HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%

Time: 15.1s

Alternatives: 15

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: 98.5% 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: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (- (/ (* sinTheta_i sinTheta_O) v)))
   (* (* cosTheta_i cosTheta_O) (/ 1.0 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_i * cosTheta_O) * (1.0f / 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_i * costheta_o) * (1.0e0 / 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(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / 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_i * cosTheta_O) * (single(1.0) / v))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 98.5% 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(Float32(sinTheta_i * sinTheta_O) / 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 98.4%

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

   1. distribute-neg-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-rgt-neg-in 98.4%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-*l/ 98.4%

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

   2. times-frac 98.4%

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

   3. exp-neg 98.4%

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

   4. associate-*l/ 98.4%

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

   5. *-lft-identity 98.4%

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

   6. associate-/l/ 98.4%

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

   7. associate-*l* 98.4%

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

   8. associate-*l* 98.4%

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

   9. *-commutative 98.4%

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

   10. *-commutative 98.4%

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

   11. associate-*l/ 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

   2. *-commutative 98.4%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

5. Applied egg-rr 98.4%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 98.4%

      \[\leadsto \color{blue}{1 \cdot \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

   2. times-frac 98.5%

      \[\leadsto 1 \cdot \color{blue}{\left(\frac{cosTheta_i}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)} \]

   3. expm1-log1p-u 98.5%

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

   4. exp-prod 98.5%

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

7. Applied egg-rr 98.5%

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

   1. *-lft-identity 98.5%

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

   2. associate-*l/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l* 98.4%

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

   5. times-frac 98.6%

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

   6. exp-prod 98.6%

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

   7. associate-*l/ 98.6%

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

   8. associate-/l* 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-*r* 98.6%

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

   8. associate-/l/ 98.6%

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

   9. exp-neg 98.6%

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

   10. associate-/l/ 98.6%

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

   11. associate-/r* 98.6%

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

   12. metadata-eval 98.6%

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

   13. associate-*l/ 98.6%

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

   14. *-commutative 98.6%

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

   15. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. associate-/r* 98.4%

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

   2. rec-exp 98.4%

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

   3. distribute-neg-frac 98.4%

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

   4. metadata-eval 98.4%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-*l/ 98.4%

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

   2. times-frac 98.4%

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

   3. exp-neg 98.4%

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

   4. associate-*l/ 98.4%

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

   5. *-lft-identity 98.4%

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

   6. associate-/l/ 98.4%

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

   7. associate-*l* 98.4%

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

   8. associate-*l* 98.4%

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

   9. *-commutative 98.4%

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

   10. *-commutative 98.4%

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

   11. associate-*l/ 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

   2. *-commutative 98.4%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

5. Applied egg-rr 98.4%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 98.4%

      \[\leadsto \color{blue}{1 \cdot \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

   2. times-frac 98.5%

      \[\leadsto 1 \cdot \color{blue}{\left(\frac{cosTheta_i}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)} \]

   3. expm1-log1p-u 98.5%

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

   4. exp-prod 98.5%

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

7. Applied egg-rr 98.5%

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

   1. *-lft-identity 98.5%

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

   2. associate-*l/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l* 98.4%

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

   5. times-frac 98.6%

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

   6. exp-prod 98.6%

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

   7. associate-*l/ 98.6%

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

   8. associate-/l* 98.6%

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

9. Simplified 98.6%

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

10. Taylor expanded in sinTheta_O around 0 98.4%

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

    1. *-commutative 98.4%

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

    2. times-frac 98.3%

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

    3. unpow2 98.3%

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

    4. rec-exp 98.3%

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

    5. distribute-neg-frac 98.3%

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

    6. metadata-eval 98.3%

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

12. Simplified 98.3%

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

13. Final simplification 98.3%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. distribute-neg-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-rgt-neg-in 98.4%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in sinTheta_O around inf 98.5%

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

   1. times-frac 97.6%

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

   2. mul-1-neg 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-/l* 97.6%

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

   5. unpow2 97.6%

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

   6. rec-exp 97.6%

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

   7. distribute-neg-frac 97.6%

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

   8. metadata-eval 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in sinTheta_O around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. unpow2 98.4%

