HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 13.6s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*r/ 98.7%

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

   3. associate-/l* 98.7%

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

   4. associate-/r/ 98.7%

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

   5. *-commutative 98.7%

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

   6. *-commutative 98.7%

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

   7. associate-*r* 98.7%

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

   8. associate-/l/ 98.7%

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

   9. exp-neg 98.7%

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

   10. associate-/l/ 98.7%

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

   11. associate-/r* 98.7%

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

   12. metadata-eval 98.7%

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

   13. associate-*l/ 98.7%

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

   14. *-commutative 98.7%

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

   15. exp-prod 98.7%

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

3. Simplified 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*l/ 98.7%

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

   3. associate-*r/ 98.7%

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

   4. expm1-log1p-u 98.7%

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

   5. expm1-udef 51.6%

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

   6. associate-*l* 51.6%

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

5. Applied egg-rr 51.6%

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

   1. expm1-def 98.6%

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

   2. expm1-log1p 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-/l/ 98.6%

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

   5. associate-*l/ 98.7%

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

7. Simplified 98.7%

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

   1. expm1-log1p-u 98.7%

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

   2. expm1-udef 61.3%

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

9. Applied egg-rr 61.3%

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

    1. expm1-def 98.7%

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

    2. expm1-log1p 98.7%

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

    3. associate-/r/ 98.6%

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

    4. associate-*l/ 98.7%

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

    5. *-rgt-identity 98.7%

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

    6. times-frac 98.7%

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

    7. rem-exp-log 44.8%

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

    8. exp-neg 44.8%

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

    9. exp-sum 44.8%

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

    10. sub-neg 44.8%

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

    11. exp-diff 44.8%

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

    12. rem-exp-log 98.7%

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

    13. associate-*l/ 98.6%

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

11. Simplified 98.9%

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

    1. div-inv 98.8%

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

13. Applied egg-rr 98.8%

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

    1. associate-*r/ 98.8%

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

    2. un-div-inv 98.9%

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

    3. associate-/r/ 98.9%

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

15. Applied egg-rr 98.9%

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

    1. associate-/l* 98.8%

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

    2. associate-/r/ 98.9%

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

    3. associate-*l/ 98.9%

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

    4. associate-/l* 98.9%

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

17. Simplified 98.9%

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

18. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*r/ 98.7%

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

   3. associate-/l* 98.7%

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

   4. associate-/r/ 98.7%

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

   5. *-commutative 98.7%

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

   6. *-commutative 98.7%

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

   7. associate-*r* 98.7%

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

   8. associate-/l/ 98.7%

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

   9. exp-neg 98.7%

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

   10. associate-/l/ 98.7%

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

   11. associate-/r* 98.7%

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

   12. metadata-eval 98.7%

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

   13. associate-*l/ 98.7%

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

   14. *-commutative 98.7%

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

   15. exp-prod 98.7%

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

3. Simplified 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*l/ 98.7%

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

   3. associate-*r/ 98.7%

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

   4. expm1-log1p-u 98.7%

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

   5. expm1-udef 51.6%

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

   6. associate-*l* 51.6%

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

5. Applied egg-rr 51.6%

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

   1. expm1-def 98.6%

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

   2. expm1-log1p 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-/l/ 98.6%

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

   5. associate-*l/ 98.7%

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

7. Simplified 98.7%

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

   1. expm1-log1p-u 98.7%

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

   2. expm1-udef 61.3%

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

9. Applied egg-rr 61.3%

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

    1. expm1-def 98.7%

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

    2. expm1-log1p 98.7%

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

    3. associate-/r/ 98.6%

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

    4. associate-*l/ 98.7%

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

    5. *-rgt-identity 98.7%

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

    6. times-frac 98.7%

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

    7. rem-exp-log 44.8%

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

    8. exp-neg 44.8%

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

    9. exp-sum 44.8%

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

    10. sub-neg 44.8%

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

    11. exp-diff 44.8%

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

    12. rem-exp-log 98.7%

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

    13. associate-*l/ 98.6%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / (v * v)) * (cosTheta_O / (expf((1.0f / v)) - expf((-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_i / (v * v)) * (costheta_o / (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*l/ 98.7%

