HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 11.8s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. distribute-neg-frac 98.5%

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

   2. *-commutative 98.5%

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

   3. distribute-rgt-neg-in 98.5%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.5%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

5. Applied egg-rr 98.5%

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

   1. associate-*l* 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*l/ 98.4%

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

   4. associate-*r/ 98.5%

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

   5. associate-/l* 98.5%

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

   6. associate-/r* 98.7%

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

   7. *-commutative 98.7%

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

7. Simplified 98.7%

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

   1. associate-/r* 98.5%

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

   2. add-exp-log 98.4%

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

   3. inv-pow 98.4%

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

   4. add-exp-log 98.5%

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

   5. *-commutative 98.5%

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

   6. unpow-prod-down 98.7%

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

   7. inv-pow 98.7%

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

   8. inv-pow 98.7%

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

9. Applied egg-rr 98.7%

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

    1. expm1-log1p-u 98.7%

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

    2. expm1-udef 53.0%

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

11. Applied egg-rr 53.0%

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

    1. expm1-def 98.5%

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

    2. expm1-log1p 98.5%

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

    3. associate-*l* 98.6%

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

    4. associate-/l/ 98.6%

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

    5. associate-/r* 98.9%

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

    6. *-commutative 98.9%

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

    7. associate-*r/ 98.9%

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

    8. neg-mul-1 98.9%

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

    9. associate-*r* 98.9%

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

    10. mul-1-neg 98.9%

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

13. Simplified 98.9%

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

14. Final simplification 98.9%

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

Alternative 2: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * (costheta_o / v)) * ((1.0e0 / v) * (0.5e0 / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l/ 98.5%

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

   9. exp-neg 98.5%

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

   10. associate-/l/ 98.5%

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

   11. associate-/r* 98.5%

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

   12. metadata-eval 98.5%

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

   13. associate-*l/ 98.5%

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

   14. *-commutative 98.5%

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

   15. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. rec-exp 98.3%

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

   2. distribute-neg-frac 98.3%

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

   3. metadata-eval 98.3%

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

6. Simplified 98.3%

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

   1. add-cube-cbrt 98.1%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \sqrt[3]{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}\right) \cdot \sqrt[3]{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}\right)}} \]

   2. pow3 98.0%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v \cdot \color{blue}{{\left(\sqrt[3]{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}\right)}^{3}}} \]

   3. div-inv 98.0%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v \cdot {\left(\sqrt[3]{e^{\frac{1}{v}} - e^{\color{blue}{-1 \cdot \frac{1}{v}}}}\right)}^{3}} \]

   4. neg-mul-1 98.0%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v \cdot {\left(\sqrt[3]{e^{\frac{1}{v}} - e^{\color{blue}{-\frac{1}{v}}}}\right)}^{3}} \]

   5. sinh-undef 98.1%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v \cdot {\left(\sqrt[3]{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}}\right)}^{3}} \]

8. Applied egg-rr 98.1%

   \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v \cdot \color{blue}{{\left(\sqrt[3]{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)}^{3}}} \]
9. Step-by-step derivation

   1. metadata-eval 98.1%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{v \cdot {\left(\sqrt[3]{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)}^{3}} \]

   2. rem-cube-cbrt 98.3%

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

   3. *-commutative 98.3%

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

   4. frac-times 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-/r* 98.6%

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

   7. metadata-eval 98.6%

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

10. Applied egg-rr 98.6%

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

11. Final simplification 98.6%

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

Alternative 3: 64.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{2 + \frac{0.3333333333333333}{v \cdot v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i (/ cosTheta_O v))
  (/ 1.0 (+ 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)) * (1.0f / (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)) * (1.0e0 / (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 * Float32(cosTheta_O / v)) * Float32(Float32(1.0) / 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(1.0) / (single(2.0) + (single(0.3333333333333333) / (v * v))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l/ 98.5%

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

   9. exp-neg 98.5%

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

   10. associate-/l/ 98.5%

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

   11. associate-/r* 98.5%

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

   12. metadata-eval 98.5%

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

   13. associate-*l/ 98.5%

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

   14. *-commutative 98.5%

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

   15. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. rec-exp 98.3%

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

   2. distribute-neg-frac 98.3%

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

   3. metadata-eval 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in v around inf 64.4%

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

   1. unpow2 64.4%

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

   2. associate-*r/ 64.4%

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

   3. metadata-eval 64.4%

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

9. Simplified 64.4%

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

10. Final simplification 64.4%

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

Alternative 4: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 * (costheta_o * (costheta_i / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l/ 98.5%

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

   9. exp-neg 98.5%

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

   10. associate-/l/ 98.5%

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

   11. associate-/r* 98.5%

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

   12. metadata-eval 98.5%

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

   13. associate-*l/ 98.5%

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

   14. *-commutative 98.5%

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

   15. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.9%

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

   1. associate-*l/ 58.9%

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

   2. *-commutative 58.9%

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

6. Simplified 58.9%

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

7. Final simplification 58.9%

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

Alternative 5: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 * ((costheta_o * costheta_i) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l/ 98.5%

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

   9. exp-neg 98.5%

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

   10. associate-/l/ 98.5%

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

   11. associate-/r* 98.5%

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

   12. metadata-eval 98.5%

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

   13. associate-*l/ 98.5%

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

   14. *-commutative 98.5%

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

   15. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.9%

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

5. Final simplification 58.9%

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

Alternative 6: 58.4% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_O cosTheta_i) (/ 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 / 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 / 0.5e0)
end function

Julia

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l/ 98.5%

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

   9. exp-neg 98.5%

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

   10. associate-/l/ 98.5%

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

   11. associate-/r* 98.5%

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

   12. metadata-eval 98.5%

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

   13. associate-*l/ 98.5%

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

   14. *-commutative 98.5%

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

   15. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.9%

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

   1. associate-*l/ 58.9%

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

   2. *-commutative 58.9%

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

6. Simplified 58.9%

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

7. Taylor expanded in cosTheta_O around 0 58.9%

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

   1. associate-*r/ 58.9%

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

   2. *-commutative 58.9%

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

   3. associate-*l* 58.9%

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

   4. *-commutative 58.9%

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

   5. associate-*l/ 58.9%

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

   6. *-commutative 58.9%

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

   7. associate-/l* 58.9%

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

9. Simplified 58.9%

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

    1. associate-*r/ 58.9%

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

11. Applied egg-rr 58.9%

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

12. Final simplification 58.9%

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

Alternative 7: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 * (costheta_o * costheta_i)) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l/ 98.5%

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

   9. exp-neg 98.5%

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

   10. associate-/l/ 98.5%

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

   11. associate-/r* 98.5%

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

   12. metadata-eval 98.5%

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

   13. associate-*l/ 98.5%

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

   14. *-commutative 98.5%

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

   15. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.9%

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

   1. associate-*l/ 58.9%

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

   2. *-commutative 58.9%

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

6. Simplified 58.9%

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

   1. *-commutative 58.9%

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

   2. associate-*r/ 58.9%

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

   3. *-commutative 58.9%

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

   4. associate-*l/ 58.9%

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

8. Applied egg-rr 58.9%

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

9. Final simplification 58.9%

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

Reproduce

herbie shell --seed 2023224 
(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)))