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

   3. associate-*l* 98.3%

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

   4. times-frac 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 7: 72.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \mathbf{if}\;v \leq 0.5120000243186951:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* cosTheta_O (/ cosTheta_i v))))
   (if (<= v 0.5120000243186951)
     (* t_0 (/ (/ 1.0 v) (+ (exp (/ 1.0 v)) -1.0)))
     (*
      t_0
      (+
       0.5
       (-
        (/ 0.009722222222222222 (pow v 4.0))
        (/ 0.08333333333333333 (* v v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cosTheta_O * (cosTheta_i / v);
	float tmp;
	if (v <= 0.5120000243186951f) {
		tmp = t_0 * ((1.0f / v) / (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = t_0 * (0.5f + ((0.009722222222222222f / powf(v, 4.0f)) - (0.08333333333333333f / (v * v))));
	}
	return tmp;
}

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = costheta_o * (costheta_i / v)
    if (v <= 0.5120000243186951e0) then
        tmp = t_0 * ((1.0e0 / v) / (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = t_0 * (0.5e0 + ((0.009722222222222222e0 / (v ** 4.0e0)) - (0.08333333333333333e0 / (v * v))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(cosTheta_O * Float32(cosTheta_i / v))
	tmp = Float32(0.0)
	if (v <= Float32(0.5120000243186951))
		tmp = Float32(t_0 * Float32(Float32(Float32(1.0) / v) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(t_0 * Float32(Float32(0.5) + Float32(Float32(Float32(0.009722222222222222) / (v ^ Float32(4.0))) - Float32(Float32(0.08333333333333333) / Float32(v * v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cosTheta_O * (cosTheta_i / v);
	tmp = single(0.0);
	if (v <= single(0.5120000243186951))
		tmp = t_0 * ((single(1.0) / v) / (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = t_0 * (single(0.5) + ((single(0.009722222222222222) / (v ^ single(4.0))) - (single(0.08333333333333333) / (v * v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.512000024

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*r/ 98.1%

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

      3. associate-/l* 98.2%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 97.8%

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

      1. associate-/r* 98.0%

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

      2. rec-exp 98.0%

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

      3. distribute-neg-frac 98.0%

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

      4. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 70.6%

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

   if 0.512000024 < v

   1. Initial program 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-*r/ 98.8%

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

      3. associate-/l* 98.8%

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

      4. associate-/r/ 98.9%

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

      5. *-commutative 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 98.9%

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

      8. associate-/l/ 98.9%

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

      9. exp-neg 98.9%

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

      10. associate-/l/ 98.9%

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

      11. associate-/r* 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-*l/ 98.9%

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

      14. *-commutative 98.9%

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

      15. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in sinTheta_O around 0 98.9%

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

      1. associate-/r* 99.0%

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

      2. rec-exp 99.0%

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

      3. distribute-neg-frac 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in v around inf 72.4%

      \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \color{blue}{\left(\left(0.5 + 0.009722222222222222 \cdot \frac{1}{{v}^{4}}\right) - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate--l+ 72.4%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \color{blue}{\left(0.5 + \left(0.009722222222222222 \cdot \frac{1}{{v}^{4}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right)} \]

      2. associate-*r/ 72.4%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(0.5 + \left(\color{blue}{\frac{0.009722222222222222 \cdot 1}{{v}^{4}}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right) \]

      3. metadata-eval 72.4%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(0.5 + \left(\frac{\color{blue}{0.009722222222222222}}{{v}^{4}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right) \]

      4. associate-*r/ 72.4%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \color{blue}{\frac{0.08333333333333333 \cdot 1}{{v}^{2}}}\right)\right) \]

      5. metadata-eval 72.4%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{\color{blue}{0.08333333333333333}}{{v}^{2}}\right)\right) \]

      6. unpow2 72.4%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{\color{blue}{v \cdot v}}\right)\right) \]

   9. Simplified 72.4%

      \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \color{blue}{\left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5120000243186951:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \]

Alternative 8: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{v}}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.46000000834465027)
   (* (* cosTheta_O (/ cosTheta_i v)) (/ (/ 1.0 v) (+ (exp (/ 1.0 v)) -1.0)))
   (/
    (* (/ cosTheta_i v) (/ cosTheta_O v))
    (+ (/ 2.0 v) (/ 0.3333333333333333 (pow v 3.0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = (cosTheta_O * (cosTheta_i / v)) * ((1.0f / v) / (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = ((cosTheta_i / v) * (cosTheta_O / v)) / ((2.0f / v) + (0.3333333333333333f / powf(v, 3.0f)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (v <= 0.46000000834465027e0) then
        tmp = (costheta_o * (costheta_i / v)) * ((1.0e0 / v) / (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = ((costheta_i / v) * (costheta_o / v)) / ((2.0e0 / v) + (0.3333333333333333e0 / (v ** 3.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		tmp = Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) * Float32(Float32(Float32(1.0) / v) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(Float32(Float32(cosTheta_i / v) * Float32(cosTheta_O / v)) / Float32(Float32(Float32(2.0) / v) + Float32(Float32(0.3333333333333333) / (v ^ Float32(3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = (cosTheta_O * (cosTheta_i / v)) * ((single(1.0) / v) / (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = ((cosTheta_i / v) * (cosTheta_O / v)) / ((single(2.0) / v) + (single(0.3333333333333333) / (v ^ single(3.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*r/ 98.1%