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

   2. times-frac 98.6%

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

   3. exp-neg 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-lft-identity 98.6%

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

   6. associate-/l/ 98.6%

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

   7. associate-*l* 98.6%

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

   8. associate-*l* 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*l/ 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.7%

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

   1. times-frac 98.7%

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

   2. unpow2 98.7%

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

   3. rec-exp 98.7%

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

   4. distribute-neg-frac 98.7%

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

   5. metadata-eval 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 4: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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)) * ((1.0e0 / v) / (sinh((1.0e0 / v)) / 0.5e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*r/ 98.7%

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

   3. associate-/l* 98.7%

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

   4. associate-/r/ 98.7%

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

   5. *-commutative 98.7%

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

   6. *-commutative 98.7%

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

   7. associate-*r* 98.7%

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

   8. associate-/l/ 98.7%

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

   9. exp-neg 98.7%

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

   10. associate-/l/ 98.7%

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

   11. associate-/r* 98.7%

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

   12. metadata-eval 98.7%

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

   13. associate-*l/ 98.7%

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

   14. *-commutative 98.7%

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

   15. exp-prod 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. associate-/r* 98.7%

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

   2. rec-exp 98.8%

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

   3. distribute-neg-frac 98.8%

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

   4. metadata-eval 98.8%

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

6. Simplified 98.8%

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

   1. sub-neg 98.8%

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

8. Applied egg-rr 98.8%

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

   1. sub-neg 98.8%

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

   2. /-rgt-identity 98.8%

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

   3. metadata-eval 98.8%

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

   4. distribute-neg-frac 98.8%

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

   5. rec-exp 98.7%

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

   6. metadata-eval 98.7%

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

   7. associate-/l/ 98.7%

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

   8. rec-exp 98.8%

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

   9. sinh-def 98.7%

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

10. Simplified 98.7%

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

11. Final simplification 98.7%

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

Alternative 5: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*r/ 98.7%

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

   3. associate-/l* 98.7%

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

   4. associate-/r/ 98.7%

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

   5. *-commutative 98.7%

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

   6. *-commutative 98.7%

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

   7. associate-*r* 98.7%

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

   8. associate-/l/ 98.7%

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

   9. exp-neg 98.7%

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

   10. associate-/l/ 98.7%

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

   11. associate-/r* 98.7%

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

   12. metadata-eval 98.7%

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

   13. associate-*l/ 98.7%

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

   14. *-commutative 98.7%

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

   15. exp-prod 98.7%

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

3. Simplified 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*l/ 98.7%

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

   3. associate-*r/ 98.7%

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

   4. expm1-log1p-u 98.7%

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

   5. expm1-udef 51.6%

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

   6. associate-*l* 51.6%

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

5. Applied egg-rr 51.6%

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

   1. expm1-def 98.6%

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

   2. expm1-log1p 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-/l/ 98.6%

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

   5. associate-*l/ 98.7%

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

7. Simplified 98.7%

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

   1. expm1-log1p-u 98.7%

