HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 22.0s

Alternatives: 19

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \left(\frac{\frac{\frac{1}{v}}{\frac{1}{cosTheta\_i}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{0.5}{v}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (*
   (/
    (/ (/ 1.0 v) (/ 1.0 cosTheta_i))
    (* (sinh (/ 1.0 v)) (exp (fma sinTheta_i (/ sinTheta_O v) 0.0))))
   (/ 0.5 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((((1.0f / v) / (1.0f / cosTheta_i)) / (sinhf((1.0f / v)) * expf(fmaf(sinTheta_i, (sinTheta_O / v), 0.0f)))) * (0.5f / v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(Float32(Float32(Float32(1.0) / v) / Float32(Float32(1.0) / cosTheta_i)) / Float32(sinh(Float32(Float32(1.0) / v)) * exp(fma(sinTheta_i, Float32(sinTheta_O / v), Float32(0.0))))) * Float32(Float32(0.5) / v)))
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. /-lowering-/.f32 N/A

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

   9. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} \cdot \frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]

6. Applied egg-rr 98.9%

   \[\leadsto cosTheta\_O \cdot \color{blue}{\left(\frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{0.5}{v}\right)} \]
7. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{\frac{1}{\frac{v}{cosTheta\_i}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   2. inv-pow N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{{\left(\frac{v}{cosTheta\_i}\right)}^{-1}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   3. div-inv N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{{\color{blue}{\left(v \cdot \frac{1}{cosTheta\_i}\right)}}^{-1}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   4. unpow-prod-down N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{{v}^{-1} \cdot {\left(\frac{1}{cosTheta\_i}\right)}^{-1}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   5. inv-pow N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{\frac{1}{v}} \cdot {\left(\frac{1}{cosTheta\_i}\right)}^{-1}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{\frac{1}{v} \cdot {\left(\frac{1}{cosTheta\_i}\right)}^{-1}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   7. 1-exp N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\color{blue}{e^{0}}}{v} \cdot {\left(\frac{1}{cosTheta\_i}\right)}^{-1}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{\frac{e^{0}}{v}} \cdot {\left(\frac{1}{cosTheta\_i}\right)}^{-1}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   9. 1-exp N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\color{blue}{1}}{v} \cdot {\left(\frac{1}{cosTheta\_i}\right)}^{-1}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   10. pow-lowering-pow.f32 N/A

       \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{1}{v} \cdot \color{blue}{{\left(\frac{1}{cosTheta\_i}\right)}^{-1}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   11. 1-exp N/A

       \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{1}{v} \cdot {\left(\frac{\color{blue}{e^{0}}}{cosTheta\_i}\right)}^{-1}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{1}{v} \cdot {\color{blue}{\left(\frac{e^{0}}{cosTheta\_i}\right)}}^{-1}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   13. 1-exp 99.0

       \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{1}{v} \cdot {\left(\frac{\color{blue}{1}}{cosTheta\_i}\right)}^{-1}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{0.5}{v}\right) \]

8. Applied egg-rr 99.0%

   \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{\frac{1}{v} \cdot {\left(\frac{1}{cosTheta\_i}\right)}^{-1}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{0.5}{v}\right) \]
9. Step-by-step derivation

   1. unpow-1 N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{1}{v} \cdot \color{blue}{\frac{1}{\frac{1}{cosTheta\_i}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   2. un-div-inv N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{\frac{\frac{1}{v}}{\frac{1}{cosTheta\_i}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{\frac{\frac{1}{v}}{\frac{1}{cosTheta\_i}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\color{blue}{\frac{1}{v}}}{\frac{1}{cosTheta\_i}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   5. clear-num N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\frac{1}{v}}{\color{blue}{\frac{1}{\frac{cosTheta\_i}{1}}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   6. clear-num N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\frac{1}{v}}{\frac{1}{\color{blue}{\frac{1}{\frac{1}{cosTheta\_i}}}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   7. unpow-1 N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\frac{1}{v}}{\frac{1}{\color{blue}{{\left(\frac{1}{cosTheta\_i}\right)}^{-1}}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\frac{1}{v}}{\color{blue}{\frac{1}{{\left(\frac{1}{cosTheta\_i}\right)}^{-1}}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   9. unpow-1 N/A