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

      3. associate-/l* 98.2%

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

      4. associate-/r/ 98.1%

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

      5. *-commutative 98.1%

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

      6. *-commutative 98.1%

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

      7. associate-*r* 98.1%

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

      8. associate-/l/ 98.1%

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

      9. exp-neg 98.1%

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

      10. associate-/l/ 98.1%

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

      11. associate-/r* 98.1%

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

      12. metadata-eval 98.1%

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

      13. associate-*l/ 98.1%

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

      14. *-commutative 98.1%

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

      15. exp-prod 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in sinTheta_O around 0 97.8%

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

      1. associate-/r* 98.0%

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

      2. rec-exp 98.0%

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

      3. distribute-neg-frac 98.0%

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

      4. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 71.0%

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

   if 0.460000008 < v

   1. Initial program 98.8%

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

      1. distribute-neg-frac 98.8%

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

      2. *-commutative 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

      4. associate-*l/ 99.1%

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

      5. *-commutative 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in sinTheta_O around 0 98.9%

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

      1. associate-/r* 98.7%

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

      2. unpow2 98.7%

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

      3. rec-exp 98.7%

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

      4. distribute-neg-frac 98.7%

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

      5. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in v around inf 69.6%

      \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\color{blue}{2 \cdot \frac{1}{v} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 69.6%

         \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\color{blue}{\frac{2 \cdot 1}{v}} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}} \]

      2. metadata-eval 69.6%

         \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\frac{\color{blue}{2}}{v} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}} \]

      3. associate-*r/ 69.6%

         \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\frac{2}{v} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{3}}}} \]

      4. metadata-eval 69.6%

         \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\frac{2}{v} + \frac{\color{blue}{0.3333333333333333}}{{v}^{3}}} \]

   9. Simplified 69.6%

      \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\color{blue}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}}} \]

   10. Taylor expanded in cosTheta_i around 0 69.6%

       \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{{v}^{2}}}}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}} \]
   11. Step-by-step derivation

       1. unpow2 69.6%

          \[\leadsto \frac{\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{v \cdot v}}}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}} \]

       2. times-frac 69.7%

          \[\leadsto \frac{\color{blue}{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{v}}}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}} \]

   12. Simplified 69.7%

       \[\leadsto \frac{\color{blue}{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{v}}}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{v}}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}}\\ \end{array} \]

Alternative 9: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot 2 + \frac{0.3333333333333333}{v}}{cosTheta_O}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.46000000834465027)
   (* (* cosTheta_O (/ cosTheta_i v)) (/ (/ 1.0 v) (+ (exp (/ 1.0 v)) -1.0)))
   (/ cosTheta_i (/ (+ (* v 2.0) (/ 0.3333333333333333 v)) cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = (cosTheta_O * (cosTheta_i / v)) * ((1.0f / v) / (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = cosTheta_i / (((v * 2.0f) + (0.3333333333333333f / v)) / cosTheta_O);
	}
	return tmp;
}

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
    real(4) :: tmp
    if (v <= 0.46000000834465027e0) then
        tmp = (costheta_o * (costheta_i / v)) * ((1.0e0 / v) / (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = costheta_i / (((v * 2.0e0) + (0.3333333333333333e0 / v)) / costheta_o)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		tmp = Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) * Float32(Float32(Float32(1.0) / v) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(cosTheta_i / Float32(Float32(Float32(v * Float32(2.0)) + Float32(Float32(0.3333333333333333) / v)) / cosTheta_O));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = (cosTheta_O * (cosTheta_i / v)) * ((single(1.0) / v) / (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = cosTheta_i / (((v * single(2.0)) + (single(0.3333333333333333) / v)) / cosTheta_O);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*r/ 98.1%

