HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 14.8s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i (- sinTheta_O)) v))
   (* (* cosTheta_O cosTheta_i) (/ 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_O * cosTheta_i) * (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_o * costheta_i) * (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(sinTheta_i * Float32(-sinTheta_O)) / v)) * Float32(Float32(cosTheta_O * cosTheta_i) * 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_O * cosTheta_i) * (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%

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

   2. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp((sinTheta_i / (v / -sinTheta_O))) / (sinh((single(1.0) / v)) * single(2.0))) * ((single(1.0) / v) * (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. times-frac 98.3%

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

   2. exp-neg 98.3%

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

   3. *-commutative 98.3%

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

   4. exp-neg 98.3%

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

   5. distribute-neg-frac 98.3%

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

   6. *-commutative 98.3%

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

   7. distribute-rgt-neg-out 98.3%

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

   8. associate-/l* 98.3%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

   1. div-inv 98.5%

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

   2. associate-/l* 98.5%

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

   3. associate-*r/ 98.5%

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

5. Applied egg-rr 98.5%

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

6. Final simplification 98.5%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp((sinTheta_O * (-sinTheta_i / v))) / (sinh((single(1.0) / v)) * single(2.0))) * (cosTheta_i / (v * (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. times-frac 98.3%

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

   2. exp-neg 98.3%

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

   3. *-commutative 98.3%

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

   4. exp-neg 98.3%

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

   5. *-commutative 98.3%

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

   6. associate-/l* 98.3%

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

   7. distribute-frac-neg 98.3%

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

   8. associate-/r/ 98.3%

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

   9. associate-/l* 98.4%

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

   10. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_i (- sinTheta_O)) v)) (* cosTheta_i (/ cosTheta_O 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 / 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 / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v)) * Float32(cosTheta_i * Float32(cosTheta_O / 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 / 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. associate-*r/ 98.4%

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

   2. *-commutative 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 5: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. clear-num 93.0%

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

   2. inv-pow 93.0%

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

5. Applied egg-rr 93.0%

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

   1. unpow-1 93.0%

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

   2. associate-/l* 93.0%

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

   3. associate-/l* 93.1%

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

   4. *-commutative 93.1%

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

7. Simplified 93.1%

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

8. Taylor expanded in sinTheta_O around 0 92.5%

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

   1. associate-/r* 92.7%

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

   2. associate-*r/ 92.5%

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

   3. rec-exp 92.5%

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

   4. distribute-neg-frac 92.5%

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

   5. metadata-eval 92.5%

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

10. Simplified 92.5%

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

    1. div-inv 92.7%

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

12. Applied egg-rr 92.7%

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

    1. expm1-log1p-u 92.7%

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

    2. expm1-udef 53.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{v}{\frac{cosTheta_O \cdot \left(cosTheta_i \cdot \frac{1}{v}\right)}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}}\right)} - 1} \]

    3. associate-/r/ 53.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{v} \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot \frac{1}{v}\right)}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}\right)} - 1 \]

    4. associate-/l* 53.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{v} \cdot \color{blue}{\frac{cosTheta_O}{\frac{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}{cosTheta_i \cdot \frac{1}{v}}}}\right)} - 1 \]

    5. un-div-inv 53.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{v} \cdot \frac{cosTheta_O}{\frac{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}{\color{blue}{\frac{cosTheta_i}{v}}}}\right)} - 1 \]

14. Applied egg-rr 53.0%

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

    1. expm1-def 97.8%

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

    2. expm1-log1p 97.8%

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

    3. associate-*r/ 98.4%

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

    4. *-commutative 98.4%

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

    5. associate-*r/ 98.1%

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

    6. *-rgt-identity 98.1%

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

16. Simplified 98.1%

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

17. Final simplification 98.1%

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

Alternative 6: 93.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. clear-num 93.0%

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

   2. inv-pow 93.0%

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

5. Applied egg-rr 93.0%

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

   1. unpow-1 93.0%

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

   2. associate-/l* 93.0%

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

   3. associate-/l* 93.1%

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

   4. *-commutative 93.1%

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

7. Simplified 93.1%

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

   1. *-un-lft-identity 93.1%

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

   2. div-inv 93.1%

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

   3. times-frac 93.1%

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

9. Applied egg-rr 93.1%

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

10. Taylor expanded in sinTheta_O around 0 93.0%

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

11. Final simplification 93.0%

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

Alternative 7: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.49000000953674316:\\ \;\;\;\;\frac{1}{\frac{v}{\frac{cosTheta_O \cdot \frac{cosTheta_i}{v}}{e^{\frac{1}{v}} + -1}}}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_i \cdot \left(-0.08333333333333333 \cdot \frac{cosTheta_O}{{v}^{3}} + \frac{cosTheta_O}{v} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.49000000953674316)
   (/ 1.0 (/ v (/ (* cosTheta_O (/ cosTheta_i v)) (+ (exp (/ 1.0 v)) -1.0))))
   (*
    cosTheta_i
    (+
     (* -0.08333333333333333 (/ cosTheta_O (pow v 3.0)))
     (* (/ cosTheta_O v) 0.5)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.49000000953674316f) {
		tmp = 1.0f / (v / ((cosTheta_O * (cosTheta_i / v)) / (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = cosTheta_i * ((-0.08333333333333333f * (cosTheta_O / powf(v, 3.0f))) + ((cosTheta_O / v) * 0.5f));
	}
	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 <= 0.49000000953674316e0) then
        tmp = 1.0e0 / (v / ((costheta_o * (costheta_i / v)) / (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = costheta_i * (((-0.08333333333333333e0) * (costheta_o / (v ** 3.0e0))) + ((costheta_o / v) * 0.5e0))
    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(0.49000000953674316))
		tmp = Float32(Float32(1.0) / Float32(v / Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(cosTheta_i * Float32(Float32(Float32(-0.08333333333333333) * Float32(cosTheta_O / (v ^ Float32(3.0)))) + Float32(Float32(cosTheta_O / v) * Float32(0.5))));
	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(0.49000000953674316))
		tmp = single(1.0) / (v / ((cosTheta_O * (cosTheta_i / v)) / (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = cosTheta_i * ((single(-0.08333333333333333) * (cosTheta_O / (v ^ single(3.0)))) + ((cosTheta_O / v) * single(0.5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.49000001