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

   2. expm1-udef 61.3%

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

9. Applied egg-rr 61.3%

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

    1. expm1-def 98.7%

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

    2. expm1-log1p 98.7%

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

    3. associate-/r/ 98.6%

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

    4. associate-*l/ 98.7%

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

    5. *-rgt-identity 98.7%

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

    6. times-frac 98.7%

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

    7. rem-exp-log 44.8%

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

    8. exp-neg 44.8%

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

    9. exp-sum 44.8%

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

    10. sub-neg 44.8%

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

    11. exp-diff 44.8%

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

    12. rem-exp-log 98.7%

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

    13. associate-*l/ 98.6%

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

11. Simplified 98.9%

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

12. Taylor expanded in sinTheta_i around 0 98.7%

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

    1. *-commutative 98.7%

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

    2. associate-/r* 98.7%

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

14. Simplified 98.7%

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

15. Final simplification 98.7%

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

Alternative 6: 64.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \frac{0.3333333333333333}{v \cdot v}\\ \frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{t_0} - \frac{sinTheta_i}{v \cdot v} \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 2.0 (/ 0.3333333333333333 (* v v)))))
   (-
    (* (/ cosTheta_O v) (/ cosTheta_i t_0))
    (*
     (/ sinTheta_i (* v v))
     (/ (* cosTheta_i (* sinTheta_O cosTheta_O)) t_0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 2.0f + (0.3333333333333333f / (v * v));
	return ((cosTheta_O / v) * (cosTheta_i / t_0)) - ((sinTheta_i / (v * v)) * ((cosTheta_i * (sinTheta_O * cosTheta_O)) / t_0));
}

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
    real(4) :: t_0
    t_0 = 2.0e0 + (0.3333333333333333e0 / (v * v))
    code = ((costheta_o / v) * (costheta_i / t_0)) - ((sintheta_i / (v * v)) * ((costheta_i * (sintheta_o * costheta_o)) / t_0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))
	return Float32(Float32(Float32(cosTheta_O / v) * Float32(cosTheta_i / t_0)) - Float32(Float32(sinTheta_i / Float32(v * v)) * Float32(Float32(cosTheta_i * Float32(sinTheta_O * cosTheta_O)) / t_0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(2.0) + (single(0.3333333333333333) / (v * v));
	tmp = ((cosTheta_O / v) * (cosTheta_i / t_0)) - ((sinTheta_i / (v * v)) * ((cosTheta_i * (sinTheta_O * cosTheta_O)) / t_0));
end

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in v around inf 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 64.9%

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

   2. metadata-eval 64.9%

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

   3. unpow2 64.9%

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

4. Simplified 64.9%

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

5. Taylor expanded in sinTheta_i around 0 64.9%

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

   1. mul-1-neg 64.9%

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

   2. unsub-neg 64.9%

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

   3. associate-/r* 64.9%

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

   4. associate-*r/ 64.9%

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

   5. associate-*r/ 64.9%

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

   6. metadata-eval 64.9%

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

   7. unpow2 64.9%

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

   8. times-frac 64.9%

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

   9. unpow2 64.9%

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

   10. *-commutative 64.9%

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

   11. associate-*r/ 64.9%

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

7. Simplified 64.9%

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

8. Taylor expanded in cosTheta_i around 0 64.9%

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

   1. *-commutative 64.9%

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

   2. associate-*r/ 64.9%

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

   3. metadata-eval 64.9%

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

   4. unpow2 64.9%

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

   5. times-frac 64.9%

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

10. Simplified 64.9%

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

11. Final simplification 64.9%

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

Alternative 7: 64.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \frac{0.3333333333333333}{v \cdot v}\\ \frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{t_0} - \frac{sinTheta_i}{v \cdot v} \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 2.0 (/ 0.3333333333333333 (* v v)))))
   (-
    (* (/ cosTheta_i v) (/ cosTheta_O t_0))
    (*
     (/ sinTheta_i (* v v))
     (/ (* cosTheta_i (* sinTheta_O cosTheta_O)) t_0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 2.0f + (0.3333333333333333f / (v * v));
	return ((cosTheta_i / v) * (cosTheta_O / t_0)) - ((sinTheta_i / (v * v)) * ((cosTheta_i * (sinTheta_O * cosTheta_O)) / t_0));
}

