HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 18.2s

Alternatives: 17

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: 98.8% accurate, 1.0× speedup

Math

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

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)))
  (/ 2.0 (/ (/ 1.0 v) (sinh (/ 1.0 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_O * cosTheta_i) * (1.0f / v))) / (2.0f / ((1.0f / v) / sinhf((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 = (exp(((sintheta_i * -sintheta_o) / v)) * ((costheta_o * costheta_i) * (1.0e0 / v))) / (2.0e0 / ((1.0e0 / v) / sinh((1.0e0 / v))))
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(Float32(2.0) / Float32(Float32(Float32(1.0) / v) / sinh(Float32(Float32(1.0) / v)))))
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))) / (single(2.0) / ((single(1.0) / v) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.6%

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

   1. expm1-log1p-u 98.4%

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

3. Applied egg-rr 98.4%

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

   1. expm1-udef 98.3%

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

   2. log1p-udef 98.2%

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

   3. rem-exp-log 98.4%

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

5. Applied egg-rr 98.4%

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

   1. associate-*l* 98.4%

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

   2. add-exp-log 98.2%

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

   3. log1p-udef 98.3%

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

   4. expm1-udef 98.4%

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

   5. expm1-log1p-u 98.6%

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

   6. sinh-def 98.5%

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

   7. sinh-undef 98.6%

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

   8. expm1-log1p-u 98.4%

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

   9. expm1-udef 98.3%

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

   10. log1p-udef 98.2%

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

   11. add-exp-log 98.4%

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

   12. *-commutative 98.4%

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

   13. associate-*l/ 98.4%

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

7. Applied egg-rr 98.6%

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

   1. associate-/l* 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-/r* 98.7%

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

   4. metadata-eval 98.7%

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

   5. associate-/l* 98.8%

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

9. Simplified 98.8%

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

    1. div-inv 98.9%

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

    2. *-commutative 98.9%

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

11. Applied egg-rr 98.9%

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

12. Final simplification 98.9%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-rgt-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.8%

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

5. Applied egg-rr 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. expm1-log1p-u 98.4%

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

3. Applied egg-rr 98.4%

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

   1. expm1-udef 98.3%

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

   2. log1p-udef 98.2%

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

   3. rem-exp-log 98.4%

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

5. Applied egg-rr 98.4%

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

   1. associate-*l* 98.4%

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

   2. add-exp-log 98.2%

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

   3. log1p-udef 98.3%

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

   4. expm1-udef 98.4%

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

   5. expm1-log1p-u 98.6%

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

   6. sinh-def 98.5%

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

   7. sinh-undef 98.6%

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

   8. expm1-log1p-u 98.4%

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

   9. expm1-udef 98.3%

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

   10. log1p-udef 98.2%

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

   11. add-exp-log 98.4%

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

   12. *-commutative 98.4%

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

   13. associate-*l/ 98.4%

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

7. Applied egg-rr 98.6%

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

   1. associate-/l* 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-/r* 98.7%

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

   4. metadata-eval 98.7%

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

   5. associate-/l* 98.8%

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

9. Simplified 98.8%

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

10. Taylor expanded in cosTheta_i around 0 98.8%

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

    1. associate-/l* 98.8%

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

12. Simplified 98.8%

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

13. Final simplification 98.8%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf((sinTheta_i / (v / -sinTheta_O))) / (2.0f * sinhf((1.0f / v)))) * (cosTheta_O / (v * (v / 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 = (exp((sintheta_i / (v / -sintheta_o))) / (2.0e0 * sinh((1.0e0 / v)))) * (costheta_o / (v * (v / costheta_i)))
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(Float32(2.0) * sinh(Float32(Float32(1.0) / v)))) * Float32(cosTheta_O / Float32(v * Float32(v / cosTheta_i))))
end

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-rgt-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.8%

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

5. Applied egg-rr 98.8%

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

   1. un-div-inv 98.6%

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

   2. *-commutative 98.6%

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

   3. associate-/r/ 98.5%

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

   4. associate-/l/ 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-rgt-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.8%

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

5. Applied egg-rr 98.8%

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

   1. associate-*r/ 98.6%

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

   2. *-rgt-identity 98.6%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-/l* 98.5%

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

7. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in sinTheta_i around 0 98.0%

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

    1. rec-exp 98.1%

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

    2. distribute-neg-frac 98.1%

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

    3. metadata-eval 98.1%

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

12. Simplified 98.1%

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

    1. div-inv 98.3%

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

    2. div-inv 98.3%

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

    3. clear-num 98.3%

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

    4. associate-*r/ 98.2%

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

14. Applied egg-rr 98.2%

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

15. Final simplification 98.2%

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

Alternative 7: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-rgt-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.8%

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

5. Applied egg-rr 98.8%

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

   1. associate-*r/ 98.6%

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

   2. *-rgt-identity 98.6%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-/l* 98.5%

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

7. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in sinTheta_i around 0 98.0%