      \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\frac{1}{v}}{\frac{1}{\color{blue}{\frac{1}{\frac{1}{cosTheta\_i}}}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   10. clear-num N/A

       \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\frac{1}{v}}{\frac{1}{\color{blue}{\frac{cosTheta\_i}{1}}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{\frac{1}{2}}{v}\right) \]

   11. /-rgt-identity 99.0

       \[\leadsto cosTheta\_O \cdot \left(\frac{\frac{\frac{1}{v}}{\frac{1}{\color{blue}{cosTheta\_i}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{0.5}{v}\right) \]

10. Applied egg-rr 99.0%

    \[\leadsto cosTheta\_O \cdot \left(\frac{\color{blue}{\frac{\frac{1}{v}}{\frac{1}{cosTheta\_i}}}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{0.5}{v}\right) \]
11. Add Preprocessing

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \left(\frac{0.5}{v} \cdot \frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (*
   (/ 0.5 v)
   (/
    (/ cosTheta_i v)
    (* (sinh (/ 1.0 v)) (exp (fma sinTheta_i (/ sinTheta_O v) 0.0)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((0.5f / v) * ((cosTheta_i / v) / (sinhf((1.0f / v)) * expf(fmaf(sinTheta_i, (sinTheta_O / v), 0.0f)))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(Float32(0.5) / v) * Float32(Float32(cosTheta_i / v) / Float32(sinh(Float32(Float32(1.0) / v)) * exp(fma(sinTheta_i, Float32(sinTheta_O / v), Float32(0.0)))))))
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. /-lowering-/.f32 N/A

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

   9. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} \cdot \frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]

6. Applied egg-rr 98.9%

   \[\leadsto cosTheta\_O \cdot \color{blue}{\left(\frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}} \cdot \frac{0.5}{v}\right)} \]

7. Final simplification 98.9%

   \[\leadsto cosTheta\_O \cdot \left(\frac{0.5}{v} \cdot \frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}}\right) \]
8. Add Preprocessing

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(cosTheta\_i, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v \cdot \left(v \cdot v\right)} \cdot -0.16666666666666666, \frac{0.5 \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v \cdot v}\right), \frac{cosTheta\_i \cdot sinTheta\_O}{-v}\right), cosTheta\_i\right)}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/
   (fma
    sinTheta_i
    (fma
     sinTheta_i
     (fma
      cosTheta_i
      (*
       (/
        (* sinTheta_i (* sinTheta_O (* sinTheta_O sinTheta_O)))
        (* v (* v v)))
       -0.16666666666666666)
      (/ (* 0.5 (* cosTheta_i (* sinTheta_O sinTheta_O))) (* v v)))
     (/ (* cosTheta_i sinTheta_O) (- v)))
    cosTheta_i)
   (* v 2.0))
  (/ (/ cosTheta_O v) (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (fmaf(sinTheta_i, fmaf(sinTheta_i, fmaf(cosTheta_i, (((sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O))) / (v * (v * v))) * -0.16666666666666666f), ((0.5f * (cosTheta_i * (sinTheta_O * sinTheta_O))) / (v * v))), ((cosTheta_i * sinTheta_O) / -v)), cosTheta_i) / (v * 2.0f)) * ((cosTheta_O / v) / sinhf((1.0f / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(fma(sinTheta_i, fma(sinTheta_i, fma(cosTheta_i, Float32(Float32(Float32(sinTheta_i * Float32(sinTheta_O * Float32(sinTheta_O * sinTheta_O))) / Float32(v * Float32(v * v))) * Float32(-0.16666666666666666)), Float32(Float32(Float32(0.5) * Float32(cosTheta_i * Float32(sinTheta_O * sinTheta_O))) / Float32(v * v))), Float32(Float32(cosTheta_i * sinTheta_O) / Float32(-v))), cosTheta_i) / Float32(v * Float32(2.0))) * Float32(Float32(cosTheta_O / v) / sinh(Float32(Float32(1.0) / v))))
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\color{blue}{cosTheta\_i + sinTheta\_i \cdot \left(-1 \cdot \frac{cosTheta\_i \cdot sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{-1}{6} \cdot \frac{cosTheta\_i \cdot \left({sinTheta\_O}^{3} \cdot sinTheta\_i\right)}{{v}^{3}} + \frac{1}{2} \cdot \frac{cosTheta\_i \cdot {sinTheta\_O}^{2}}{{v}^{2}}\right)\right)}}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{sinTheta\_i \cdot \left(-1 \cdot \frac{cosTheta\_i \cdot sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{-1}{6} \cdot \frac{cosTheta\_i \cdot \left({sinTheta\_O}^{3} \cdot sinTheta\_i\right)}{{v}^{3}} + \frac{1}{2} \cdot \frac{cosTheta\_i \cdot {sinTheta\_O}^{2}}{{v}^{2}}\right)\right) + cosTheta\_i}}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_i, -1 \cdot \frac{cosTheta\_i \cdot sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{-1}{6} \cdot \frac{cosTheta\_i \cdot \left({sinTheta\_O}^{3} \cdot sinTheta\_i\right)}{{v}^{3}} + \frac{1}{2} \cdot \frac{cosTheta\_i \cdot {sinTheta\_O}^{2}}{{v}^{2}}\right), cosTheta\_i\right)}}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(cosTheta\_i, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v \cdot \left(v \cdot v\right)} \cdot -0.16666666666666666, \frac{0.5 \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v \cdot v}\right), 0 - \frac{cosTheta\_i \cdot sinTheta\_O}{v}\right), cosTheta\_i\right)}}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