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

      3. associate-/l* 98.2%

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

      4. associate-/r/ 98.1%

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

      5. *-commutative 98.1%

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

      6. *-commutative 98.1%

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

      7. associate-*r* 98.1%

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

      8. associate-/l/ 98.1%

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

      9. exp-neg 98.1%

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

      10. associate-/l/ 98.1%

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

      11. associate-/r* 98.1%

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

      12. metadata-eval 98.1%

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

      13. associate-*l/ 98.1%

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

      14. *-commutative 98.1%

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

      15. exp-prod 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in sinTheta_O around 0 97.8%

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

      1. associate-/r* 98.0%

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

      2. rec-exp 98.0%

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

      3. distribute-neg-frac 98.0%

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

      4. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 71.0%

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

   if 0.460000008 < v

   1. Initial program 98.8%

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

      1. associate-*l/ 98.7%

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

      2. times-frac 98.7%

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

      3. exp-neg 98.7%

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

      4. associate-*l/ 98.7%

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

      5. *-lft-identity 98.7%

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

      6. associate-/l/ 98.7%

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

      7. associate-*l* 98.7%

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

      8. associate-*l* 98.8%

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

      9. *-commutative 98.8%

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

      10. *-commutative 98.8%

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

      11. associate-*l/ 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in v around inf 69.6%

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

   5. Taylor expanded in sinTheta_O around 0 69.6%

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

      1. associate-/l* 69.7%

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

      2. *-commutative 69.7%

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

      3. associate-*r/ 69.7%

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

      4. metadata-eval 69.7%

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

   7. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot 2 + \frac{0.3333333333333333}{v}}{cosTheta_O}}\\ \end{array} \]

Alternative 10: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot 2 + \frac{0.3333333333333333}{v}}{cosTheta_O}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.46000000834465027)
   (/ (/ (* cosTheta_i cosTheta_O) (* v v)) (+ (exp (/ 1.0 v)) -1.0))
   (/ cosTheta_i (/ (+ (* v 2.0) (/ 0.3333333333333333 v)) cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = ((cosTheta_i * cosTheta_O) / (v * v)) / (expf((1.0f / v)) + -1.0f);
	} else {
		tmp = cosTheta_i / (((v * 2.0f) + (0.3333333333333333f / v)) / cosTheta_O);
	}
	return tmp;
}

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
    real(4) :: tmp
    if (v <= 0.46000000834465027e0) then
        tmp = ((costheta_i * costheta_o) / (v * v)) / (exp((1.0e0 / v)) + (-1.0e0))
    else
        tmp = costheta_i / (((v * 2.0e0) + (0.3333333333333333e0 / v)) / costheta_o)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		tmp = Float32(Float32(Float32(cosTheta_i * cosTheta_O) / Float32(v * v)) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)));
	else
		tmp = Float32(cosTheta_i / Float32(Float32(Float32(v * Float32(2.0)) + Float32(Float32(0.3333333333333333) / v)) / cosTheta_O));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = ((cosTheta_i * cosTheta_O) / (v * v)) / (exp((single(1.0) / v)) + single(-1.0));
	else
		tmp = cosTheta_i / (((v * single(2.0)) + (single(0.3333333333333333) / v)) / cosTheta_O);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   1. Initial program 98.1%

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

      1. distribute-neg-frac 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. associate-*l/ 98.2%

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

      5. *-commutative 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 97.8%

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

      1. associate-/r* 97.8%

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

      2. unpow2 97.8%

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

      3. rec-exp 97.8%

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

      4. distribute-neg-frac 97.8%

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

      5. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in v around inf 71.0%

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

   if 0.460000008 < v

   1. Initial program 98.8%

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

      1. associate-*l/ 98.7%

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

      2. times-frac 98.7%

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

      3. exp-neg 98.7%

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

      4. associate-*l/ 98.7%

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

      5. *-lft-identity 98.7%

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

      6. associate-/l/ 98.7%

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

      7. associate-*l* 98.7%

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

      8. associate-*l* 98.8%

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

      9. *-commutative 98.8%

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

      10. *-commutative 98.8%

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

      11. associate-*l/ 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in v around inf 69.6%

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

   5. Taylor expanded in sinTheta_O around 0 69.6%

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

      1. associate-/l* 69.7%

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

      2. *-commutative 69.7%

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

      3. associate-*r/ 69.7%

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

      4. metadata-eval 69.7%

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

   7. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot 2 + \frac{0.3333333333333333}{v}}{cosTheta_O}}\\ \end{array} \]