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
    real(4) :: t_0
    t_0 = 2.0e0 + (0.3333333333333333e0 / (v * v))
    code = ((costheta_i / v) * (costheta_o / t_0)) - ((sintheta_i / (v * v)) * ((costheta_i * (sintheta_o * costheta_o)) / t_0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))
	return Float32(Float32(Float32(cosTheta_i / v) * Float32(cosTheta_O / t_0)) - Float32(Float32(sinTheta_i / Float32(v * v)) * Float32(Float32(cosTheta_i * Float32(sinTheta_O * cosTheta_O)) / t_0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(2.0) + (single(0.3333333333333333) / (v * v));
	tmp = ((cosTheta_i / v) * (cosTheta_O / t_0)) - ((sinTheta_i / (v * v)) * ((cosTheta_i * (sinTheta_O * cosTheta_O)) / t_0));
end

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in v around inf 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 64.9%

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

   2. metadata-eval 64.9%

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

   3. unpow2 64.9%

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

4. Simplified 64.9%

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

5. Taylor expanded in sinTheta_i around 0 64.9%

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

   1. mul-1-neg 64.9%

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

   2. unsub-neg 64.9%

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

   3. associate-/r* 64.9%

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

   4. associate-*r/ 64.9%

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

   5. associate-*r/ 64.9%

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

   6. metadata-eval 64.9%

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

   7. unpow2 64.9%

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

   8. times-frac 64.9%

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

   9. unpow2 64.9%

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

   10. *-commutative 64.9%

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

   11. associate-*r/ 64.9%

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

7. Simplified 64.9%

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

8. Taylor expanded in cosTheta_i around 0 64.9%

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

   1. times-frac 64.9%

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

   2. associate-*r/ 64.9%

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

   3. metadata-eval 64.9%

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

   4. unpow2 64.9%

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

10. Simplified 64.9%

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

11. Final simplification 64.9%

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

Alternative 8: 64.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i \cdot cosTheta_O}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_i cosTheta_O) (* v (+ 2.0 (/ 0.3333333333333333 (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) / (v * (2.0f + (0.3333333333333333f / (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 * costheta_o) / (v * (2.0e0 + (0.3333333333333333e0 / (v * v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) / (v * (single(2.0) + (single(0.3333333333333333) / (v * v))));
end

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in v around inf 64.9%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 64.9%

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

   2. metadata-eval 64.9%

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

   3. unpow2 64.9%

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

4. Simplified 64.9%

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

5. Taylor expanded in sinTheta_i around 0 64.9%

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

   1. associate-*r/ 64.9%

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

   2. metadata-eval 64.9%

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

   3. unpow2 64.9%

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

7. Simplified 64.9%

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

8. Final simplification 64.9%

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

Alternative 9: 58.2% 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 98.7%

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

   1. associate-*l/ 98.7%

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

   2. times-frac 98.6%

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

   3. exp-neg 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-lft-identity 98.6%

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

   6. associate-/l/ 98.6%

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

   7. associate-*l* 98.6%

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

   8. associate-*l* 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*l/ 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 58.6%

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

5. Final simplification 58.6%

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

Alternative 10: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

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) * (costheta_i * 0.5e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*l/ 98.7%

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

   2. times-frac 98.6%

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

   3. exp-neg 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-lft-identity 98.6%

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

   6. associate-/l/ 98.6%

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

   7. associate-*l* 98.6%

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

   8. associate-*l* 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*l/ 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.6%

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

   2. associate-*r* 58.6%

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

6. Simplified 58.6%

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

7. Final simplification 58.6%

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

Alternative 11: 58.7% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (/ 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 / (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 / (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(v / Float32(cosTheta_i * cosTheta_O)))
end

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*l/ 98.7%

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

   2. times-frac 98.6%

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

   3. exp-neg 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-lft-identity 98.6%

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

   6. associate-/l/ 98.6%

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

   7. associate-*l* 98.6%

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

   8. associate-*l* 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*l/ 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.6%

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

   2. associate-*r* 58.6%

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

6. Simplified 58.6%

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

7. Taylor expanded in cosTheta_i around 0 58.6%

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

   1. associate-*r/ 58.6%

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

   2. associate-/l* 59.3%

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

   3. *-commutative 59.3%

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

9. Simplified 59.3%

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

10. Final simplification 59.3%

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

Reproduce

herbie shell --seed 2023181 
(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)))