HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 17.6s

Alternatives: 11

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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(\frac{1}{v} \cdot \left(\frac{1}{v} \cdot cosTheta\_i\right)\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.7%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}} \]

   2. associate-*l/ 98.7%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-*r/ 98.7%

      \[\leadsto \color{blue}{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   4. distribute-frac-neg2 98.7%

      \[\leadsto e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. associate-/l* 98.7%

      \[\leadsto e^{\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. exp-prod 98.7%

      \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. *-commutative 98.7%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\frac{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. associate-/l* 98.8%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\frac{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. associate-/l* 98.7%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\color{blue}{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 98.8%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\frac{1}{\frac{v}{\frac{cosTheta\_i}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. inv-pow 98.8%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{{\left(\frac{v}{\frac{cosTheta\_i}{v}}\right)}^{-1}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Applied egg-rr 98.8%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{{\left(\frac{v}{\frac{cosTheta\_i}{v}}\right)}^{-1}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
7. Step-by-step derivation

   1. unpow-1 98.8%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\frac{1}{\frac{v}{\frac{cosTheta\_i}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. associate-/r/ 98.8%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i} \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

8. Simplified 98.8%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i} \cdot v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
9. Step-by-step derivation

   1. associate-/r* 98.7%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\frac{\frac{1}{\frac{v}{cosTheta\_i}}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   2. clear-num 98.7%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{\color{blue}{\frac{cosTheta\_i}{v}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. un-div-inv 99.0%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. *-commutative 99.0%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{cosTheta\_i}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. div-inv 99.1%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{1}{v}\right)}\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. associate-*r* 99.1%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\left(\frac{1}{v} \cdot cosTheta\_i\right) \cdot \frac{1}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

10. Applied egg-rr 99.1%

    \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\left(\frac{1}{v} \cdot cosTheta\_i\right) \cdot \frac{1}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

11. Final simplification 99.1%

    \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(\frac{1}{v} \cdot \left(\frac{1}{v} \cdot cosTheta\_i\right)\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
12. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(\frac{1}{v} \cdot \frac{cosTheta\_i}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.7%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}} \]

   2. associate-*l/ 98.7%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. associate-*r/ 98.7%

      \[\leadsto \color{blue}{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   4. distribute-frac-neg2 98.7%

      \[\leadsto e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. associate-/l* 98.7%

      \[\leadsto e^{\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. exp-prod 98.7%

      \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)}} \cdot \frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. *-commutative 98.7%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\frac{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. associate-/l* 98.8%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\frac{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. associate-/l* 98.7%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\color{blue}{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 99.0%

      \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

6. Applied egg-rr 99.0%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

7. Final simplification 99.0%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(\frac{1}{v} \cdot \frac{cosTheta\_i}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
8. Add Preprocessing

Alternative 3: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)} \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)}} \]

   2. times-frac 98.9%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)}} \]

   3. *-commutative 98.9%

      \[\leadsto \frac{cosTheta\_i}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)}} \]

6. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)}} \]

7. Final simplification 98.9%

   \[\leadsto \frac{cosTheta\_i}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \]
8. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.9%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied egg-rr 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Final simplification 98.9%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
6. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0 98.7%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
4. Step-by-step derivation

   1. associate-*r/ 98.8%

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Simplified 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

6. Final simplification 98.8%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
7. Add Preprocessing

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / sinh((single(1.0) / v))) * (single(0.5) * (cosTheta_O / (v ^ single(2.0))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.8%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(\left(v \cdot 2\right) \cdot v\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

   3. *-commutative 98.8%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]

7. Taylor expanded in v around inf 98.7%

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

8. Final simplification 98.7%

   \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \left(0.5 \cdot \frac{cosTheta\_O}{{v}^{2}}\right) \]
9. Add Preprocessing

Alternative 7: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 63.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
6. Step-by-step derivation

   1. clear-num 63.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_i \cdot cosTheta\_O}}} \]

   2. inv-pow 63.8%

      \[\leadsto \color{blue}{{\left(\frac{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}} \]

   3. times-frac 63.8%

      \[\leadsto {\color{blue}{\left(\frac{v}{cosTheta\_i} \cdot \frac{2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{cosTheta\_O}\right)}}^{-1} \]

   4. +-commutative 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\color{blue}{2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 2}}{cosTheta\_O}\right)}^{-1} \]