8. Final simplification 98.8%

   \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(cosTheta\_i, \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v \cdot \left(v \cdot v\right)} \cdot -0.16666666666666666, \frac{0.5 \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v \cdot v}\right), \frac{cosTheta\_i \cdot sinTheta\_O}{-v}\right), cosTheta\_i\right)}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]
9. Add Preprocessing

Alternative 4: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \mathsf{fma}\left(0.5, \frac{cosTheta\_i}{v}, \frac{\left(cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right) \cdot -0.5}{v \cdot v}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. /-lowering-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. /-lowering-/.f32 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f32 98.7

       \[\leadsto \mathsf{fma}\left(0.5, \frac{cosTheta\_i}{v}, \frac{\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right) \cdot -0.5}{\color{blue}{v \cdot v}}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{cosTheta\_i}{v}, \frac{\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right) \cdot -0.5}{v \cdot v}\right)} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

8. Final simplification 98.7%

   \[\leadsto \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \mathsf{fma}\left(0.5, \frac{cosTheta\_i}{v}, \frac{\left(cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right) \cdot -0.5}{v \cdot v}\right) \]
9. Add Preprocessing

Alternative 5: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)} + cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)\right) + cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)} + cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   5. mul-1-neg N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(cosTheta\_i, \color{blue}{\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}, cosTheta\_i\right)}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   8. neg-sub0 N/A

      \[\leadsto \frac{\mathsf{fma}\left(cosTheta\_i, \color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}, cosTheta\_i\right)}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   9. --lowering--.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(cosTheta\_i, \color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}, cosTheta\_i\right)}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(cosTheta\_i, 0 - \color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}, cosTheta\_i\right)}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   11. *-lowering-*.f32 98.6

       \[\leadsto \frac{\mathsf{fma}\left(cosTheta\_i, 0 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}, cosTheta\_i\right)}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(cosTheta\_i, 0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}, cosTheta\_i\right)}}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

8. Final simplification 98.6%

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

Alternative 6: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta\_i \cdot \frac{0.5}{v}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (/ cosTheta_O v) (sinh (/ 1.0 v))) (* 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 / v) / sinhf((1.0f / v))) * (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 / v) / sinh((1.0e0 / v))) * (costheta_i * (0.5e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 98.2

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot 0.5}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot 0.5}{v}} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]
8. Step-by-step derivation

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 N/A

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

   4. /-lowering-/.f32 98.4

      \[\leadsto \left(\color{blue}{\frac{0.5}{v}} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

9. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot cosTheta\_i\right)} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

10. Final simplification 98.4%

    \[\leadsto \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta\_i \cdot \frac{0.5}{v}\right) \]
11. Add Preprocessing

Alternative 7: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ cosTheta\_i \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta\_O}{v \cdot \sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 98.2

