HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 17.0s

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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

   1. associate-*r/ 98.5%

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

   2. clear-num 95.5%

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

   3. *-commutative 95.5%

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

5. Applied egg-rr 95.5%

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

   1. associate-/l/ 95.6%

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

   2. associate-/r/ 98.7%

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

   3. times-frac 98.7%

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

   4. pow-exp 98.7%

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

   5. clear-num 98.7%

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

   6. associate-*l/ 98.7%

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

   7. *-un-lft-identity 98.7%

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

7. Applied egg-rr 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*l* 98.7%

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

   3. sinh-def 98.7%

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

   4. sinh-undef 98.7%

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

   5. *-commutative 98.7%

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

   6. associate-*l/ 98.7%

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

9. Applied egg-rr 98.7%

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

    1. associate-/l* 98.8%

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

    2. associate-/r* 98.8%

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

    3. metadata-eval 98.8%

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

    4. *-commutative 98.8%

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

    5. associate-/l* 98.9%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

   1. associate-*r/ 98.5%

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

   2. clear-num 95.5%

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

   3. *-commutative 95.5%

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

5. Applied egg-rr 95.5%

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

   1. associate-/l/ 95.6%

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

   2. associate-/r/ 98.7%

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

   3. times-frac 98.7%

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

   4. pow-exp 98.7%

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

   5. clear-num 98.7%

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

   6. associate-*l/ 98.7%

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

   7. *-un-lft-identity 98.7%

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

7. Applied egg-rr 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*l* 98.7%

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

   3. sinh-def 98.7%

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

   4. sinh-undef 98.7%

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

   5. *-commutative 98.7%

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

   6. associate-*l/ 98.7%

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

9. Applied egg-rr 98.7%

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

    1. associate-/l* 98.8%

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

    2. associate-/r* 98.8%

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

    3. metadata-eval 98.8%

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

    4. *-commutative 98.8%

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

    5. associate-/l* 98.9%

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

11. Simplified 98.9%

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

    1. associate-/r/ 98.8%

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

13. Applied egg-rr 98.8%

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

14. Final simplification 98.8%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

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))))
  (* (/ 1.0 v) (/ cosTheta_i (/ 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)))) * ((1.0f / v) * (cosTheta_i / (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)))) * ((1.0e0 / v) * (costheta_i / (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(Float32(Float32(1.0) / v) * Float32(cosTheta_i / 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)))) * ((single(1.0) / v) * (cosTheta_i / (v / cosTheta_O)));
end

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-/l/ 98.5%

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

   2. div-inv 98.8%

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

   3. associate-/l* 98.6%

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

   4. associate-*r/ 98.7%

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

5. Applied egg-rr 98.7%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l* 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

   1. associate-*r/ 98.5%

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

   2. clear-num 95.5%

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

   3. *-commutative 95.5%

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

5. Applied egg-rr 95.5%

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

   1. associate-/l/ 95.6%

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

   2. associate-/r/ 98.7%

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

   3. times-frac 98.7%

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

   4. pow-exp 98.7%

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

   5. clear-num 98.7%

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

   6. associate-*l/ 98.7%

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

   7. *-un-lft-identity 98.7%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

   1. associate-*r/ 98.5%

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

   2. clear-num 95.5%

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

   3. *-commutative 95.5%

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

5. Applied egg-rr 95.5%

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

   1. associate-/l/ 95.6%

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

   2. associate-/r/ 98.7%

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

   3. times-frac 98.7%

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

   4. pow-exp 98.7%

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

   5. clear-num 98.7%

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

   6. associate-*l/ 98.7%

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

   7. *-un-lft-identity 98.7%

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

7. Applied egg-rr 98.7%

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

   1. associate-*l/ 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-/r/ 98.7%

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

   4. exp-prod 98.7%

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

9. Applied egg-rr 98.7%

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

    1. associate-*r/ 98.7%

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

    2. add-exp-log 98.7%

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

    3. log-pow 98.7%

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

    4. add-log-exp 98.7%

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

11. Applied egg-rr 98.7%

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

12. Final simplification 98.7%

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

Alternative 6: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

   1. associate-*r/ 98.5%

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

   2. clear-num 95.5%

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

   3. *-commutative 95.5%

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

5. Applied egg-rr 95.5%

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

   1. frac-2neg 95.5%

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

   2. div-inv 95.6%

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

   3. distribute-neg-frac 95.6%

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

   4. clear-num 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-*r/ 98.5%

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

   7. pow-exp 98.5%

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

   8. clear-num 98.5%

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

   9. associate-*l/ 98.5%

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

   10. *-un-lft-identity 98.5%

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

   11. associate-*r* 98.5%

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

   12. distribute-rgt-neg-in 98.5%

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

   13. metadata-eval 98.5%

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

7. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-/r/ 98.5%

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

   3. associate-/r/ 98.5%

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

   4. associate-/r* 98.5%

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

   5. associate-/r* 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-/l/ 98.5%

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

   2. div-inv 98.8%

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

   3. associate-/l* 98.6%

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

   4. associate-*r/ 98.7%

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

5. Applied egg-rr 98.7%

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

6. Taylor expanded in sinTheta_i around 0 98.5%

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

   1. rec-exp 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. metadata-eval 98.4%