   5. fma-define 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{v}, 2\right)}}{cosTheta\_O}\right)}^{-1} \]

   6. associate-/l* 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}, 2\right)}{cosTheta\_O}\right)}^{-1} \]

7. Applied egg-rr 63.8%

   \[\leadsto \color{blue}{{\left(\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 63.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]

9. Simplified 63.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]

10. Final simplification 63.8%

    \[\leadsto \frac{1}{\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}} \]
11. Add Preprocessing

Alternative 8: 59.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{v \cdot \frac{\frac{\mathsf{fma}\left(sinTheta\_O \cdot 2, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{cosTheta\_i}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  1.0
  (*
   v
   (/
    (/ (fma (* sinTheta_O 2.0) (/ sinTheta_i v) 2.0) 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 * ((fmaf((sinTheta_O * 2.0f), (sinTheta_i / v), 2.0f) / cosTheta_O) / cosTheta_i));
}

Julia

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 63.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
6. Step-by-step derivation

   1. clear-num 63.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_i \cdot cosTheta\_O}}} \]

   2. inv-pow 63.8%

      \[\leadsto \color{blue}{{\left(\frac{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}} \]

   3. times-frac 63.8%

      \[\leadsto {\color{blue}{\left(\frac{v}{cosTheta\_i} \cdot \frac{2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{cosTheta\_O}\right)}}^{-1} \]

   4. +-commutative 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\color{blue}{2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 2}}{cosTheta\_O}\right)}^{-1} \]

   5. fma-define 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{v}, 2\right)}}{cosTheta\_O}\right)}^{-1} \]

   6. associate-/l* 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}, 2\right)}{cosTheta\_O}\right)}^{-1} \]

7. Applied egg-rr 63.8%

   \[\leadsto \color{blue}{{\left(\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 63.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]

   2. associate-/r* 62.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]

   3. fma-undefine 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{2 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right) + 2}}{cosTheta\_O}} \]

   4. associate-*r/ 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{2 \cdot \color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} + 2}{cosTheta\_O}} \]

   5. associate-*r/ 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\frac{2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}} + 2}{cosTheta\_O}} \]

   6. associate-*l* 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\frac{\color{blue}{\left(2 \cdot sinTheta\_O\right) \cdot sinTheta\_i}}{v} + 2}{cosTheta\_O}} \]

   7. associate-/l* 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\left(2 \cdot sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}} + 2}{cosTheta\_O}} \]

   8. fma-define 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}}{cosTheta\_O}} \]

9. Simplified 62.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. associate-/r/ 62.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{v} \cdot cosTheta\_i}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}} \]

11. Applied egg-rr 62.9%

    \[\leadsto \frac{\color{blue}{\frac{1}{v} \cdot cosTheta\_i}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}} \]
12. Step-by-step derivation

    1. clear-num 63.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{\frac{1}{v} \cdot cosTheta\_i}}} \]

    2. inv-pow 63.8%

       \[\leadsto \color{blue}{{\left(\frac{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{\frac{1}{v} \cdot cosTheta\_i}\right)}^{-1}} \]

    3. associate-*l/ 63.8%

       \[\leadsto {\left(\frac{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{\color{blue}{\frac{1 \cdot cosTheta\_i}{v}}}\right)}^{-1} \]

    4. *-un-lft-identity 63.8%

       \[\leadsto {\left(\frac{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{\frac{\color{blue}{cosTheta\_i}}{v}}\right)}^{-1} \]

13. Applied egg-rr 63.8%

    \[\leadsto \color{blue}{{\left(\frac{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{\frac{cosTheta\_i}{v}}\right)}^{-1}} \]
14. Step-by-step derivation

    1. unpow-1 63.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{\frac{cosTheta\_i}{v}}}} \]

    2. associate-/r/ 63.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{cosTheta\_i} \cdot v}} \]

15. Simplified 63.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{cosTheta\_i} \cdot v}} \]

16. Final simplification 63.8%

    \[\leadsto \frac{1}{v \cdot \frac{\frac{\mathsf{fma}\left(sinTheta\_O \cdot 2, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}{cosTheta\_i}} \]
17. Add Preprocessing

Alternative 9: 58.8% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 63.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

6. Final simplification 63.0%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \]
7. Add Preprocessing

Alternative 10: 58.8% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 63.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
6. Step-by-step derivation

   1. clear-num 63.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_i \cdot cosTheta\_O}}} \]