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot 0.5}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot 0.5}{v}} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]
8. Step-by-step derivation

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. /-lowering-/.f32 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f32 N/A

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

   10. sinh-lowering-sinh.f32 N/A

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

   11. 1-exp N/A

       \[\leadsto cosTheta\_i \cdot \left(\frac{\frac{1}{2}}{v} \cdot \frac{cosTheta\_O}{v \cdot \sinh \left(\frac{\color{blue}{e^{0}}}{v}\right)}\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto cosTheta\_i \cdot \left(\frac{\frac{1}{2}}{v} \cdot \frac{cosTheta\_O}{v \cdot \sinh \color{blue}{\left(\frac{e^{0}}{v}\right)}}\right) \]

   13. 1-exp 98.2

       \[\leadsto cosTheta\_i \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta\_O}{v \cdot \sinh \left(\frac{\color{blue}{1}}{v}\right)}\right) \]

9. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta\_O}{v \cdot \sinh \left(\frac{1}{v}\right)}\right)} \]
10. Add Preprocessing

Alternative 8: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 98.2

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot 0.5}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot 0.5}{v}} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

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

   2. associate-*r/ N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. /-lowering-/.f32 N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. *-lowering-*.f32 N/A

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

   14. *-commutative N/A

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

   15. 1-exp N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{e^{0}}}{\sinh \left(\frac{1}{v}\right) \cdot v} \]

   16. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{e^{0}}{\sinh \left(\frac{1}{v}\right) \cdot v}} \]

   17. 1-exp N/A

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

   18. *-commutative N/A

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

   19. *-lowering-*.f32 N/A

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

   20. sinh-lowering-sinh.f32 N/A

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

   21. 1-exp N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \sinh \left(\frac{\color{blue}{e^{0}}}{v}\right)} \]

   22. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \sinh \color{blue}{\left(\frac{e^{0}}{v}\right)}} \]

   23. 1-exp 98.0

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

9. Applied egg-rr 98.0%

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

    1. un-div-inv N/A

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

    2. associate-*l/ N/A

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

    3. associate-/l/ N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. /-lowering-/.f32 N/A

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

    7. /-rgt-identity N/A

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

    8. clear-num N/A

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

    9. unpow-1 N/A

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

    10. *-lowering-*.f32 N/A

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

    11. unpow-1 N/A

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

    12. clear-num N/A

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

    13. /-rgt-identity N/A

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

    14. *-lowering-*.f32 N/A

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

    15. sinh-lowering-sinh.f32 N/A

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

    16. /-lowering-/.f32 N/A

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

    17. *-lowering-*.f32 N/A

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

    18. *-lowering-*.f32 98.0

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

11. Applied egg-rr 98.0%

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

12. Final simplification 98.0%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \]
13. Add Preprocessing

Alternative 9: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v \cdot \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 98.2

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot 0.5}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot 0.5}{v}} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l/ N/A

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

   3. frac-times N/A

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

   4. remove-double-neg N/A

      \[\leadsto \frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(cosTheta\_i \cdot \frac{1}{2}\right)\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot v\right) \cdot v} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot \frac{1}{2}\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot v\right) \cdot v} \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\mathsf{neg}\left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot v\right) \cdot v} \]

   7. metadata-eval N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f32 N/A

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

   14. *-lowering-*.f32 N/A

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

   15. *-lowering-*.f32 N/A

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

   16. *-lowering-*.f32 N/A

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

9. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v}} \]

10. Final simplification 97.8%

    \[\leadsto \frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v \cdot \left(v \cdot \sinh \left(\frac{1}{v}\right)\right)} \]
11. Add Preprocessing

Alternative 10: 70.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot 0.5}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{-1 + \frac{\mathsf{fma}\left(-1, \frac{0.008333333333333333}{v \cdot v}, -0.16666666666666666\right)}{v \cdot v}}{-v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (* cosTheta_i 0.5) v)
  (/
   (/ cosTheta_O v)
   (/
    (+
     -1.0
     (/
      (fma -1.0 (/ 0.008333333333333333 (* v v)) -0.16666666666666666)
      (* v v)))
    (- v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * 0.5f) / v) * ((cosTheta_O / v) / ((-1.0f + (fmaf(-1.0f, (0.008333333333333333f / (v * v)), -0.16666666666666666f) / (v * v))) / -v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * Float32(0.5)) / v) * Float32(Float32(cosTheta_O / v) / Float32(Float32(Float32(-1.0) + Float32(fma(Float32(-1.0), Float32(Float32(0.008333333333333333) / Float32(v * v)), Float32(-0.16666666666666666)) / Float32(v * v))) / Float32(-v))))
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 98.2