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

8. Simplified 98.5%

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

9. Final simplification 98.5%

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

Alternative 8: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

   1. associate-*r/ 98.5%

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

   2. clear-num 95.5%

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

   3. *-commutative 95.5%

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

5. Applied egg-rr 95.5%

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

   1. expm1-log1p-u 95.5%

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

   2. expm1-udef 53.1%

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

7. Applied egg-rr 53.1%

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

   1. expm1-def 98.4%

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

   2. expm1-log1p 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-/r/ 98.5%

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

   5. associate-*l* 98.5%

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

   6. associate-/r/ 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 9: 98.3% accurate, 1.0× speedup

Math

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

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 (* v (/ v cosTheta_O)))))

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 / (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 = (1.0e0 / (exp((1.0e0 / v)) - exp(((-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(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) - exp(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 = (single(1.0) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * (cosTheta_i / (v * (v / cosTheta_O)));
end

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_i around 0 98.4%

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

   1. rec-exp 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. metadata-eval 98.4%

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

6. Simplified 98.4%

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

7. Final simplification 98.4%

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

Alternative 10: 64.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in v around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. metadata-eval 64.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

4. Simplified 64.4%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

5. Taylor expanded in sinTheta_i around 0 64.4%

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

   1. times-frac 64.4%

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

   2. associate-*r/ 64.4%

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

   3. metadata-eval 64.4%

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

7. Simplified 64.4%

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

8. Final simplification 64.4%

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

Alternative 11: 64.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O}{\frac{v \cdot \left(2 + \frac{0.3333333333333333}{{v}^{2}}\right)}{cosTheta_i}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  cosTheta_O
  (/ (* v (+ 2.0 (/ 0.3333333333333333 (pow v 2.0)))) 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 / powf(v, 2.0f)))) / 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 / (v ** 2.0e0)))) / costheta_i)
end function

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in v around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. metadata-eval 64.4%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

4. Simplified 64.4%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

5. Taylor expanded in sinTheta_i around 0 64.4%

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

   1. associate-/l* 64.5%

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

   2. associate-*r/ 64.5%

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

   3. metadata-eval 64.5%

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

7. Simplified 64.5%

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

8. Final simplification 64.5%

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

Alternative 12: 59.3% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.5%

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

6. Simplified 58.5%

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

   1. associate-*r/ 58.6%

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

   2. clear-num 58.8%

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

   3. div-inv 59.1%

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

   4. associate-/r* 59.1%

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

8. Applied egg-rr 59.1%

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

9. Final simplification 59.1%

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

Alternative 13: 58.7% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.5%

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

6. Simplified 58.5%

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

   1. associate-*r/ 58.6%

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

   2. *-commutative 58.6%

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

   3. associate-*r/ 58.5%

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

   4. add-exp-log 47.4%

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

8. Applied egg-rr 47.4%

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

   1. add-exp-log 58.5%

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

   2. associate-*r/ 58.6%

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

   3. div-inv 58.6%

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

   4. *-commutative 58.6%

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

10. Applied egg-rr 58.6%

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

11. Final simplification 58.6%

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

Alternative 14: 59.1% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.5%

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

6. Simplified 58.5%

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

   1. associate-*r/ 58.6%

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

   2. clear-num 58.8%

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

8. Applied egg-rr 58.8%

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

9. Final simplification 58.8%

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

Alternative 15: 58.6% 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 98.5%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.5%

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

6. Simplified 58.5%

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

7. Final simplification 58.5%

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

Alternative 16: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.5%

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

6. Simplified 58.5%

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

7. Taylor expanded in cosTheta_O around 0 58.6%

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

   1. associate-/l* 58.5%

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

9. Simplified 58.5%

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

    1. associate-/r/ 58.5%

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

11. Applied egg-rr 58.5%

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

12. Final simplification 58.5%

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

Alternative 17: 58.7% 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 98.5%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.6%

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

5. Final simplification 58.6%

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

Reproduce

herbie shell --seed 2023319 
(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)))