HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 14.3s

Alternatives: 10

Speedup: 0.7×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. times-frac 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

   1. times-frac 98.9%

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

   2. un-div-inv 98.7%

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

8. Applied egg-rr 98.7%

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

   1. div-inv 98.7%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 2: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. times-frac 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

   1. times-frac 98.9%

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

   2. un-div-inv 98.7%

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

8. Applied egg-rr 98.7%

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

   1. div-inv 98.7%

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

10. Applied egg-rr 98.9%

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

11. Taylor expanded in sinTheta_i around 0 98.8%

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

12. Final simplification 98.8%

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

Alternative 3: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. times-frac 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

   1. times-frac 98.9%

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

   2. un-div-inv 98.7%

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

8. Applied egg-rr 98.7%

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

9. Taylor expanded in sinTheta_i around 0 98.6%

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

10. Final simplification 98.6%

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

Alternative 4: 64.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot \left(2 - \frac{\frac{-0.3333333333333333 \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v} - 0.3333333333333333}{v} - 2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right)} \end{array} \]

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Simplified 98.4%

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

   1. associate-/l* 98.5%

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

   2. associate-*l* 98.5%

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

   3. *-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in sinTheta_i around 0 98.5%

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

8. Taylor expanded in v around -inf 64.1%

   \[\leadsto cosTheta\_i \cdot \frac{cosTheta\_O}{\color{blue}{-1 \cdot \left(v \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-0.3333333333333333 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} - 0.3333333333333333}{v} + 2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v} - 2\right)\right)}} \]

9. Final simplification 64.1%

   \[\leadsto cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot \left(2 - \frac{\frac{-0.3333333333333333 \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v} - 0.3333333333333333}{v} - 2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right)} \]
10. Add Preprocessing

Alternative 5: 64.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot \left(2 + \frac{2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right) + \frac{1}{v} \cdot 0.3333333333333333}{v}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Simplified 98.4%

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

   1. associate-/l* 98.5%

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

   2. associate-*l* 98.5%

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

   3. *-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in sinTheta_i around 0 98.5%

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

8. Taylor expanded in v around -inf 64.1%

   \[\leadsto cosTheta\_i \cdot \frac{cosTheta\_O}{\color{blue}{-1 \cdot \left(v \cdot \left(-1 \cdot \frac{2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 0.3333333333333333 \cdot \frac{1}{v}}{v} - 2\right)\right)}} \]

9. Final simplification 64.1%

   \[\leadsto cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot \left(2 + \frac{2 \cdot \left(sinTheta\_i \cdot sinTheta\_O\right) + \frac{1}{v} \cdot 0.3333333333333333}{v}\right)} \]
10. Add Preprocessing

Alternative 6: 58.2% accurate, 11.6× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (+
   (* -0.5 (/ (* cosTheta_O (* cosTheta_i (* sinTheta_i sinTheta_O))) v))
   (* 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 * (sinTheta_i * sinTheta_O))) / v)) + (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 * (sintheta_i * sintheta_o))) / v)) + (0.5e0 * (costheta_o * costheta_i))) / v
end function

Julia

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

Derivation

1. Initial program 98.3%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 58.8%

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

6. Final simplification 58.8%

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

Alternative 7: 58.2% accurate, 14.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   0.5
   (*
    cosTheta_O
    (+ cosTheta_i (/ (* cosTheta_i (* sinTheta_i sinTheta_O)) v))))
  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 + ((cosTheta_i * (sinTheta_i * sinTheta_O)) / v)))) / 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 + ((costheta_i * (sintheta_i * sintheta_o)) / v)))) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-*r/ 98.3%

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

   2. associate-/l/ 98.5%

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

   3. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.3%

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

   9. associate-*r/ 98.3%

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

3. Simplified 98.4%

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

   1. *-un-lft-identity 98.4%

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

6. Applied egg-rr 98.4%

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

   1. *-lft-identity 98.4%

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

   2. times-frac 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-/l* 98.4%

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

   5. associate-/l/ 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*r* 98.4%

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

   8. unpow2 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in sinTheta_i around 0 98.4%

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

    1. add-exp-log 45.8%

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

    2. associate-/r* 45.8%

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

11. Applied egg-rr 45.8%

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

12. Taylor expanded in v around inf 58.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + 0.5 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}} \]
13. Step-by-step derivation

    1. distribute-lft-out 58.8%

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

    2. associate-/l* 58.8%

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

    3. distribute-lft-out 58.8%

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

14. Simplified 58.8%

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

15. Final simplification 58.8%

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

Alternative 8: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Simplified 98.4%

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

   1. associate-/l* 98.5%

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

   2. associate-*l* 98.5%

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

   3. *-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in v around inf 58.8%

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

   1. associate-*r/ 58.8%

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

   2. *-commutative 58.8%

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

9. Simplified 58.8%

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

    1. associate-*r/ 58.8%

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

11. Applied egg-rr 58.8%

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

Alternative 9: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Simplified 98.4%

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

   1. associate-/l* 98.5%

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

   2. associate-*l* 98.5%

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

   3. *-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in v around inf 58.8%

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

   1. associate-*r/ 58.8%

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

   2. *-commutative 58.8%

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

9. Simplified 58.8%

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

Alternative 10: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Simplified 98.4%

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

   1. associate-/l* 98.5%

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

   2. associate-*l* 98.5%

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

   3. *-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in v around inf 58.8%

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

   1. associate-*r/ 58.8%

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

   2. *-commutative 58.8%

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

9. Simplified 58.8%

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

    1. associate-/l* 58.8%

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

11. Applied egg-rr 58.8%

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

Reproduce

herbie shell --seed 2024132 
(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)))