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot 0.5}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot 0.5}{v}} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

8. Taylor expanded in v around -inf

   \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}}} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}}} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{-1 \cdot v}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}}} \]

10. Simplified 72.7%

    \[\leadsto \frac{cosTheta\_i \cdot 0.5}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{\frac{\mathsf{fma}\left(-1, \frac{0.008333333333333333}{v \cdot v}, -0.16666666666666666\right)}{v \cdot v} + -1}{-v}}} \]

11. Final simplification 72.7%

    \[\leadsto \frac{cosTheta\_i \cdot 0.5}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{-1 + \frac{\mathsf{fma}\left(-1, \frac{0.008333333333333333}{v \cdot v}, -0.16666666666666666\right)}{v \cdot v}}{-v}} \]
12. Add Preprocessing

Alternative 11: 70.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot 2}\right) \cdot \frac{-1}{v \cdot \frac{-1 + \frac{\mathsf{fma}\left(-1, \frac{0.008333333333333333}{v \cdot v}, -0.16666666666666666\right)}{v \cdot v}}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_O (/ cosTheta_i (* v 2.0)))
  (/
   -1.0
   (*
    v
    (/
     (+
      -1.0
      (/
       (fma -1.0 (/ 0.008333333333333333 (* v v)) -0.16666666666666666)
       (* v 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))) * (-1.0f / (v * ((-1.0f + (fmaf(-1.0f, (0.008333333333333333f / (v * v)), -0.16666666666666666f) / (v * v))) / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(cosTheta_i / Float32(v * Float32(2.0)))) * Float32(Float32(-1.0) / Float32(v * Float32(Float32(Float32(-1.0) + Float32(fma(Float32(-1.0), Float32(Float32(0.008333333333333333) / Float32(v * v)), Float32(-0.16666666666666666)) / Float32(v * v))) / v))))
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 98.2

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot 0.5}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot 0.5}{v}} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

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

   2. associate-*r/ N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. /-lowering-/.f32 N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. *-lowering-*.f32 N/A

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

   14. *-commutative N/A

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

   15. 1-exp N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{e^{0}}}{\sinh \left(\frac{1}{v}\right) \cdot v} \]

   16. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{e^{0}}{\sinh \left(\frac{1}{v}\right) \cdot v}} \]

   17. 1-exp N/A

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

   18. *-commutative N/A

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

   19. *-lowering-*.f32 N/A

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

   20. sinh-lowering-sinh.f32 N/A

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

   21. 1-exp N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \sinh \left(\frac{\color{blue}{e^{0}}}{v}\right)} \]

   22. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \sinh \color{blue}{\left(\frac{e^{0}}{v}\right)}} \]

   23. 1-exp 98.0

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

9. Applied egg-rr 98.0%

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

10. Taylor expanded in v around -inf

    \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right)}} \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}}} \]

    3. mul-1-neg N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{-1 \cdot v}}} \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}}} \]

12. Simplified 72.6%

    \[\leadsto \left(\frac{cosTheta\_i}{v \cdot 2} \cdot cosTheta\_O\right) \cdot \frac{1}{v \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, \frac{0.008333333333333333}{v \cdot v}, -0.16666666666666666\right)}{v \cdot v} + -1}{-v}}} \]

13. Final simplification 72.6%

    \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot 2}\right) \cdot \frac{-1}{v \cdot \frac{-1 + \frac{\mathsf{fma}\left(-1, \frac{0.008333333333333333}{v \cdot v}, -0.16666666666666666\right)}{v \cdot v}}{v}} \]
14. Add Preprocessing