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

    1. rec-exp 98.1%

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

    2. distribute-neg-frac 98.1%

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

    3. metadata-eval 98.1%

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

12. Simplified 98.1%

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

    1. associate-/l* 98.0%

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

    2. div-inv 98.2%

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

    3. *-commutative 98.2%

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

    4. associate-*r* 98.4%

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

14. Applied egg-rr 98.4%

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

15. Final simplification 98.4%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in sinTheta_i around 0 98.1%

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

   1. rec-exp 98.2%

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

   2. distribute-neg-frac 98.2%

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

   3. metadata-eval 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 9: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-rgt-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.8%

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

5. Applied egg-rr 98.8%

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

   1. associate-*r/ 98.6%

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

   2. *-rgt-identity 98.6%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-/l* 98.5%

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

7. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in sinTheta_i around 0 98.0%

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

    1. rec-exp 98.1%

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

    2. distribute-neg-frac 98.1%

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

    3. metadata-eval 98.1%

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

12. Simplified 98.1%

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

    1. associate-/r/ 57.5%

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

14. Applied egg-rr 98.2%

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

15. Final simplification 98.2%

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

Alternative 10: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-rgt-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.8%

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

5. Applied egg-rr 98.8%

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

   1. associate-*r/ 98.6%

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

   2. *-rgt-identity 98.6%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-/l* 98.5%

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

7. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in sinTheta_i around 0 98.0%

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

    1. rec-exp 98.1%

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

    2. distribute-neg-frac 98.1%

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

    3. metadata-eval 98.1%

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

12. Simplified 98.1%

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

    1. *-un-lft-identity 98.1%

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

    2. associate-/l/ 98.1%

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

14. Applied egg-rr 98.1%

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

15. Final simplification 98.1%

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

Alternative 11: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v}\\ \mathbf{if}\;v \leq 0.5130000114440918:\\ \;\;\;\;\frac{1}{e^{\frac{1}{v}} + \left(\frac{1}{v} + -1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \frac{0.08333333333333333}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (/ cosTheta_O (/ v cosTheta_i)) v)))
   (if (<= v 0.5130000114440918)
     (* (/ 1.0 (+ (exp (/ 1.0 v)) (+ (/ 1.0 v) -1.0))) t_0)
     (*
      t_0
      (+
       (/ 0.009722222222222222 (pow v 3.0))
       (- (* v 0.5) (/ 0.08333333333333333 v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_O / (v / cosTheta_i)) / v;
	float tmp;
	if (v <= 0.5130000114440918f) {
		tmp = (1.0f / (expf((1.0f / v)) + ((1.0f / v) + -1.0f))) * t_0;
	} else {
		tmp = t_0 * ((0.009722222222222222f / powf(v, 3.0f)) + ((v * 0.5f) - (0.08333333333333333f / 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 / (v / costheta_i)) / v
    if (v <= 0.5130000114440918e0) then
        tmp = (1.0e0 / (exp((1.0e0 / v)) + ((1.0e0 / v) + (-1.0e0)))) * t_0
    else
        tmp = t_0 * ((0.009722222222222222e0 / (v ** 3.0e0)) + ((v * 0.5e0) - (0.08333333333333333e0 / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_O / Float32(v / cosTheta_i)) / v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5130000114440918))
		tmp = Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(Float32(Float32(1.0) / v) + Float32(-1.0)))) * t_0);
	else
		tmp = Float32(t_0 * Float32(Float32(Float32(0.009722222222222222) / (v ^ Float32(3.0))) + Float32(Float32(v * Float32(0.5)) - Float32(Float32(0.08333333333333333) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_O / (v / cosTheta_i)) / v;
	tmp = single(0.0);
	if (v <= single(0.5130000114440918))
		tmp = (single(1.0) / (exp((single(1.0) / v)) + ((single(1.0) / v) + single(-1.0)))) * t_0;
	else
		tmp = t_0 * ((single(0.009722222222222222) / (v ^ single(3.0))) + ((v * single(0.5)) - (single(0.08333333333333333) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.513000011

   1. Initial program 98.3%

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

      1. times-frac 98.2%

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

      2. exp-neg 98.2%

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

      3. *-commutative 98.2%

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

      4. exp-neg 98.2%

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

      5. distribute-neg-frac 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-rgt-neg-out 98.2%

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

      8. associate-/l* 98.2%

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

      9. associate-/l* 98.3%

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

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
   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* 98.4%

