HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 14.3s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (* cosTheta_O (/ cosTheta_i v)) (exp (* (/ sinTheta_i v) sinTheta_O)))
  (* (/ 0.5 v) (/ 1.0 (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 / v)) / expf(((sinTheta_i / v) * sinTheta_O))) * ((0.5f / v) * (1.0f / 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 / v)) / exp(((sintheta_i / v) * sintheta_o))) * ((0.5e0 / v) * (1.0e0 / sinh((1.0e0 / v))))
end function

Julia

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

Derivation

1. Initial program 98.3%

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

   1. div-inv 98.5%

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

3. Applied egg-rr 98.5%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 49.6%

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

   3. *-commutative 49.6%

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

   4. div-inv 49.6%

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

   5. associate-/l* 49.6%

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

   6. distribute-neg-frac 49.6%

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

   7. *-commutative 49.6%

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

5. Applied egg-rr 49.6%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 98.5%

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

   3. *-rgt-identity 98.5%

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

   4. associate-*r/ 98.4%

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

   5. associate-/r* 98.6%

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

7. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot \frac{cosTheta_i}{v}}{e^{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}} \]
8. Step-by-step derivation

   1. div-inv 98.7%

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (* cosTheta_O (/ cosTheta_i v)) (exp (* (/ sinTheta_i v) sinTheta_O)))
  (/ (/ 0.5 v) (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 / v)) / expf(((sinTheta_i / v) * sinTheta_O))) * ((0.5f / 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 = ((costheta_o * (costheta_i / v)) / exp(((sintheta_i / v) * sintheta_o))) * ((0.5e0 / v) / sinh((1.0e0 / v)))
end function

Julia

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

Derivation

1. Initial program 98.3%

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

   1. div-inv 98.5%

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

3. Applied egg-rr 98.5%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 49.6%

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

   3. *-commutative 49.6%

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

   4. div-inv 49.6%

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

   5. associate-/l* 49.6%

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

   6. distribute-neg-frac 49.6%

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

   7. *-commutative 49.6%

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

5. Applied egg-rr 49.6%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 98.5%

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

   3. *-rgt-identity 98.5%

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

   4. associate-*r/ 98.4%

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

   5. associate-/r* 98.6%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*r/ 98.4%

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

   3. associate-/l* 98.4%

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

   4. associate-/r/ 98.5%

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

   5. *-commutative 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

   8. associate-/l/ 98.5%

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

   9. exp-neg 98.5%

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

   10. associate-/l/ 98.5%

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

   11. associate-/r* 98.5%

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

   12. metadata-eval 98.5%

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

   13. associate-*l/ 98.5%

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

   14. *-commutative 98.5%

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

   15. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_O around 0 98.4%

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

   1. rec-exp 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. metadata-eval 98.4%

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

6. Simplified 98.4%

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

   1. expm1-log1p-u 98.3%

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

   2. expm1-udef 96.0%

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

8. Applied egg-rr 96.0%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 98.4%

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

   3. associate-/r* 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.4%

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

   5. *-commutative 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*r* 98.4%

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

   8. associate-/l/ 98.4%

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

   9. exp-neg 98.4%

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

   10. associate-/l/ 98.4%

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

   11. associate-/r* 98.4%

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

   12. metadata-eval 98.4%

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

   13. associate-*l/ 98.4%

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

   14. *-commutative 98.4%

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

   15. exp-prod 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. rec-exp 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. metadata-eval 98.4%

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

6. Simplified 98.3%

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

   1. expm1-log1p-u 98.3%

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

   2. expm1-udef 96.0%

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

8. Applied egg-rr 96.1%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 98.4%

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

   3. associate-/r* 98.6%

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

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-*l/ 98.4%

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

   5. *-commutative 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*r* 98.4%

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

   8. associate-/l/ 98.4%

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

   9. exp-neg 98.4%

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

   10. associate-/l/ 98.4%

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

   11. associate-/r* 98.4%

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

   12. metadata-eval 98.4%

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

   13. associate-*l/ 98.4%

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

   14. *-commutative 98.4%

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

   15. exp-prod 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_O around 0 98.4%

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

   1. times-frac 98.4%

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

   2. unpow2 98.4%

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

   3. rec-exp 98.4%

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

   4. distribute-neg-frac 98.4%

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

   5. metadata-eval 98.4%

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

6. Simplified 98.4%

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

7. Final simplification 98.4%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. div-inv 98.5%

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

3. Applied egg-rr 98.5%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 49.6%

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

   3. *-commutative 49.6%

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

   4. div-inv 49.6%

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

   5. associate-/l* 49.6%

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

   6. distribute-neg-frac 49.6%

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

   7. *-commutative 49.6%

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

5. Applied egg-rr 49.6%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 98.5%

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

   3. *-rgt-identity 98.5%

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

   4. associate-*r/ 98.4%

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

   5. associate-/r* 98.6%

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

7. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot \frac{cosTheta_i}{v}}{e^{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}} \]
8. Step-by-step derivation

