HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 18.8s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. expm1-log1p-u 98.5%

      \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)}} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

6. Applied egg-rr 98.5%

   \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)}} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
7. Step-by-step derivation

   1. expm1-log1p-u 98.7%

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

   2. associate-*l/ 98.9%

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

   3. *-commutative 98.9%

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

   4. *-commutative 98.9%

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

   5. exp-prod 98.9%

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

8. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

   2. times-frac 98.9%

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

   3. /-rgt-identity 98.9%

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

   4. distribute-frac-neg 98.9%

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

10. Simplified 98.9%

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

    1. associate-*r/ 98.9%

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

    2. associate-/l* 98.8%

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

    3. clear-num 94.6%

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

12. Applied egg-rr 94.6%

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

    1. associate-/l/ 94.6%

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

    2. associate-/r/ 99.0%

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

    3. *-commutative 99.0%

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

14. Simplified 99.0%

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

15. Final simplification 99.0%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. expm1-log1p-u 98.5%

      \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)}} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

6. Applied egg-rr 98.5%

   \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)}} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
7. Step-by-step derivation

   1. expm1-log1p-u 98.7%

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

   2. expm1-log1p-u 98.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   3. expm1-udef 53.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   4. associate-/l* 53.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta\_i}}}\right)} - 1\right) \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

8. Applied egg-rr 53.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta\_i}}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
9. Step-by-step derivation

   1. expm1-def 98.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta\_O}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta\_i}}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   2. expm1-log1p 98.3%

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

   3. associate-/r/ 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. associate-/l* 98.7%

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

   2. associate-/r/ 98.9%

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

4. Applied egg-rr 98.9%

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

   1. expm1-log1p-u 98.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right)} \]

   2. expm1-udef 53.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)} - 1} \]

   3. times-frac 53.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta\_i}{v} \cdot cosTheta\_O}{v}}\right)} - 1 \]

   4. associate-/l* 53.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{-\color{blue}{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta\_i}{v} \cdot cosTheta\_O}{v}\right)} - 1 \]

   5. *-commutative 53.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{-\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}}{v}\right)} - 1 \]

   6. associate-*r/ 53.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{-\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}}{v}\right)} - 1 \]

   7. associate-*l/ 53.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{-\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}}{v}\right)} - 1 \]

   8. *-commutative 53.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{-\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}{v}\right)} - 1 \]

6. Applied egg-rr 53.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{-\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 98.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{-\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v}\right)\right)} \]

   2. expm1-log1p 98.7%

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

   3. associate-/r/ 98.7%

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

   4. distribute-rgt-neg-out 98.7%

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

   5. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in cosTheta_O around 0 98.7%

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

   1. associate-*r/ 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. associate-/l* 98.7%

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

   2. associate-/r/ 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 6: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. expm1-log1p-u 98.5%

      \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)}} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

6. Applied egg-rr 98.5%

   \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)}} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
7. Step-by-step derivation

   1. expm1-log1p-u 98.7%

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

   2. associate-*l/ 98.9%

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

   3. *-commutative 98.9%

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

   4. *-commutative 98.9%

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

   5. exp-prod 98.9%

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

8. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

   2. times-frac 98.9%

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

   3. /-rgt-identity 98.9%

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

   4. distribute-frac-neg 98.9%

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

10. Simplified 98.9%

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

    1. associate-*r/ 98.9%

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

    2. associate-/l* 98.8%

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

    3. clear-num 94.6%

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

12. Applied egg-rr 94.6%

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

    1. associate-/l/ 94.6%

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

    2. associate-/r/ 99.0%

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

    3. *-commutative 99.0%

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

14. Simplified 99.0%

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

15. Taylor expanded in sinTheta_O around 0 98.5%

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

    1. associate-/r* 98.5%

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

    2. rec-exp 98.4%

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

    3. distribute-neg-frac 98.4%

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

    4. metadata-eval 98.4%

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

17. Simplified 98.5%

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

18. Final simplification 98.5%

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

Alternative 7: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. expm1-log1p-u 98.5%

      \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)}} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

6. Applied egg-rr 98.5%

   \[\leadsto \frac{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)}} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
7. Step-by-step derivation

   1. expm1-log1p-u 98.7%

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

   2. associate-*l/ 98.9%

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

   3. *-commutative 98.9%

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

   4. *-commutative 98.9%

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

   5. exp-prod 98.9%

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

8. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

   2. times-frac 98.9%

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

   3. /-rgt-identity 98.9%

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

   4. distribute-frac-neg 98.9%

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

10. Simplified 98.9%

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

11. Taylor expanded in sinTheta_O around 0 98.4%

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

    1. associate-/r* 98.5%

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

    2. rec-exp 98.4%

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

    3. distribute-neg-frac 98.4%

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

    4. metadata-eval 98.4%

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

13. Simplified 98.4%

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

14. Final simplification 98.4%

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

Alternative 8: 64.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * expf(((sinTheta_i * -sinTheta_O) / v))) / ((2.0f * (v / cosTheta_i)) + (0.3333333333333333f * (1.0f / (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 = (costheta_o * exp(((sintheta_i * -sintheta_o) / v))) / ((2.0e0 * (v / costheta_i)) + (0.3333333333333333e0 * (1.0e0 / (v * costheta_i))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. associate-/l* 92.7%

