \cdot v}}} \cdot \frac{\frac{cosTheta_i}{v}}{2} \]

15. Final simplification 65.0%

    \[\leadsto \frac{\frac{cosTheta_i}{v}}{2} \cdot \frac{cosTheta_O}{1 + \frac{0.16666666666666666}{v \cdot v}} \]

Alternative 9: 59.4% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ 1.0 (/ v (* cosTheta_i cosTheta_O)))))

C

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

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 = 0.5e0 * (1.0e0 / (v / (costheta_i * costheta_o)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(Float32(1.0) / Float32(v / Float32(cosTheta_i * cosTheta_O))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (single(1.0) / (v / (cosTheta_i * cosTheta_O)));
end

Derivation

1. Initial program 98.6%

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

   1. 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 59.5%

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

   1. *-commutative 59.5%

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

   2. associate-*l/ 59.4%

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

   3. *-commutative 59.4%

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

6. Simplified 59.4%

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

7. Taylor expanded in cosTheta_O around 0 59.5%

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

   1. associate-/l* 59.4%

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

9. Simplified 59.4%

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

    1. clear-num 60.1%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O}}} \]

    2. inv-pow 60.1%

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

11. Applied egg-rr 60.1%

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

    1. unpow-1 60.1%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O}}} \]

    2. associate-/l/ 60.1%

       \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta_O \cdot cosTheta_i}}} \]

13. Simplified 60.1%

    \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta_O \cdot cosTheta_i}}} \]

14. Final simplification 60.1%

    \[\leadsto 0.5 \cdot \frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \]

Alternative 10: 59.0% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* cosTheta_O (/ cosTheta_i v))))

C

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

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

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

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 98.6%

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

   1. 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 59.5%

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

   1. *-commutative 59.5%

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

   2. associate-*l/ 59.4%

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

   3. *-commutative 59.4%

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

6. Simplified 59.4%

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

7. Final simplification 59.4%

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

Alternative 11: 59.0% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* cosTheta_i (/ cosTheta_O v))))

C

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

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

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

MATLAB

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 98.6%

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

   1. 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 59.5%

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

   1. *-commutative 59.5%

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

   2. associate-*l/ 59.4%

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

   3. *-commutative 59.4%

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

6. Simplified 59.4%

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

7. Taylor expanded in cosTheta_O around 0 59.5%

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

   1. associate-/l* 59.4%

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

9. Simplified 59.4%

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

    1. associate-/r/ 59.4%

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

11. Applied egg-rr 59.4%

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

12. Final simplification 59.4%

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

Alternative 12: 59.0% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (* cosTheta_i cosTheta_O) v)))

C

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

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

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

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 98.6%

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

   1. 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 59.5%

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

5. Final simplification 59.5%

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

Alternative 13: 59.0% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (* cosTheta_i cosTheta_O)) v))

C

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

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

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

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 98.6%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 59.5%

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

5. Final simplification 59.5%

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

Reproduce

herbie shell --seed 2023272 
(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)))