   1. div-inv 98.7%

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

9. Applied egg-rr 98.7%

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

10. Taylor expanded in sinTheta_i around 0 98.4%

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

    1. *-commutative 98.4%

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

    2. associate-/l* 98.2%

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

    3. *-commutative 98.2%

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

    4. rec-exp 98.3%

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

    5. distribute-neg-frac 98.3%

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

    6. metadata-eval 98.3%

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

    7. unpow2 98.3%

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

12. Simplified 98.3%

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

13. Final simplification 98.3%

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

Alternative 7: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \mathbf{if}\;v \leq 0.5149999856948853:\\ \;\;\;\;t_0 \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* cosTheta_O (/ cosTheta_i v))))
   (if (<= v 0.5149999856948853)
     (* t_0 (/ 1.0 (* v (+ (exp (/ 1.0 v)) -1.0))))
     (*
      t_0
      (+
       0.5
       (-
        (/ 0.009722222222222222 (pow v 4.0))
        (/ 0.08333333333333333 (* v v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cosTheta_O * (cosTheta_i / v);
	float tmp;
	if (v <= 0.5149999856948853f) {
		tmp = t_0 * (1.0f / (v * (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = t_0 * (0.5f + ((0.009722222222222222f / powf(v, 4.0f)) - (0.08333333333333333f / (v * v))));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = costheta_o * (costheta_i / v)
    if (v <= 0.5149999856948853e0) then
        tmp = t_0 * (1.0e0 / (v * (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = t_0 * (0.5e0 + ((0.009722222222222222e0 / (v ** 4.0e0)) - (0.08333333333333333e0 / (v * v))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(cosTheta_O * Float32(cosTheta_i / v))
	tmp = Float32(0.0)
	if (v <= Float32(0.5149999856948853))
		tmp = Float32(t_0 * Float32(Float32(1.0) / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(t_0 * Float32(Float32(0.5) + Float32(Float32(Float32(0.009722222222222222) / (v ^ Float32(4.0))) - Float32(Float32(0.08333333333333333) / Float32(v * v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cosTheta_O * (cosTheta_i / v);
	tmp = single(0.0);
	if (v <= single(0.5149999856948853))
		tmp = t_0 * (single(1.0) / (v * (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = t_0 * (single(0.5) + ((single(0.009722222222222222) / (v ^ single(4.0))) - (single(0.08333333333333333) / (v * v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.514999986

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*r/ 98.0%

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

      3. associate-/l* 98.2%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 98.2%

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

      1. rec-exp 98.2%

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

      2. distribute-neg-frac 98.2%

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

      3. metadata-eval 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in v around inf 69.3%

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

   if 0.514999986 < v

   1. Initial program 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-*r/ 98.9%

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

      3. associate-/l* 98.9%

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

      4. associate-/r/ 98.9%

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

      5. *-commutative 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 98.9%

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

      8. associate-/l/ 98.9%

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

      9. exp-neg 98.9%

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

      10. associate-/l/ 98.9%

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

      11. associate-/r* 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-*l/ 98.9%

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

      14. *-commutative 98.9%

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

      15. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in sinTheta_O around 0 98.7%

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

      1. rec-exp 98.7%

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

      2. distribute-neg-frac 98.7%

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

      3. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in v around inf 72.0%

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

      1. associate--l+ 72.0%

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

      2. associate-*r/ 72.0%

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

      3. metadata-eval 72.0%

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

      4. associate-*r/ 72.0%

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

      5. metadata-eval 72.0%

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

      6. unpow2 72.0%

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

   9. Simplified 72.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5149999856948853:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \]

Alternative 8: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5149999856948853:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.5149999856948853)
   (* (* cosTheta_O (/ cosTheta_i v)) (/ 1.0 (* v (+ (exp (/ 1.0 v)) -1.0))))
   (*
    (* cosTheta_i (/ cosTheta_O v))
    (+
     0.5
     (-
      (/ 0.009722222222222222 (pow v 4.0))
      (/ 0.08333333333333333 (* v v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.5149999856948853f) {
		tmp = (cosTheta_O * (cosTheta_i / v)) * (1.0f / (v * (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = (cosTheta_i * (cosTheta_O / v)) * (0.5f + ((0.009722222222222222f / powf(v, 4.0f)) - (0.08333333333333333f / (v * v))));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.5149999856948853e0) then
        tmp = (costheta_o * (costheta_i / v)) * (1.0e0 / (v * (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = (costheta_i * (costheta_o / v)) * (0.5e0 + ((0.009722222222222222e0 / (v ** 4.0e0)) - (0.08333333333333333e0 / (v * v))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5149999856948853))
		tmp = Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) * Float32(Float32(1.0) / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(0.5) + Float32(Float32(Float32(0.009722222222222222) / (v ^ Float32(4.0))) - Float32(Float32(0.08333333333333333) / Float32(v * v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.5149999856948853))
		tmp = (cosTheta_O * (cosTheta_i / v)) * (single(1.0) / (v * (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = (cosTheta_i * (cosTheta_O / v)) * (single(0.5) + ((single(0.009722222222222222) / (v ^ single(4.0))) - (single(0.08333333333333333) / (v * v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.514999986