Alternative 12: 64.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right), 0.16666666666666666\right)}{v \cdot v}\right), 2\right)}}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   cosTheta_O
   (/
    cosTheta_i
    (fma
     2.0
     (fma
      sinTheta_i
      (/ sinTheta_O v)
      (/
       (fma
        0.5
        (* sinTheta_O (* sinTheta_O (* sinTheta_i sinTheta_i)))
        0.16666666666666666)
       (* v v)))
     2.0)))
  v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * (cosTheta_i / fmaf(2.0f, fmaf(sinTheta_i, (sinTheta_O / v), (fmaf(0.5f, (sinTheta_O * (sinTheta_O * (sinTheta_i * sinTheta_i))), 0.16666666666666666f) / (v * v))), 2.0f))) / v;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(cosTheta_i / fma(Float32(2.0), fma(sinTheta_i, Float32(sinTheta_O / v), Float32(fma(Float32(0.5), Float32(sinTheta_O * Float32(sinTheta_O * Float32(sinTheta_i * sinTheta_i))), Float32(0.16666666666666666)) / Float32(v * v))), Float32(2.0)))) / v)
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}\right)\right) \cdot v}} \]

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(\left(2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 2 \cdot \frac{\frac{1}{6} + \frac{1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)}{{v}^{2}}\right) + 2\right)} \cdot v} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\color{blue}{2 \cdot \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} + \frac{\frac{1}{6} + \frac{1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)}{{v}^{2}}\right)} + 2\right) \cdot v} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{v} + \frac{\frac{1}{6} + \frac{1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)}{{v}^{2}}, 2\right)} \cdot v} \]

7. Simplified 67.1%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{\mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, \left(sinTheta\_i \cdot sinTheta\_i\right) \cdot 0.5, 0.16666666666666666\right)}{v \cdot v}\right), 2\right)} \cdot v} \]
8. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{2 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v} + \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot \frac{1}{2}\right) + \frac{1}{6}}{v \cdot v}\right) + 2} \cdot \frac{cosTheta\_O}{v}} \]

   2. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i}{2 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v} + \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot \frac{1}{2}\right) + \frac{1}{6}}{v \cdot v}\right) + 2} \cdot cosTheta\_O}{v}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i}{2 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v} + \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot \frac{1}{2}\right) + \frac{1}{6}}{v \cdot v}\right) + 2} \cdot cosTheta\_O}{v}} \]

9. Applied egg-rr 67.2%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right), 0.16666666666666666\right)}{v \cdot v}\right), 2\right)} \cdot cosTheta\_O}{v}} \]

10. Final simplification 67.2%

    \[\leadsto \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right), 0.16666666666666666\right)}{v \cdot v}\right), 2\right)}}{v} \]
11. Add Preprocessing

Alternative 13: 64.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot 0.5}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (* cosTheta_i 0.5) v)
  (/ (/ cosTheta_O v) (/ (+ 1.0 (/ 0.16666666666666666 (* v v))) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * 0.5f) / v) * ((cosTheta_O / v) / ((1.0f + (0.16666666666666666f / (v * 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_i * 0.5e0) / v) * ((costheta_o / v) / ((1.0e0 + (0.16666666666666666e0 / (v * v))) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * Float32(0.5)) / v) * Float32(Float32(cosTheta_O / v) / Float32(Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v))) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * single(0.5)) / v) * ((cosTheta_O / v) / ((single(1.0) + (single(0.16666666666666666) / (v * v))) / v));
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i}{v \cdot 2} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 98.2

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot 0.5}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot 0.5}{v}} \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)} \]

8. Taylor expanded in v around inf

   \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}}} \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}}} \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{\color{blue}{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}}{v}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{1 + \color{blue}{\frac{\frac{1}{6} \cdot 1}{{v}^{2}}}}{v}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{1 + \frac{\color{blue}{\frac{1}{6}}}{{v}^{2}}}{v}} \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{1 + \color{blue}{\frac{\frac{1}{6}}{{v}^{2}}}}{v}} \]

   6. unpow2 N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{1 + \frac{\frac{1}{6}}{\color{blue}{v \cdot v}}}{v}} \]

   7. *-lowering-*.f32 67.1

      \[\leadsto \frac{cosTheta\_i \cdot 0.5}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\frac{1 + \frac{0.16666666666666666}{\color{blue}{v \cdot v}}}{v}} \]