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

   2. *-commutative 92.7%

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

   3. distribute-neg-frac 92.7%

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

   4. distribute-rgt-neg-out 92.7%

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

   5. associate-/l* 92.7%

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

   6. *-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-/l* 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in v around inf 64.3%

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

6. Taylor expanded in cosTheta_O around 0 65.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{2 \cdot \frac{v}{cosTheta\_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta\_i \cdot v}}} \]

7. Final simplification 65.2%

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

Alternative 9: 64.5% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \frac{-cosTheta\_O}{\frac{v \cdot -2 + 0.3333333333333333 \cdot \frac{-1}{v}}{cosTheta\_i}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return -cosTheta_O / (((v * -2.0f) + (0.3333333333333333f * (-1.0f / 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 = -costheta_o / (((v * (-2.0e0)) + (0.3333333333333333e0 * ((-1.0e0) / v))) / costheta_i)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. associate-/l* 92.7%

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

   2. *-commutative 92.7%

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

   3. distribute-neg-frac 92.7%

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

   4. distribute-rgt-neg-out 92.7%

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

   5. associate-/l* 92.7%

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

   6. *-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-/l* 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in v around inf 64.3%

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

6. Taylor expanded in cosTheta_i around -inf 65.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]
7. Step-by-step derivation

   1. mul-1-neg 65.2%

      \[\leadsto \color{blue}{-\frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]

   2. associate-/l* 65.2%

      \[\leadsto -\color{blue}{\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

   3. associate-*r/ 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{cosTheta\_O \cdot v}}}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   4. metadata-eval 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{\color{blue}{0.3333333333333333}}{cosTheta\_O \cdot v}}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   5. *-commutative 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{\color{blue}{v \cdot cosTheta\_O}}}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   6. neg-mul-1 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{v \cdot cosTheta\_O}}{e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

   7. distribute-neg-frac 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{v \cdot cosTheta\_O}}{e^{\color{blue}{\frac{-sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

8. Simplified 65.2%

   \[\leadsto \color{blue}{-\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{v \cdot cosTheta\_O}}{e^{\frac{-sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

9. Taylor expanded in cosTheta_O around 0 65.2%

   \[\leadsto -\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)}{-2 \cdot v - 0.3333333333333333 \cdot \frac{1}{v}}} \]
10. Step-by-step derivation

    1. associate-/l* 65.2%

       \[\leadsto -\color{blue}{\frac{cosTheta\_O}{\frac{-2 \cdot v - 0.3333333333333333 \cdot \frac{1}{v}}{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

    2. *-commutative 65.2%

       \[\leadsto -\frac{cosTheta\_O}{\frac{\color{blue}{v \cdot -2} - 0.3333333333333333 \cdot \frac{1}{v}}{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

    3. associate-*r/ 65.2%

       \[\leadsto -\frac{cosTheta\_O}{\frac{v \cdot -2 - \color{blue}{\frac{0.3333333333333333 \cdot 1}{v}}}{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

    4. metadata-eval 65.2%

       \[\leadsto -\frac{cosTheta\_O}{\frac{v \cdot -2 - \frac{\color{blue}{0.3333333333333333}}{v}}{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

    5. *-commutative 65.2%

       \[\leadsto -\frac{cosTheta\_O}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{\color{blue}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \cdot cosTheta\_i}}} \]

    6. neg-mul-1 65.2%

       \[\leadsto -\frac{cosTheta\_O}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \cdot cosTheta\_i}} \]

    7. distribute-neg-frac 65.2%

       \[\leadsto -\frac{cosTheta\_O}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{e^{\color{blue}{\frac{-sinTheta\_O \cdot sinTheta\_i}{v}}} \cdot cosTheta\_i}} \]

    8. distribute-rgt-neg-out 65.2%

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

    9. associate-*r/ 65.2%

       \[\leadsto -\frac{cosTheta\_O}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{e^{\color{blue}{sinTheta\_O \cdot \frac{-sinTheta\_i}{v}}} \cdot cosTheta\_i}} \]

11. Simplified 65.2%

    \[\leadsto -\color{blue}{\frac{cosTheta\_O}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{e^{sinTheta\_O \cdot \frac{-sinTheta\_i}{v}} \cdot cosTheta\_i}}} \]

