HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 20.8s

Alternatives: 18

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

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

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

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

   15. neg-lowering-neg.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. remove-double-div N/A

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

Alternative 2: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / v) * ((single(0.5) / v) * (cosTheta_O / 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. *-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. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

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

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

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

   15. neg-lowering-neg.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. remove-double-div N/A

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

   1. associate-/l* N/A

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

   2. times-frac N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r/ N/A

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

   5. associate-/r/ N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. *-commutative N/A

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

   10. times-frac N/A

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

   11. associate-*l* N/A

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in sinTheta_i around 0

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

    1. /-lowering-/.f32 98.8

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

11. Simplified 98.8%

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

Alternative 3: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

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

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

   2. *-lowering-*.f32 98.5

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

5. Simplified 98.5%

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. /-lowering-/.f32 98.7

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / v) * (cosTheta_O / (sinhf((1.0f / 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_i / v) * (costheta_o / (sinh((1.0e0 / v)) * (v * 2.0e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / v) * (cosTheta_O / (sinh((single(1.0) / 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. *-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. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

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

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

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

   15. neg-lowering-neg.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. remove-double-div N/A

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. /-lowering-/.f32 98.8

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

9. Simplified 98.8%

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

    1. associate-/r/ N/A

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

    2. div-inv N/A

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

    3. associate-/r* N/A

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

    4. *-commutative N/A

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

    5. associate-/l* N/A

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

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

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

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

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

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

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

    9. associate-*l* N/A

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

    10. *-commutative N/A

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

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

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

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

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

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

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

    14. *-lowering-*.f32 98.6

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

11. Applied egg-rr 98.6%

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

Alternative 5: 98.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (/ cosTheta_O (* (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_i * (cosTheta_O / (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_i * (costheta_o / (sinh((1.0e0 / v)) * (v * (v * 2.0e0))))
end function

Julia

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

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

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

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

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

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

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

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

   7. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   11. associate-/l/ N/A

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

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

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

   13. associate-*l* N/A

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

   14. associate-*l* N/A

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

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

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 6: 70.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}{\frac{2 \cdot \frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}{\frac{1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_O (/ cosTheta_i v))
  (/
   (*
    2.0
    (/
     (-
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      -1.0)
     v))
   (/ 1.0 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * (cosTheta_i / v)) / ((2.0f * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) - -1.0f) / v)) / (1.0f / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o * (costheta_i / v)) / ((2.0e0 * ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / v)) / (1.0e0 / v))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * (cosTheta_i / v)) / ((single(2.0) * ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - single(-1.0)) / v)) / (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. *-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. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

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

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

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

   15. neg-lowering-neg.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. remove-double-div N/A

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. /-lowering-/.f32 98.8

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

9. Simplified 98.8%

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

10. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

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

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

12. Simplified 71.4%

    \[\leadsto \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}{\frac{\color{blue}{\frac{-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{-v}} \cdot 2}{\frac{1}{v}}} \]

13. Final simplification 71.4%

    \[\leadsto \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}{\frac{2 \cdot \frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}{\frac{1}{v}}} \]
14. Add Preprocessing

Alternative 7: 70.3% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ (* cosTheta_O cosTheta_i) v)
  (*
   v
   (*
    2.0
    (/
     (-
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      -1.0)
     v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * cosTheta_i) / v) / (v * (2.0f * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) - -1.0f) / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_o * costheta_i) / v) / (v * (2.0e0 * ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O * cosTheta_i) / v) / (v * (single(2.0) * ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - 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. Taylor expanded in sinTheta_i around 0

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

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

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

   2. *-lowering-*.f32 98.5

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

5. Simplified 98.5%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

8. Simplified 71.4%

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

9. Final simplification 71.4%

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

Alternative 8: 64.0% accurate, 5.8× speedup

Math

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

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 * Float32(cosTheta_i / 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. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

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

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

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

   15. neg-lowering-neg.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. remove-double-div N/A

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. /-lowering-/.f32 98.8

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

9. Simplified 98.8%

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

10. Taylor expanded in v around inf

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

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

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

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

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

    5. unpow2 N/A

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

    6. *-lowering-*.f32 65.6

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

12. Simplified 65.6%

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

Alternative 9: 58.8% accurate, 7.0× speedup

Math

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

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(Float32(1.0) / Float32(Float32(2.0) / 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 60.1

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

5. Simplified 60.1%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

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

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

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

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

   7. *-lowering-*.f32 60.8

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

7. Applied egg-rr 60.8%

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

8. Final simplification 60.8%

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

Alternative 10: 58.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. 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 60.1

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

5. Simplified 60.1%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. *-lowering-*.f32 60.0

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

7. Applied egg-rr 60.0%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. clear-num N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f32 60.5

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

9. Applied egg-rr 60.5%

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

Alternative 11: 58.6% 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 60.1

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

5. Simplified 60.1%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. *-lowering-*.f32 60.0

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

7. Applied egg-rr 60.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f32 60.0

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

9. Applied egg-rr 60.0%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. associate-/l* N/A

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

    4. associate-*l/ N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. div-inv N/A

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

    8. clear-num N/A

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

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

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

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

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

    11. associate-*r/ N/A

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

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

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

    13. *-lowering-*.f32 60.5

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

11. Applied egg-rr 60.5%

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

Alternative 12: 58.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 60.1

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

5. Simplified 60.1%

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

   1. clear-num N/A

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

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

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

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

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

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

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. *-lowering-*.f32 60.5

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

7. Applied egg-rr 60.5%

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

Alternative 13: 58.5% 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 60.1

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

5. Simplified 60.1%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/r/ N/A

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

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

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

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

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

   7. *-lowering-*.f32 60.3

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

7. Applied egg-rr 60.3%

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

8. Final simplification 60.3%

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

Alternative 14: 58.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 60.1

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

5. Simplified 60.1%

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

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

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

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

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

   5. *-lowering-*.f32 60.1

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

7. Applied egg-rr 60.1%

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

8. Final simplification 60.1%

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

Alternative 15: 58.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * single(0.5)) * (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 60.1

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

5. Simplified 60.1%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. associate-/r/ N/A

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

   5. clear-num N/A

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

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

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

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

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

   8. *-lowering-*.f32 60.0

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

7. Applied egg-rr 60.0%

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

8. Final simplification 60.0%

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

Alternative 16: 58.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 60.1

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

5. Simplified 60.1%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. *-lowering-*.f32 60.0

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

7. Applied egg-rr 60.0%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. /-lowering-/.f32 60.0

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

9. Applied egg-rr 60.0%

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

10. Final simplification 60.0%

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

Alternative 17: 58.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 60.1

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

5. Simplified 60.1%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. *-lowering-*.f32 60.0

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

7. Applied egg-rr 60.0%

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

Alternative 18: 58.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * 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 * (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. 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 60.1

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

5. Simplified 60.1%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. *-lowering-*.f32 60.0

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

7. Applied egg-rr 60.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f32 60.0

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

9. Applied egg-rr 60.0%

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

10. Final simplification 60.0%

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

Reproduce

herbie shell --seed 2024198 
(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)))