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*r/ 98.0%

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

      3. associate-/l* 98.2%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 98.2%

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

      1. rec-exp 98.2%

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

      2. distribute-neg-frac 98.2%

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

      3. metadata-eval 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in v around inf 69.3%

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

   if 0.514999986 < v

   1. Initial program 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-*r/ 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l/ 98.9%

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

      5. *-commutative 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 98.9%

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

      8. associate-/l/ 98.9%

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

      9. exp-neg 98.9%

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

      10. associate-/l/ 98.9%

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

      11. associate-/r* 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-*l/ 98.9%

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

      14. *-commutative 98.9%

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

      15. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in sinTheta_O around 0 98.8%

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

      1. rec-exp 98.7%

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

      2. distribute-neg-frac 98.7%

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

      3. metadata-eval 98.7%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in v around inf 72.0%

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

      1. associate--l+ 72.0%

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

      2. associate-*r/ 72.0%

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

      3. metadata-eval 72.0%

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

      4. associate-*r/ 72.0%

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

      5. metadata-eval 72.0%

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

      6. unpow2 72.0%

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

   9. Simplified 72.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5149999856948853:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \]

Alternative 9: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4399999976158142:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{v \cdot \left(2 + 0.3333333333333333 \cdot {v}^{-2}\right)} \cdot \left(cosTheta_O \cdot cosTheta_i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.4399999976158142)
   (* (* cosTheta_O (/ cosTheta_i v)) (/ 1.0 (* v (+ (exp (/ 1.0 v)) -1.0))))
   (*
    (/ 1.0 (* v (+ 2.0 (* 0.3333333333333333 (pow v -2.0)))))
    (* cosTheta_O cosTheta_i))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.4399999976158142f) {
		tmp = (cosTheta_O * (cosTheta_i / v)) * (1.0f / (v * (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = (1.0f / (v * (2.0f + (0.3333333333333333f * powf(v, -2.0f))))) * (cosTheta_O * cosTheta_i);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4399999976158142e0) then
        tmp = (costheta_o * (costheta_i / v)) * (1.0e0 / (v * (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = (1.0e0 / (v * (2.0e0 + (0.3333333333333333e0 * (v ** (-2.0e0)))))) * (costheta_o * costheta_i)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4399999976158142))
		tmp = Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) * Float32(Float32(1.0) / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) * (v ^ Float32(-2.0)))))) * Float32(cosTheta_O * cosTheta_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.4399999976158142))
		tmp = (cosTheta_O * (cosTheta_i / v)) * (single(1.0) / (v * (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = (single(1.0) / (v * (single(2.0) + (single(0.3333333333333333) * (v ^ single(-2.0)))))) * (cosTheta_O * cosTheta_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.439999998

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*r/ 98.0%

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

      3. associate-/l* 98.1%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 98.2%

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

      1. rec-exp 98.2%

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

      2. distribute-neg-frac 98.2%

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

      3. metadata-eval 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in v around inf 70.0%

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

   if 0.439999998 < v

   1. Initial program 98.8%

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

      1. distribute-neg-frac 98.8%

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

      2. *-commutative 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

      4. associate-*l/ 98.9%

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

      5. *-commutative 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in v around inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. metadata-eval 68.9%

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

      3. unpow2 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in sinTheta_O around 0 68.9%

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

      1. div-inv 68.9%

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

      2. *-commutative 68.9%

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

      3. pow-flip 68.9%

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

      4. metadata-eval 68.9%

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

   9. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\left(cosTheta_O \cdot cosTheta_i\right) \cdot \frac{1}{v \cdot \left(2 + 0.3333333333333333 \cdot {v}^{-2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4399999976158142:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{v \cdot \left(2 + 0.3333333333333333 \cdot {v}^{-2}\right)} \cdot \left(cosTheta_O \cdot cosTheta_i\right)\\ \end{array} \]

Alternative 10: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{v \cdot \left(2 + 0.3333333333333333 \cdot {v}^{-2}\right)} \cdot \left(cosTheta_O \cdot cosTheta_i\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. distribute-neg-frac 98.3%

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

   2. *-commutative 98.3%

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

   3. distribute-rgt-neg-in 98.3%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 61.0%