   2. inv-pow 63.8%

      \[\leadsto \color{blue}{{\left(\frac{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}} \]

   3. times-frac 63.8%

      \[\leadsto {\color{blue}{\left(\frac{v}{cosTheta\_i} \cdot \frac{2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{cosTheta\_O}\right)}}^{-1} \]

   4. +-commutative 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\color{blue}{2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 2}}{cosTheta\_O}\right)}^{-1} \]

   5. fma-define 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{v}, 2\right)}}{cosTheta\_O}\right)}^{-1} \]

   6. associate-/l* 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}, 2\right)}{cosTheta\_O}\right)}^{-1} \]

7. Applied egg-rr 63.8%

   \[\leadsto \color{blue}{{\left(\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 63.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]

   2. associate-/r* 62.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]

   3. fma-undefine 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{2 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right) + 2}}{cosTheta\_O}} \]

   4. associate-*r/ 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{2 \cdot \color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} + 2}{cosTheta\_O}} \]

   5. associate-*r/ 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\frac{2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}} + 2}{cosTheta\_O}} \]

   6. associate-*l* 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\frac{\color{blue}{\left(2 \cdot sinTheta\_O\right) \cdot sinTheta\_i}}{v} + 2}{cosTheta\_O}} \]

   7. associate-/l* 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\left(2 \cdot sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}} + 2}{cosTheta\_O}} \]

   8. fma-define 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}}{cosTheta\_O}} \]

9. Simplified 62.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. associate-/r/ 62.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{v} \cdot cosTheta\_i}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}} \]

11. Applied egg-rr 62.9%

    \[\leadsto \frac{\color{blue}{\frac{1}{v} \cdot cosTheta\_i}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}} \]

12. Taylor expanded in v around inf 63.0%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
13. Step-by-step derivation

    1. associate-/l* 62.9%

       \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

14. Simplified 62.9%

    \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

15. Final simplification 62.9%

    \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot 0.5 \]
16. Add Preprocessing

Alternative 11: 58.8% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 63.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
6. Step-by-step derivation

   1. clear-num 63.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_i \cdot cosTheta\_O}}} \]

   2. inv-pow 63.8%

      \[\leadsto \color{blue}{{\left(\frac{v \cdot \left(2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}} \]

   3. times-frac 63.8%

      \[\leadsto {\color{blue}{\left(\frac{v}{cosTheta\_i} \cdot \frac{2 + 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{cosTheta\_O}\right)}}^{-1} \]

   4. +-commutative 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\color{blue}{2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 2}}{cosTheta\_O}\right)}^{-1} \]

   5. fma-define 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{v}, 2\right)}}{cosTheta\_O}\right)}^{-1} \]

   6. associate-/l* 63.8%

      \[\leadsto {\left(\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}, 2\right)}{cosTheta\_O}\right)}^{-1} \]

7. Applied egg-rr 63.8%

   \[\leadsto \color{blue}{{\left(\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 63.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta\_i} \cdot \frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]

   2. associate-/r* 62.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\mathsf{fma}\left(2, sinTheta\_O \cdot \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]

   3. fma-undefine 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{2 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right) + 2}}{cosTheta\_O}} \]

   4. associate-*r/ 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{2 \cdot \color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} + 2}{cosTheta\_O}} \]

   5. associate-*r/ 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\frac{2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}} + 2}{cosTheta\_O}} \]

   6. associate-*l* 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\frac{\color{blue}{\left(2 \cdot sinTheta\_O\right) \cdot sinTheta\_i}}{v} + 2}{cosTheta\_O}} \]

   7. associate-/l* 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\left(2 \cdot sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}} + 2}{cosTheta\_O}} \]

   8. fma-define 62.9%

      \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\color{blue}{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}}{cosTheta\_O}} \]

9. Simplified 62.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{\frac{v}{cosTheta\_i}}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. associate-/r/ 62.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{v} \cdot cosTheta\_i}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}} \]

11. Applied egg-rr 62.9%

    \[\leadsto \frac{\color{blue}{\frac{1}{v} \cdot cosTheta\_i}}{\frac{\mathsf{fma}\left(2 \cdot sinTheta\_O, \frac{sinTheta\_i}{v}, 2\right)}{cosTheta\_O}} \]

12. Taylor expanded in v around inf 63.0%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

13. Final simplification 63.0%

    \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(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)))