12. Taylor expanded in sinTheta_O around 0 65.2%

    \[\leadsto -\frac{cosTheta\_O}{\color{blue}{\frac{-2 \cdot v - 0.3333333333333333 \cdot \frac{1}{v}}{cosTheta\_i}}} \]

13. Final simplification 65.2%

    \[\leadsto \frac{-cosTheta\_O}{\frac{v \cdot -2 + 0.3333333333333333 \cdot \frac{-1}{v}}{cosTheta\_i}} \]
14. Add Preprocessing

Alternative 10: 64.5% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \frac{-cosTheta\_i}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{cosTheta\_O}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. associate-/l* 92.7%

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

   2. *-commutative 92.7%

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

   3. distribute-neg-frac 92.7%

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

   4. distribute-rgt-neg-out 92.7%

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

   5. associate-/l* 92.7%

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

   6. *-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-/l* 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in v around inf 64.3%

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

6. Taylor expanded in cosTheta_i around -inf 65.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]
7. Step-by-step derivation

   1. mul-1-neg 65.2%

      \[\leadsto \color{blue}{-\frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]

   2. associate-/l* 65.2%

      \[\leadsto -\color{blue}{\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

   3. associate-*r/ 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{cosTheta\_O \cdot v}}}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   4. metadata-eval 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{\color{blue}{0.3333333333333333}}{cosTheta\_O \cdot v}}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   5. *-commutative 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{\color{blue}{v \cdot cosTheta\_O}}}{e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   6. neg-mul-1 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{v \cdot cosTheta\_O}}{e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

   7. distribute-neg-frac 65.2%

      \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{v \cdot cosTheta\_O}}{e^{\color{blue}{\frac{-sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

8. Simplified 65.2%

   \[\leadsto \color{blue}{-\frac{cosTheta\_i}{\frac{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{v \cdot cosTheta\_O}}{e^{\frac{-sinTheta\_O \cdot sinTheta\_i}{v}}}}} \]

9. Taylor expanded in sinTheta_O around 0 65.2%

   \[\leadsto -\color{blue}{\frac{cosTheta\_i}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]
10. Step-by-step derivation

    1. associate-*r/ 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\color{blue}{\frac{-2 \cdot v}{cosTheta\_O}} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}} \]

    2. associate-*r/ 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot v}{cosTheta\_O} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{cosTheta\_O \cdot v}}} \]

    3. metadata-eval 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot v}{cosTheta\_O} - \frac{\color{blue}{0.3333333333333333}}{cosTheta\_O \cdot v}} \]

    4. associate-/l/ 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot v}{cosTheta\_O} - \color{blue}{\frac{\frac{0.3333333333333333}{v}}{cosTheta\_O}}} \]

    5. metadata-eval 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot v}{cosTheta\_O} - \frac{\frac{\color{blue}{0.3333333333333333 \cdot 1}}{v}}{cosTheta\_O}} \]

    6. associate-*r/ 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\frac{-2 \cdot v}{cosTheta\_O} - \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{v}}}{cosTheta\_O}} \]

    7. div-sub 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\color{blue}{\frac{-2 \cdot v - 0.3333333333333333 \cdot \frac{1}{v}}{cosTheta\_O}}} \]

    8. *-commutative 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\frac{\color{blue}{v \cdot -2} - 0.3333333333333333 \cdot \frac{1}{v}}{cosTheta\_O}} \]

    9. associate-*r/ 65.2%

       \[\leadsto -\frac{cosTheta\_i}{\frac{v \cdot -2 - \color{blue}{\frac{0.3333333333333333 \cdot 1}{v}}}{cosTheta\_O}} \]

    10. metadata-eval 65.2%

        \[\leadsto -\frac{cosTheta\_i}{\frac{v \cdot -2 - \frac{\color{blue}{0.3333333333333333}}{v}}{cosTheta\_O}} \]

11. Simplified 65.2%

    \[\leadsto -\color{blue}{\frac{cosTheta\_i}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{cosTheta\_O}}} \]

12. Final simplification 65.2%

    \[\leadsto \frac{-cosTheta\_i}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{cosTheta\_O}} \]
13. Add Preprocessing

Alternative 11: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 58.9%

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

   1. associate-*r/ 58.9%

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

7. Simplified 58.9%

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

8. Final simplification 58.9%

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

Alternative 12: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 58.9%

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

6. Final simplification 58.9%

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

Alternative 13: 59.3% 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 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.7%

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

   6. distribute-neg-frac 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. associate-*l/ 98.7%

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

   9. *-commutative 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 58.9%

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

   1. associate-*r/ 58.9%

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

   2. associate-/l* 59.6%

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

7. Simplified 59.6%

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

8. Final simplification 59.6%

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

Reproduce

herbie shell --seed 2024031 
(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)))