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

      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. Step-by-step derivation

      1. associate-*r/ 98.3%

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

      2. *-rgt-identity 98.3%

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

      3. associate-*r/ 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-/l* 98.3%

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

   7. Simplified 98.3%

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

      1. expm1-log1p-u 98.0%

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

   9. Applied egg-rr 97.9%

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

   10. Taylor expanded in sinTheta_i around 0 98.0%

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

       1. rec-exp 98.0%

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

       2. distribute-neg-frac 98.0%

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

       3. metadata-eval 98.0%

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

   12. Simplified 98.0%

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

   13. Taylor expanded in v around inf 69.4%

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

   if 0.513000011 < v

   1. Initial program 98.9%

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

      1. times-frac 98.9%

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

      2. exp-neg 98.9%

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

      3. *-commutative 98.9%

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

      4. exp-neg 98.9%

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

      5. distribute-neg-frac 98.9%

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

      6. *-commutative 98.9%

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

      7. distribute-rgt-neg-out 98.9%

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

      8. associate-/l* 98.9%

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

      9. associate-/l* 98.8%

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

   3. Simplified 98.8%

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

      1. div-inv 99.0%

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

      2. associate-/l* 99.0%

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

      3. associate-*r/ 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

      3. associate-*r/ 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-/l* 98.7%

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

   7. Simplified 98.7%

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

      1. expm1-log1p-u 98.8%

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

   9. Applied egg-rr 98.9%

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

   10. Taylor expanded in sinTheta_i around 0 98.1%

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

       1. rec-exp 98.2%

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

       2. distribute-neg-frac 98.2%

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

       3. metadata-eval 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in v around inf 72.1%

       \[\leadsto \color{blue}{\left(\left(0.009722222222222222 \cdot \frac{1}{{v}^{3}} + 0.5 \cdot v\right) - 0.08333333333333333 \cdot \frac{1}{v}\right)} \cdot \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v} \]
   14. Step-by-step derivation

       1. associate--l+ 72.1%

          \[\leadsto \color{blue}{\left(0.009722222222222222 \cdot \frac{1}{{v}^{3}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right)} \cdot \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v} \]

       2. associate-*r/ 72.1%

          \[\leadsto \left(\color{blue}{\frac{0.009722222222222222 \cdot 1}{{v}^{3}}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v} \]

       3. metadata-eval 72.1%

          \[\leadsto \left(\frac{\color{blue}{0.009722222222222222}}{{v}^{3}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v} \]

       4. *-commutative 72.1%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(\color{blue}{v \cdot 0.5} - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v} \]

       5. associate-*r/ 72.1%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \color{blue}{\frac{0.08333333333333333 \cdot 1}{v}}\right)\right) \cdot \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v} \]

       6. metadata-eval 72.1%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \frac{\color{blue}{0.08333333333333333}}{v}\right)\right) \cdot \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v} \]

   15. Simplified 72.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5130000114440918:\\ \;\;\;\;\frac{1}{e^{\frac{1}{v}} + \left(\frac{1}{v} + -1\right)} \cdot \frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}{v} \cdot \left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \frac{0.08333333333333333}{v}\right)\right)\\ \end{array} \]

Alternative 12: 67.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-rgt-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.8%

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

5. Applied egg-rr 98.8%

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

   1. associate-*r/ 98.6%

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

   2. *-rgt-identity 98.6%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-/l* 98.5%

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

7. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in sinTheta_i around 0 98.0%

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

    1. rec-exp 98.1%

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

    2. distribute-neg-frac 98.1%

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

    3. metadata-eval 98.1%

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

12. Simplified 98.1%

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

13. Taylor expanded in v around inf 66.9%

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

14. Final simplification 66.9%

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

Alternative 13: 58.5% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.5%

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

   1. *-commutative 57.5%

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

   2. associate-*l/ 57.5%

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

   3. *-commutative 57.5%

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

6. Simplified 57.5%

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

   1. clear-num 57.5%

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

   2. div-inv 57.5%

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

   3. clear-num 58.3%

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

8. Applied egg-rr 58.3%

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

9. Final simplification 58.3%

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

Alternative 14: 58.6% accurate, 24.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / ((v / cosTheta_i) / (cosTheta_O * 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 = 1.0e0 / ((v / costheta_i) / (costheta_o * 0.5e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.5%

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

   1. associate-/l* 57.5%

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

6. Simplified 57.5%

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

   1. associate-*r/ 57.5%

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

   2. clear-num 58.3%

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

   3. *-commutative 58.3%

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

8. Applied egg-rr 58.3%

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

9. Final simplification 58.3%

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

Alternative 15: 58.0% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.5%

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

   1. *-commutative 57.5%

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

   2. associate-*l/ 57.5%

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

   3. *-commutative 57.5%

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

6. Simplified 57.5%

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

7. Final simplification 57.5%

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

Alternative 16: 58.0% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.5%

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

   1. associate-/l* 57.5%

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

6. Simplified 57.5%

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

   1. associate-/r/ 57.5%

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

8. Applied egg-rr 57.5%

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

9. Final simplification 57.5%

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

Alternative 17: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.5%

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

   10. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.5%

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

   1. associate-/l* 57.5%

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

6. Simplified 57.5%

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

   1. associate-/l* 57.5%

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

   2. clear-num 58.3%

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

   3. un-div-inv 58.3%

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

8. Applied egg-rr 58.3%

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

9. Final simplification 58.3%

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

Reproduce

herbie shell --seed 2023315 
(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)))