HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.6%

Time: 16.4s

Alternatives: 9

Speedup: 0.7×

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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. associate-*l/ 98.8%

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

   2. exp-prod 98.8%

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

6. Applied egg-rr 98.8%

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

   1. associate-*l/ 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-/r* 98.8%

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

   4. associate-/l* 98.8%

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

   5. frac-times 98.8%

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

   6. pow-exp 98.8%

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

   7. *-commutative 98.8%

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

   8. exp-prod 98.8%

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

   9. add-sqr-sqrt 91.1%

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

   10. sqrt-unprod 98.8%

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

   11. sqr-neg 98.8%

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

   12. sqrt-unprod 89.2%

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

   13. add-sqr-sqrt 98.8%

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

8. Applied egg-rr 98.8%

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

   1. div-inv 98.7%

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

   2. associate-*r/ 98.7%

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

   3. *-commutative 98.7%

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

10. Applied egg-rr 98.7%

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

    1. associate-*l* 98.8%

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

    2. associate-/r/ 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 2: 98.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. associate-*l/ 98.8%

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

   2. exp-prod 98.8%

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

6. Applied egg-rr 98.8%

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

   1. associate-*l/ 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-/r* 98.8%

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

   4. associate-/l* 98.8%

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

   5. frac-times 98.8%

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

   6. pow-exp 98.8%

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

   7. *-commutative 98.8%

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

   8. exp-prod 98.8%

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

   9. add-sqr-sqrt 91.1%

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

   10. sqrt-unprod 98.8%

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

   11. sqr-neg 98.8%

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

   12. sqrt-unprod 89.2%

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

   13. add-sqr-sqrt 98.8%

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

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta_i}{{v}^{2}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. associate-*l/ 98.8%

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

   2. exp-prod 98.8%

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

6. Applied egg-rr 98.8%

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

   1. associate-*l/ 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-/r* 98.8%

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

   4. associate-/l* 98.8%

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

   5. frac-times 98.8%

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

   6. pow-exp 98.8%

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

   7. *-commutative 98.8%

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

   8. exp-prod 98.8%

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

   9. add-sqr-sqrt 91.1%

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

   10. sqrt-unprod 98.8%

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

   11. sqr-neg 98.8%

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

   12. sqrt-unprod 89.2%

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

   13. add-sqr-sqrt 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in sinTheta_O around 0 98.7%

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

    1. *-commutative 98.7%

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

    2. times-frac 98.7%

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

    3. rec-exp 98.7%

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

    4. distribute-neg-frac 98.7%

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

    5. metadata-eval 98.7%

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

11. Simplified 98.7%

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

12. Final simplification 98.7%

    \[\leadsto \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta_i}{{v}^{2}} \]
13. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. associate-*l/ 98.8%

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

   2. exp-prod 98.8%

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

6. Applied egg-rr 98.8%

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

   1. associate-*l/ 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-/r* 98.8%

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

   4. associate-/l* 98.8%

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

   5. frac-times 98.8%

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

   6. pow-exp 98.8%

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

   7. *-commutative 98.8%

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

   8. exp-prod 98.8%

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

   9. add-sqr-sqrt 91.1%

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

   10. sqrt-unprod 98.8%

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

   11. sqr-neg 98.8%

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

   12. sqrt-unprod 89.2%

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

   13. add-sqr-sqrt 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in sinTheta_O around inf 98.7%

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

10. Final simplification 98.7%

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

Alternative 5: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. associate-*l/ 98.8%

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

   2. exp-prod 98.8%

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

6. Applied egg-rr 98.8%

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

   1. associate-*l/ 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-/r* 98.8%

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

   4. associate-/l* 98.8%

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

   5. frac-times 98.8%

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

   6. pow-exp 98.8%

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

   7. *-commutative 98.8%

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

   8. exp-prod 98.8%

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

   9. add-sqr-sqrt 91.1%

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

   10. sqrt-unprod 98.8%

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

   11. sqr-neg 98.8%

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

   12. sqrt-unprod 89.2%

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

   13. add-sqr-sqrt 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in sinTheta_O around 0 98.7%

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

10. Final simplification 98.7%

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

Alternative 6: 58.9% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 60.3%

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

   1. associate-*l/ 60.3%

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

7. Applied egg-rr 60.3%

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

   1. *-commutative 60.3%

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

   2. associate-*r/ 60.3%

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

   3. associate-/l* 60.3%

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

   4. clear-num 60.3%

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

9. Applied egg-rr 60.3%

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

10. Final simplification 60.3%

    \[\leadsto 0.5 \cdot \frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \]
11. Add Preprocessing

Alternative 7: 58.9% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 60.3%

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

   1. associate-*r/ 60.3%

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

   2. clear-num 60.3%

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

7. Applied egg-rr 60.3%

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

8. Final simplification 60.3%

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

Alternative 8: 58.3% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 60.3%

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

   1. div-inv 60.3%

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

   2. associate-*l* 60.3%

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

   3. un-div-inv 60.3%

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

7. Applied egg-rr 60.3%

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

8. Final simplification 60.3%

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

Alternative 9: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 60.3%

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

   1. clear-num 60.3%

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

   2. un-div-inv 60.3%

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

7. Applied egg-rr 60.3%

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

8. Final simplification 60.3%

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

Reproduce

herbie shell --seed 2024024 
(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)))