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

   1. associate-*r/ 61.0%

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

   2. metadata-eval 61.0%

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

   3. unpow2 61.0%

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

6. Simplified 61.0%

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

7. Taylor expanded in sinTheta_O around 0 61.0%

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

   1. div-inv 61.0%

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

   2. *-commutative 61.0%

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

   3. pow-flip 61.0%

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

   4. metadata-eval 61.0%

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

9. Applied egg-rr 61.0%

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

10. Final simplification 61.0%

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

Alternative 11: 64.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{v \cdot v}\right)} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * cosTheta_i) / (v * (2.0f + (0.3333333333333333f * (1.0f / (v * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o * costheta_i) / (v * (2.0e0 + (0.3333333333333333e0 * (1.0e0 / (v * v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. distribute-neg-frac 98.3%

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

   2. *-commutative 98.3%

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

   3. distribute-rgt-neg-in 98.3%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 61.0%

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

   1. associate-*r/ 61.0%

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

   2. metadata-eval 61.0%

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

   3. unpow2 61.0%

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

6. Simplified 61.0%

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

7. Taylor expanded in sinTheta_O around 0 61.0%

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

   1. inv-pow 61.0%

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

   2. unpow2 61.0%

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

9. Applied egg-rr 61.0%

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

    1. unpow2 61.0%

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

    2. unpow-1 61.0%

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

    3. unpow2 61.0%

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

11. Simplified 61.0%

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

12. Final simplification 61.0%

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

Alternative 12: 64.1% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. distribute-neg-frac 98.3%

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

   2. *-commutative 98.3%

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

   3. distribute-rgt-neg-in 98.3%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 61.0%

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

   1. associate-*r/ 61.0%

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

   2. metadata-eval 61.0%

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

   3. unpow2 61.0%

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

6. Simplified 61.0%

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

7. Taylor expanded in sinTheta_O around 0 61.0%

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

   1. distribute-lft-in 61.0%

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

   2. associate-*r/ 61.0%

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

   3. metadata-eval 61.0%

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

   4. unpow2 61.0%

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

   5. distribute-lft-in 61.0%

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

9. Simplified 61.0%

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

10. Final simplification 61.0%

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

Alternative 13: 64.1% accurate, 20.0× speedup

Math

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

FPCore

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. distribute-neg-frac 98.3%

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

   2. *-commutative 98.3%

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

   3. distribute-rgt-neg-in 98.3%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 61.0%

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

   1. associate-*r/ 61.0%

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

   2. metadata-eval 61.0%

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

   3. unpow2 61.0%

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

6. Simplified 61.0%

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

7. Taylor expanded in sinTheta_O around 0 61.0%

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

8. Taylor expanded in v around 0 61.0%

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

   1. associate-*r/ 61.0%

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

   2. metadata-eval 61.0%

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

   3. *-commutative 61.0%

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

10. Simplified 61.0%

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

11. Final simplification 61.0%

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

Alternative 14: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. div-inv 98.5%

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

3. Applied egg-rr 98.5%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 49.6%

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

   3. *-commutative 49.6%

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

   4. div-inv 49.6%

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

   5. associate-/l* 49.6%

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

   6. distribute-neg-frac 49.6%

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

   7. *-commutative 49.6%

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

5. Applied egg-rr 49.6%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 98.5%

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

   3. *-rgt-identity 98.5%

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

   4. associate-*r/ 98.4%

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

   5. associate-/r* 98.6%

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

7. Simplified 98.7%

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

8. Taylor expanded in v around inf 54.8%

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

   1. associate-/l* 54.8%

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

10. Simplified 54.8%

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

11. Final simplification 54.8%

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

Alternative 15: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. div-inv 98.5%

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

3. Applied egg-rr 98.5%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 49.6%

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

   3. *-commutative 49.6%

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

   4. div-inv 49.6%

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

   5. associate-/l* 49.6%

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

   6. distribute-neg-frac 49.6%

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

   7. *-commutative 49.6%

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

5. Applied egg-rr 49.6%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 98.5%

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

   3. *-rgt-identity 98.5%

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

   4. associate-*r/ 98.4%

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

   5. associate-/r* 98.6%

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

7. Simplified 98.7%

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

8. Taylor expanded in v around inf 54.8%

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

   1. *-commutative 54.8%

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

   2. associate-*l/ 54.8%

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

   3. associate-*r/ 54.8%

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

   4. associate-*l* 54.8%

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

10. Simplified 54.8%

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

11. Final simplification 54.8%

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

Alternative 16: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. div-inv 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in v around inf 54.8%

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

   1. associate-*r/ 54.8%

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

   2. associate-*r* 54.8%

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

6. Simplified 54.8%

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

7. Final simplification 54.8%

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

Reproduce

herbie shell --seed 2023238 
(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)))