HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 16.1s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l* 98.6%

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

6. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. div-inv 98.6%

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

   3. times-frac 98.8%

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

8. Applied egg-rr 98.8%

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

   1. associate-/l/ 98.8%

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

   2. times-frac 98.8%

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

   3. div-inv 98.8%

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

   4. inv-pow 98.8%

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

   5. pow-flip 98.9%

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

   6. metadata-eval 98.9%

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

   7. pow1 98.9%

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

   8. pow-exp 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O / v) * (cosTheta_i / sinh((single(1.0) / v)))) * ((exp((sinTheta_O * (sinTheta_i / v))) / single(2.0)) / 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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.9%

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

   2. *-commutative 98.9%

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

4. Applied egg-rr 98.9%

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

   1. associate-*l* 98.9%

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

   2. div-inv 98.7%

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

   3. *-commutative 98.7%

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

   4. associate-/r/ 98.6%

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

6. Applied egg-rr 98.6%

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

   1. add-cube-cbrt 98.4%

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

   2. pow3 98.4%

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

8. Applied egg-rr 98.1%

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

   1. rem-cube-cbrt 98.4%

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

   2. associate-/r* 98.4%

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

   3. associate-/l* 98.4%

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

   4. frac-times 98.4%

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

   5. associate-/r/ 98.4%

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

   6. exp-prod 98.4%

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

10. Applied egg-rr 98.4%

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

    1. times-frac 98.4%

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

    2. associate-/r/ 98.4%

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

    3. exp-prod 98.4%

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

    4. *-commutative 98.4%

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

12. Simplified 98.4%

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

13. Final simplification 98.4%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l* 98.6%

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

6. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. div-inv 98.6%

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

   3. times-frac 98.8%

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

8. Applied egg-rr 98.8%

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

   1. associate-/l/ 98.8%

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

   2. frac-times 98.6%

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

   3. *-un-lft-identity 98.6%

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

   4. div-inv 98.6%

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

   5. div-inv 98.6%

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

   6. clear-num 98.6%

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

   7. times-frac 98.7%

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

   8. pow-exp 98.7%

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

10. Applied egg-rr 98.7%

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

11. Final simplification 98.7%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * (cosTheta_i / (v * (v / cosTheta_O)));
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. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-/l* 98.6%

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

   7. distribute-frac-neg 98.6%

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

   8. associate-/r/ 98.6%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in sinTheta_i around 0 98.4%

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

   1. rec-exp 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. metadata-eval 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 5: 58.8% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (/ 1.0 (/ (/ 1.0 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 * ((1.0f / ((1.0f / 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 * ((1.0e0 / ((1.0e0 / 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(Float32(1.0) / Float32(Float32(Float32(1.0) / cosTheta_O) / cosTheta_i)) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((single(1.0) / ((single(1.0) / 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. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-/l* 98.6%

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

   7. distribute-frac-neg 98.6%

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

   8. associate-/r/ 98.6%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 58.9%

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

   1. *-commutative 58.9%

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

   2. pow1 58.9%

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

   3. metadata-eval 58.9%

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

   4. pow-flip 58.9%

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

   5. inv-pow 58.9%

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

   6. div-inv 58.9%

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

   7. clear-num 59.6%

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

7. Applied egg-rr 59.6%

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

8. Final simplification 59.6%

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

Alternative 6: 58.7% 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. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-/l* 98.6%

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

   7. distribute-frac-neg 98.6%

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

   8. associate-/r/ 98.6%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 58.9%

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

   1. *-commutative 58.9%

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

   2. associate-*l/ 58.9%

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

   3. *-commutative 58.9%

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

7. Simplified 58.9%

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

   1. add-sqr-sqrt 49.4%

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

   2. pow2 49.4%

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

   3. *-commutative 49.4%

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

   4. associate-*r/ 49.4%

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

   5. associate-*l/ 49.4%

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

   6. *-commutative 49.4%

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

9. Applied egg-rr 49.4%

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

    1. unpow2 49.4%

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

    2. add-sqr-sqrt 58.9%

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

11. Applied egg-rr 58.9%

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

    1. associate-*r/ 58.9%

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

    2. associate-/l* 58.9%

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

    3. clear-num 59.4%

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

13. Applied egg-rr 59.4%

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

14. Final simplification 59.4%

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

Alternative 7: 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. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-/l* 98.6%

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

   7. distribute-frac-neg 98.6%

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

   8. associate-/r/ 98.6%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 58.9%

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

   1. *-commutative 58.9%

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

   2. associate-*l/ 58.9%

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

   3. *-commutative 58.9%

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

7. Simplified 58.9%

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

8. Final simplification 58.9%

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

Alternative 8: 58.3% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-/l* 98.6%

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

   7. distribute-frac-neg 98.6%

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

   8. associate-/r/ 98.6%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 58.9%

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

6. Final simplification 58.9%

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

Alternative 9: 58.7% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l* 98.6%

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

6. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. div-inv 98.6%

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

   3. times-frac 98.8%

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

8. Applied egg-rr 98.8%

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

   1. associate-/l/ 98.8%

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

   2. times-frac 98.8%

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

   3. div-inv 98.8%

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

   4. inv-pow 98.8%

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

   5. pow-flip 98.9%

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

   6. metadata-eval 98.9%

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

   7. pow1 98.9%

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

   8. pow-exp 98.9%

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

10. Applied egg-rr 98.9%

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

11. Taylor expanded in v around inf 58.9%

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

    1. associate-*r/ 58.9%

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

    2. *-commutative 58.9%

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

    3. associate-/l* 59.4%

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

    4. *-commutative 59.4%

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

13. Simplified 59.4%

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

14. Final simplification 59.4%

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

Reproduce

herbie shell --seed 2024010 
(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)))