Alternative 11: 63.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot 2 + \frac{0.3333333333333333}{v}\\ \frac{cosTheta_i}{\frac{t_0}{cosTheta_O}} - \frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ (* v 2.0) (/ 0.3333333333333333 v))))
   (-
    (/ cosTheta_i (/ t_0 cosTheta_O))
    (* (/ sinTheta_i v) (/ (* cosTheta_i (* sinTheta_O cosTheta_O)) t_0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (v * 2.0f) + (0.3333333333333333f / v);
	return (cosTheta_i / (t_0 / cosTheta_O)) - ((sinTheta_i / v) * ((cosTheta_i * (sinTheta_O * cosTheta_O)) / t_0));
}

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
    real(4) :: t_0
    t_0 = (v * 2.0e0) + (0.3333333333333333e0 / v)
    code = (costheta_i / (t_0 / costheta_o)) - ((sintheta_i / v) * ((costheta_i * (sintheta_o * costheta_o)) / t_0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(v * Float32(2.0)) + Float32(Float32(0.3333333333333333) / v))
	return Float32(Float32(cosTheta_i / Float32(t_0 / cosTheta_O)) - Float32(Float32(sinTheta_i / v) * Float32(Float32(cosTheta_i * Float32(sinTheta_O * cosTheta_O)) / t_0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (v * single(2.0)) + (single(0.3333333333333333) / v);
	tmp = (cosTheta_i / (t_0 / cosTheta_O)) - ((sinTheta_i / v) * ((cosTheta_i * (sinTheta_O * cosTheta_O)) / t_0));
end

Derivation

1. Initial program 98.4%

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

   1. associate-*l/ 98.4%

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

   2. times-frac 98.4%

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

   3. exp-neg 98.4%

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

   4. associate-*l/ 98.4%

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

   5. *-lft-identity 98.4%

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

   6. associate-/l/ 98.4%

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

   7. associate-*l* 98.4%

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

   8. associate-*l* 98.4%

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

   9. *-commutative 98.4%

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

   10. *-commutative 98.4%

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

   11. associate-*l/ 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 62.8%

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

5. Taylor expanded in sinTheta_O around 0 62.8%

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

   1. associate-/l* 62.8%

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

   2. *-commutative 62.8%

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

   3. associate-*r/ 62.8%

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

   4. metadata-eval 62.8%

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

   5. mul-1-neg 62.8%

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

   6. times-frac 62.8%

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

   7. *-commutative 62.8%

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

   8. *-commutative 62.8%

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

   9. associate-*r/ 62.8%

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

   10. metadata-eval 62.8%

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

7. Simplified 62.8%

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

8. Final simplification 62.8%

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

Alternative 12: 63.7% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-*l/ 98.4%

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

   2. times-frac 98.4%

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

   3. exp-neg 98.4%

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

   4. associate-*l/ 98.4%

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

   5. *-lft-identity 98.4%

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

   6. associate-/l/ 98.4%

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

   7. associate-*l* 98.4%

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

   8. associate-*l* 98.4%

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

   9. *-commutative 98.4%

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

   10. *-commutative 98.4%

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

   11. associate-*l/ 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 62.8%

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

5. Taylor expanded in sinTheta_O around 0 62.8%

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

   1. associate-/l* 62.8%

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

   2. *-commutative 62.8%

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

   3. associate-*r/ 62.8%

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

   4. metadata-eval 62.8%

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

7. Simplified 62.8%

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

8. Final simplification 62.8%

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

Alternative 13: 58.4% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*r/ 98.4%

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

   3. associate-/l* 98.5%

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

   4. associate-/r/ 98.5%

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

   5. *-commutative 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l/ 98.5%

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

   9. exp-neg 98.5%

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

   10. associate-/l/ 98.5%

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

   11. associate-/r* 98.5%

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

   12. metadata-eval 98.5%

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

   13. associate-*l/ 98.5%

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

   14. *-commutative 98.5%

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

   15. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 56.5%

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

   1. *-commutative 56.5%

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

   2. associate-/r/ 56.5%

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

   3. clear-num 56.7%

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

6. Applied egg-rr 56.7%

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

7. Final simplification 56.7%

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

Alternative 14: 57.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. distribute-neg-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-rgt-neg-in 98.4%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 56.5%

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

5. Final simplification 56.5%

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

Alternative 15: 58.0% accurate, 31.4× speedup

Math

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

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(Float32(Float32(cosTheta_i * 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 98.4%

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

   1. distribute-neg-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-rgt-neg-in 98.4%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 56.5%

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

   1. associate-*r/ 56.6%

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

6. Applied egg-rr 56.6%

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

7. Final simplification 56.6%

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

Reproduce

herbie shell --seed 2023228 
(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)))