HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 17.9s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_O / v) * cosTheta_i) * ((exp((-sinTheta_i / v)) ^ sinTheta_O) / (v * single(2.0)))) / sinh((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. *-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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. associate-*l/ 98.9%

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

   2. exp-prod 98.9%

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((single(1.0) / v) * (cosTheta_O * cosTheta_i)) / sinh((single(1.0) / v))) * (exp(((-sinTheta_i / v) * sinTheta_O)) / (v * single(2.0)));
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. 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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in cosTheta_O around 0 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-/r* 98.5%

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

   3. associate-*l/ 98.6%

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

   4. associate-/r/ 98.8%

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

   5. associate-*l/ 98.7%

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

   6. associate-*r/ 98.7%

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

   7. rec-exp 98.7%

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

   8. sinh-def 98.7%

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

7. Simplified 98.7%

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

   1. associate-*r/ 98.8%

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

   2. div-inv 98.8%

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

   3. *-un-lft-identity 98.8%

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

   4. times-frac 98.7%

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

9. Applied egg-rr 98.7%

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

    1. /-rgt-identity 98.7%

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

    2. associate-*r/ 98.8%

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

11. Applied egg-rr 98.8%

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

12. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(((-sinTheta_i / v) * sinTheta_O)) / (v * 2.0f)) * ((cosTheta_O / v) * (cosTheta_i / 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 = (exp(((-sintheta_i / v) * sintheta_o)) / (v * 2.0e0)) * ((costheta_o / v) * (costheta_i / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((-sinTheta_i / v) * sinTheta_O)) / (v * single(2.0))) * ((cosTheta_O / v) * (cosTheta_i / sinh((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. *-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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in cosTheta_O around 0 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-/r* 98.5%

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

   3. associate-*l/ 98.6%

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

   4. associate-/r/ 98.8%

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

   5. associate-*l/ 98.7%

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

   6. associate-*r/ 98.7%

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

   7. rec-exp 98.7%

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

   8. sinh-def 98.7%

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

7. Simplified 98.7%

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

   1. associate-*r/ 98.8%

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

   2. div-inv 98.8%

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

   3. *-un-lft-identity 98.8%

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

   4. times-frac 98.7%

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

9. Applied egg-rr 98.7%

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

    1. expm1-log1p-u 98.7%

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

    2. expm1-udef 57.4%

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

    3. /-rgt-identity 57.4%

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

11. Applied egg-rr 57.4%

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

    1. expm1-def 98.7%

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

    2. expm1-log1p 98.7%

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

    3. associate-*r/ 98.8%

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

    4. associate-/l* 98.8%

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

    5. *-commutative 98.8%

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

    6. associate-/r/ 98.7%

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

    7. times-frac 98.7%

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

    8. /-rgt-identity 98.7%

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

13. Simplified 98.7%

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

14. Final simplification 98.7%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (/ cosTheta_O (* (pow v 2.0) (* 2.0 (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 / (powf(v, 2.0f) * (2.0f * 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 ** 2.0e0) * (2.0e0 * sinh((1.0e0 / v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * Float32(cosTheta_O / Float32((v ^ Float32(2.0)) * Float32(Float32(2.0) * 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(2.0)) * (single(2.0) * sinh((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. *-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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. add-cube-cbrt 98.3%

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

   2. pow3 98.2%

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

   3. rec-exp 98.2%

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

   4. sinh-undef 98.2%

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

7. Applied egg-rr 98.2%

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

   1. expm1-log1p-u 98.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_O \cdot cosTheta_i}{{\left(\sqrt[3]{{v}^{2} \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)}^{3}}\right)\right)} \]

   2. expm1-udef 57.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{cosTheta_O \cdot cosTheta_i}{{\left(\sqrt[3]{{v}^{2} \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)}^{3}}\right)} - 1} \]

   3. rem-cube-cbrt 57.5%

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

   4. associate-/l* 57.5%

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

   5. *-commutative 57.5%

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

9. Applied egg-rr 57.5%

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

    1. expm1-def 98.6%

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

    2. expm1-log1p 98.6%

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

    3. associate-/r/ 98.6%

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

    4. *-commutative 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

    \[\leadsto cosTheta_i \cdot \frac{cosTheta_O}{{v}^{2} \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)} \]
13. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot {v}^{-2}\right)}{\sinh \left(\frac{1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (* (* cosTheta_O cosTheta_i) (pow v -2.0))) (sinh (/ 1.0 v))))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 * ((costheta_o * costheta_i) * (v ** (-2.0e0)))) / sinh((1.0e0 / v))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * ((cosTheta_O * cosTheta_i) * (v ^ single(-2.0)))) / sinh((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. *-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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. associate-*l/ 98.9%

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

   2. exp-prod 98.9%

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in v around inf 98.6%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2}}\right)\right)}}{\sinh \left(\frac{1}{v}\right)} \]

   2. expm1-udef 57.9%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2}}\right)} - 1\right)}}{\sinh \left(\frac{1}{v}\right)} \]

   3. div-inv 57.9%

      \[\leadsto \frac{0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(cosTheta_O \cdot cosTheta_i\right) \cdot \frac{1}{{v}^{2}}}\right)} - 1\right)}{\sinh \left(\frac{1}{v}\right)} \]

   4. pow-flip 57.9%

      \[\leadsto \frac{0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot \color{blue}{{v}^{\left(-2\right)}}\right)} - 1\right)}{\sinh \left(\frac{1}{v}\right)} \]

