HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.6%

Time: 15.2s

Alternatives: 15

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in sinTheta_O around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. mul-1-neg 99.0%

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

   3. unsub-neg 99.0%

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

   4. associate-/l* 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in sinTheta_O around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. mul-1-neg 99.0%

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

   3. unsub-neg 99.0%

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

   4. associate-/l* 99.0%

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

6. Simplified 99.0%

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

   1. add-log-exp 97.1%

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

   2. associate-*r* 97.1%

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

   3. *-commutative 97.1%

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

   4. associate-*l* 97.1%

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

   5. exp-prod 95.4%

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

   6. *-commutative 95.4%

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

   7. exp-prod 95.1%

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

8. Applied egg-rr 95.3%

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

   1. log-pow 95.2%

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

   2. log-pow 98.6%

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

   3. rem-log-exp 98.8%

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

   4. *-commutative 98.8%

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

   5. *-commutative 98.8%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 3: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. *-commutative 98.9%

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

   2. times-frac 98.7%

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

   3. distribute-neg-frac 98.7%

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

   4. distribute-rgt-neg-out 98.7%

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

   5. associate-*l/ 98.7%

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

   6. *-commutative 98.7%

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

   7. associate-*l/ 98.7%

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

3. Simplified 98.7%

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

   1. times-frac 98.6%

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

5. Applied egg-rr 98.6%

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

   1. *-commutative 98.6%

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

   2. add-exp-log 71.5%

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

   3. log-prod 71.2%

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

   4. div-inv 71.2%

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

   5. metadata-eval 71.2%

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

   6. log-div 71.2%

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

   7. add-log-exp 71.2%

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

7. Applied egg-rr 71.2%

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

   1. exp-sum 71.4%

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

   2. rem-exp-log 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-*l* 98.5%

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

   5. associate-*l* 99.0%

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

   6. associate-*r/ 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in sinTheta_i around 0 98.8%

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

    1. log-rec 98.9%

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

    2. rem-exp-log 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 4: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in sinTheta_O around 0 98.8%

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

   1. times-frac 98.7%

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

   2. *-commutative 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 5: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o * costheta_i) / (v * (v * (sinh((1.0e0 / 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(v * Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)))))
end

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in sinTheta_O around 0 98.8%

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

   1. add-log-exp 97.1%

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

   2. associate-*r* 97.1%

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

   3. *-commutative 97.1%

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

   4. associate-*l* 97.1%

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

   5. exp-prod 95.4%

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

   6. *-commutative 95.4%

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

   7. exp-prod 95.1%

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

6. Applied egg-rr 95.1%

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

   1. log-pow 95.2%

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

   2. log-pow 98.6%

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

   3. rem-log-exp 98.8%

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

   4. *-commutative 98.8%

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

   5. *-commutative 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 6: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in sinTheta_O around 0 98.8%

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

5. Final simplification 98.8%

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

Alternative 7: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in sinTheta_O around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. mul-1-neg 99.0%

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

   3. unsub-neg 99.0%

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

   4. associate-/l* 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in v around inf 67.5%

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

8. Taylor expanded in sinTheta_O around 0 67.5%

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

   1. associate-/l* 67.5%

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

   2. associate-*r/ 67.5%

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

   3. metadata-eval 67.5%

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

   4. *-commutative 67.5%

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

   5. fma-udef 67.5%

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

10. Simplified 67.5%

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

11. Final simplification 67.5%

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

Alternative 8: 64.2% accurate, 10.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (-
   (* cosTheta_O cosTheta_i)
   (/ cosTheta_O (/ v (* cosTheta_i (* sinTheta_O sinTheta_i)))))
  (+ (/ 0.3333333333333333 v) (* 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) - (cosTheta_O / (v / (cosTheta_i * (sinTheta_O * sinTheta_i))))) / ((0.3333333333333333f / v) + (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) - (costheta_o / (v / (costheta_i * (sintheta_o * sintheta_i))))) / ((0.3333333333333333e0 / v) + (v * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in sinTheta_O around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. mul-1-neg 99.0%

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

   3. unsub-neg 99.0%

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

   4. associate-/l* 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in v around inf 67.5%

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

8. Taylor expanded in v around 0 67.5%

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

9. Final simplification 67.5%

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

Alternative 9: 64.2% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

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 * 2.0e0) + ((1.0e0 / v) * 0.3333333333333333e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * cosTheta_i) / ((v * single(2.0)) + ((single(1.0) / v) * single(0.3333333333333333)));
end

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in sinTheta_O around 0 98.8%

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

5. Taylor expanded in v around inf 67.5%

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

6. Final simplification 67.5%

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

Alternative 10: 59.0% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in v around inf 61.9%

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

   1. clear-num 62.1%

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

   2. inv-pow 62.1%

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

   3. *-commutative 62.1%

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

6. Applied egg-rr 62.1%

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

   1. unpow-1 62.1%

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

   2. associate-/r* 62.1%

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

8. Simplified 62.1%

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

9. Final simplification 62.1%

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

Alternative 11: 58.5% 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 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in v around inf 61.9%

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

   1. *-commutative 61.9%

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

   2. associate-*l/ 61.8%

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

   3. *-commutative 61.8%

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

6. Simplified 61.8%

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

7. Final simplification 61.8%

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

Alternative 12: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. associate-/l* 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in v around inf 61.9%

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

   1. associate-/l* 61.8%

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

   2. associate-/r/ 61.8%

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

8. Simplified 61.8%

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

9. Final simplification 61.8%

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

Alternative 13: 58.5% 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 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. associate-/l* 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in v around inf 61.9%

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

   1. associate-/l* 61.8%

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

8. Simplified 61.8%

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

9. Final simplification 61.8%

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

Alternative 14: 58.5% 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 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in v around inf 61.9%

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

5. Final simplification 61.9%

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

Alternative 15: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. associate-*r/ 98.9%

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

   2. associate-/l/ 99.0%

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

   3. *-commutative 99.0%

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

   4. /-rgt-identity 99.0%

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

   5. associate-/r/ 99.0%

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

   6. exp-neg 99.0%

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

   7. remove-double-div 99.0%

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

   8. *-commutative 99.0%

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

   9. associate-*l/ 99.0%

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

   10. exp-prod 99.0%

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

   11. associate-*l* 99.0%

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

   12. associate-*l* 99.0%

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

   13. *-commutative 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in v around inf 61.9%

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

   1. associate-*r/ 61.9%

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

   2. *-commutative 61.9%

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

6. Applied egg-rr 61.9%

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

7. Final simplification 61.9%

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

Reproduce

herbie shell --seed 2023306 
(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)))