10. Simplified 67.1%

    \[\leadsto \frac{cosTheta\_i \cdot 0.5}{v} \cdot \frac{\frac{cosTheta\_O}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}}} \]
11. Add Preprocessing

Alternative 14: 64.6% accurate, 6.6× 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 98.5%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}\right)\right) \cdot v}} \]

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(\left(2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 2 \cdot \frac{\frac{1}{6} + \frac{1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)}{{v}^{2}}\right) + 2\right)} \cdot v} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\color{blue}{2 \cdot \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} + \frac{\frac{1}{6} + \frac{1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)}{{v}^{2}}\right)} + 2\right) \cdot v} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{v} + \frac{\frac{1}{6} + \frac{1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)}{{v}^{2}}, 2\right)} \cdot v} \]

7. Simplified 67.1%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{\mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, \left(sinTheta\_i \cdot sinTheta\_i\right) \cdot 0.5, 0.16666666666666666\right)}{v \cdot v}\right), 2\right)} \cdot v} \]

8. Taylor expanded in sinTheta_O around 0

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \cdot v} \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \cdot v} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}\right) \cdot v} \]

   3. metadata-eval N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}\right) \cdot v} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}\right) \cdot v} \]

   5. unpow2 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}\right) \cdot v} \]

   6. *-lowering-*.f32 67.1

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

10. Simplified 67.1%

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

11. Final simplification 67.1%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \]
12. Add Preprocessing

Alternative 15: 59.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i \cdot 0.5}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 62.0

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

5. Simplified 62.0%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. *-lowering-*.f32 61.9

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

7. Applied egg-rr 61.9%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. clear-num N/A

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

   7. 1-exp N/A

      \[\leadsto \frac{\color{blue}{e^{0}}}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i \cdot \frac{1}{2}}} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{0}}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i \cdot \frac{1}{2}}}} \]

   9. 1-exp N/A

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

   10. /-lowering-/.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. *-lowering-*.f32 62.2

       \[\leadsto \frac{1}{\frac{\frac{v}{cosTheta\_O}}{\color{blue}{cosTheta\_i \cdot 0.5}}} \]

9. Applied egg-rr 62.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i \cdot 0.5}}} \]
10. Add Preprocessing

Alternative 16: 59.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 62.0

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

5. Simplified 62.0%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. *-lowering-*.f32 61.9

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

7. Applied egg-rr 61.9%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 N/A

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

   4. /-lowering-/.f32 61.9

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

9. Applied egg-rr 61.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)} \]
10. Step-by-step derivation

    1. associate-*r* N/A

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

    2. /-rgt-identity N/A

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

    3. clear-num N/A

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

    4. unpow-1 N/A

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

    5. *-commutative N/A

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

    6. *-commutative N/A

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

    7. associate-*l* N/A

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

    8. div-inv N/A

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

    9. metadata-eval N/A

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

    10. div-inv N/A

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

    11. clear-num N/A

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

    12. /-lowering-/.f32 N/A

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

    13. /-lowering-/.f32 N/A

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

    14. div-inv N/A

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

    15. associate-*r/ N/A

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

    16. *-commutative N/A

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

    17. unpow-1 N/A

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

    18. clear-num N/A

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

    19. /-rgt-identity N/A

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

    20. *-commutative N/A

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

    21. /-lowering-/.f32 N/A

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

11. Applied egg-rr 62.2%

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

12. Final simplification 62.2%

    \[\leadsto \frac{1}{\frac{2}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}} \]
13. Add Preprocessing

Alternative 17: 59.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 62.0

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

5. Simplified 62.0%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. /-lowering-/.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

   8. *-lowering-*.f32 62.2

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

7. Applied egg-rr 62.2%

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

8. Final simplification 62.2%

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

Alternative 18: 58.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 62.0

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

5. Simplified 62.0%

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

6. Final simplification 62.0%

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

Alternative 19: 58.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 62.0

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

5. Simplified 62.0%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. *-lowering-*.f32 61.9

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

7. Applied egg-rr 61.9%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 N/A

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

   4. /-lowering-/.f32 61.9

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

9. Applied egg-rr 61.9%

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

10. Final simplification 61.9%

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

Reproduce

herbie shell --seed 2024194 
(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)))