   5. metadata-eval 57.9%

      \[\leadsto \frac{0.5 \cdot \left(e^{\mathsf{log1p}\left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot {v}^{\color{blue}{-2}}\right)} - 1\right)}{\sinh \left(\frac{1}{v}\right)} \]

9. Applied egg-rr 57.9%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot {v}^{-2}\right)} - 1\right)}}{\sinh \left(\frac{1}{v}\right)} \]
10. Step-by-step derivation

    1. expm1-def 98.7%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot {v}^{-2}\right)\right)}}{\sinh \left(\frac{1}{v}\right)} \]

    2. expm1-log1p 98.7%

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

11. Simplified 98.7%

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

12. Final simplification 98.7%

    \[\leadsto \frac{0.5 \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot {v}^{-2}\right)}{\sinh \left(\frac{1}{v}\right)} \]
13. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i \cdot 0.5}{{v}^{2} \cdot \frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_i 0.5) (* (pow v 2.0) (/ (sinh (/ 1.0 v)) cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * 0.5f) / (powf(v, 2.0f) * (sinhf((1.0f / v)) / 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 ** 2.0e0) * (sinh((1.0e0 / v)) / costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * Float32(0.5)) / Float32((v ^ Float32(2.0)) * Float32(sinh(Float32(Float32(1.0) / v)) / cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * single(0.5)) / ((v ^ single(2.0)) * (sinh((single(1.0) / v)) / 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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. associate-*l/ 98.9%

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

   2. exp-prod 98.9%

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in v around inf 98.6%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{{v}^{2}}}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \]

   2. expm1-udef 57.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{{v}^{2}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1} \]

   3. associate-*r/ 57.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{0.5 \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{{v}^{2}}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

9. Applied egg-rr 57.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.5 \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{{v}^{2}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 98.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.5 \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{{v}^{2}}}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \]

    2. expm1-log1p 98.5%

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

    3. associate-/l/ 98.5%

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

    4. associate-*r* 98.6%

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

    5. times-frac 98.7%

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

11. Simplified 98.7%

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

    1. *-commutative 98.7%

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

    2. associate-/l* 98.7%

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

    3. frac-times 98.6%

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

13. Applied egg-rr 98.6%

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

14. Final simplification 98.6%

    \[\leadsto \frac{cosTheta_i \cdot 0.5}{{v}^{2} \cdot \frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}} \]
15. Add Preprocessing

Alternative 7: 70.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_O cosTheta_i)
  (+
   (* 0.016666666666666666 (/ 1.0 (pow v 3.0)))
   (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * cosTheta_i) / ((0.016666666666666666f * (1.0f / powf(v, 3.0f))) + (((1.0f / v) * 0.3333333333333333f) + (v * 2.0f)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o * costheta_i) / ((0.016666666666666666e0 * (1.0e0 / (v ** 3.0e0))) + (((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * cosTheta_i) / Float32(Float32(Float32(0.016666666666666666) * Float32(Float32(1.0) / (v ^ Float32(3.0)))) + Float32(Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * cosTheta_i) / ((single(0.016666666666666666) * (single(1.0) / (v ^ single(3.0)))) + (((single(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0))));
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. 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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

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

6. Taylor expanded in v around inf 73.3%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]

7. Final simplification 73.3%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)} \]
8. Add Preprocessing

Alternative 8: 64.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * cosTheta_i) / fmaf(v, 2.0f, (0.3333333333333333f / v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * cosTheta_i) / fma(v, Float32(2.0), Float32(Float32(0.3333333333333333) / 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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

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

6. Taylor expanded in v around inf 67.9%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]
7. Step-by-step derivation

   1. *-commutative 67.9%

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

   2. fma-def 67.9%

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

   3. associate-*r/ 67.9%

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

   4. metadata-eval 67.9%

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

8. Simplified 67.9%

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

9. Final simplification 67.9%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \]
10. Add Preprocessing

Alternative 9: 64.3% accurate, 16.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_O cosTheta_i) (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * cosTheta_i) / (((1.0f / v) * 0.3333333333333333f) + (v * 2.0f));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o * costheta_i) / (((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * cosTheta_i) / Float32(Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * cosTheta_i) / (((single(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0)));
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. 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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in sinTheta_i around 0 98.6%

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

6. Taylor expanded in v around inf 67.9%

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

7. Final simplification 67.9%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} \]
8. Add Preprocessing

Alternative 10: 58.6% accurate, 31.4× speedup

Math

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

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(Float32(cosTheta_O / v) * 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. *-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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 62.8%

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

   1. associate-*l/ 62.8%

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

7. Applied egg-rr 62.8%

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

8. Final simplification 62.8%

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

Alternative 11: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_O / (v / cosTheta_i));
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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 62.8%

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

   1. associate-*l/ 62.8%

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

7. Applied egg-rr 62.8%

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

   1. associate-*l/ 62.8%

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

   2. associate-/l* 62.8%

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

9. Applied egg-rr 62.8%

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

10. Final simplification 62.8%

    \[\leadsto 0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}} \]
11. Add Preprocessing

Alternative 12: 59.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

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_o * costheta_i))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v / (cosTheta_O * cosTheta_i));
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-*l* 98.7%

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

   3. times-frac 98.8%

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

   4. *-commutative 98.8%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 62.8%

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

   1. associate-*r/ 62.8%

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

   2. associate-/l* 63.3%

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

7. Simplified 63.3%

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

8. Final simplification 63.3%

   \[\leadsto \frac{0.5}{\frac{v}{cosTheta_O \cdot cosTheta_i}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))