   1. Initial program 98.0%

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

   3. Simplified 98.1%

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

      1. clear-num 94.3%

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

      2. inv-pow 94.3%

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

   5. Applied egg-rr 94.3%

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

      1. unpow-1 94.3%

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

      2. associate-/l* 94.2%

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

      3. associate-/l* 94.3%

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

      4. *-commutative 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in sinTheta_O around 0 93.9%

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

      1. associate-/r* 94.0%

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

      2. associate-*r/ 94.0%

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

      3. rec-exp 94.0%

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

      4. distribute-neg-frac 94.0%

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

      5. metadata-eval 94.0%

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

   10. Simplified 94.0%

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

   11. Taylor expanded in v around inf 70.8%

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

   if 0.49000001 < v

   1. Initial program 99.0%

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

   3. Simplified 98.8%

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

      1. clear-num 91.1%

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

      2. inv-pow 91.1%

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

   5. Applied egg-rr 91.1%

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

      1. unpow-1 91.1%

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

      2. associate-/l* 91.1%

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

      3. associate-/l* 91.2%

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

      4. *-commutative 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in sinTheta_O around 0 90.6%

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

      1. associate-/r* 90.7%

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

      2. associate-*r/ 90.4%

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

      3. rec-exp 90.4%

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

      4. distribute-neg-frac 90.4%

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

      5. metadata-eval 90.4%

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

   10. Simplified 90.4%

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

   11. Taylor expanded in v around inf 68.4%

       \[\leadsto \color{blue}{-0.08333333333333333 \cdot \frac{cosTheta_O \cdot cosTheta_i}{{v}^{3}} + 0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]

   12. Taylor expanded in cosTheta_i around 0 68.5%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.49000000953674316:\\ \;\;\;\;\frac{1}{\frac{v}{\frac{cosTheta_O \cdot \frac{cosTheta_i}{v}}{e^{\frac{1}{v}} + -1}}}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_i \cdot \left(-0.08333333333333333 \cdot \frac{cosTheta_O}{{v}^{3}} + \frac{cosTheta_O}{v} \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 93.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. clear-num 93.0%

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

   2. inv-pow 93.0%

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

5. Applied egg-rr 93.0%

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

   1. unpow-1 93.0%

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

   2. associate-/l* 93.0%

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

   3. associate-/l* 93.1%

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

   4. *-commutative 93.1%

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

7. Simplified 93.1%

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

   1. *-un-lft-identity 93.1%

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

   2. div-inv 93.1%

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

   3. times-frac 93.1%

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

9. Applied egg-rr 93.1%

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

10. Taylor expanded in sinTheta_O around 0 92.6%

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

11. Final simplification 92.6%

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

Alternative 9: 93.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. clear-num 93.0%

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

   2. inv-pow 93.0%

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

5. Applied egg-rr 93.0%

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

   1. unpow-1 93.0%

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

   2. associate-/l* 93.0%

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

   3. associate-/l* 93.1%

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

   4. *-commutative 93.1%

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

7. Simplified 93.1%

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

8. Taylor expanded in sinTheta_O around 0 92.7%

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

9. Final simplification 92.7%

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

Alternative 10: 59.1% 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. times-frac 98.3%

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

   2. exp-neg 98.3%

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

   3. *-commutative 98.3%

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

   4. exp-neg 98.3%

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

   5. *-commutative 98.3%

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

   6. associate-/l* 98.3%

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

   7. distribute-frac-neg 98.3%

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

   8. associate-/r/ 98.3%

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

   9. associate-/l* 98.4%

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

   10. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*l/ 58.6%

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

   2. *-commutative 58.6%

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

6. Simplified 58.6%

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

   1. associate-*r/ 58.6%

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

   2. associate-/l* 58.6%

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

   3. clear-num 58.8%

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

8. Applied egg-rr 58.8%

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

9. Final simplification 58.8%

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

Alternative 11: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. exp-neg 98.3%

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

   3. *-commutative 98.3%

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

   4. exp-neg 98.3%

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

   5. *-commutative 98.3%

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

   6. associate-/l* 98.3%

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

   7. distribute-frac-neg 98.3%

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

   8. associate-/r/ 98.3%

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

   9. associate-/l* 98.4%

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

   10. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*l/ 58.6%

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

   2. *-commutative 58.6%

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

6. Simplified 58.6%

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

7. Taylor expanded in cosTheta_i around 0 58.6%

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

   1. associate-*r/ 58.6%

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

9. Simplified 58.6%

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

10. Final simplification 58.6%

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

Alternative 12: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. exp-neg 98.3%

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

   3. *-commutative 98.3%

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

   4. exp-neg 98.3%

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

   5. *-commutative 98.3%

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

   6. associate-/l* 98.3%

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

   7. distribute-frac-neg 98.3%

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

   8. associate-/r/ 98.3%

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

   9. associate-/l* 98.4%

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

   10. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.6%

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

5. Final simplification 58.6%

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

Reproduce

herbie shell --seed 2023333 
(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)))