HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 16.0s

Alternatives: 18

Speedup: 1.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.6

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 4: 98.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.5%

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

11. Applied rewrites 98.7%

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

Alternative 5: 98.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* cosTheta_O cosTheta_i) (* (/ (/ 0.5 v) (sinh (/ 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) * (((0.5f / v) / sinhf((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) * (((0.5e0 / v) / sinh((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 * cosTheta_i) * Float32(Float32(Float32(Float32(0.5) / v) / sinh(Float32(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) * (((single(0.5) / v) / sinh((single(1.0) / v))) * (single(1.0) / v));
end

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.5%

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

Alternative 6: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.5%

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

11. Applied rewrites 98.4%

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

Alternative 7: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.5%

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

11. Applied rewrites 98.4%

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

Alternative 8: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.5%

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

11. Applied rewrites 98.4%

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

Alternative 9: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.2%

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

Alternative 10: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.1%

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

Alternative 11: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.5%

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

11. Applied rewrites 98.2%

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

Alternative 12: 64.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.16666666666666666}{v \cdot v}\\ \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\frac{\frac{1}{v}}{v}}{\frac{\left(\left(2 + t\_0\right) + \left(\frac{0.5}{v} + t\_0\right)\right) - \frac{0.5}{v}}{v}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ 0.16666666666666666 (* v v))))
   (*
    (* cosTheta_O cosTheta_i)
    (/
     (/ (/ 1.0 v) v)
     (/ (- (+ (+ 2.0 t_0) (+ (/ 0.5 v) t_0)) (/ 0.5 v)) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 0.16666666666666666f / (v * v);
	return (cosTheta_O * cosTheta_i) * (((1.0f / v) / v) / ((((2.0f + t_0) + ((0.5f / v) + t_0)) - (0.5f / 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
    real(4) :: t_0
    t_0 = 0.16666666666666666e0 / (v * v)
    code = (costheta_o * costheta_i) * (((1.0e0 / v) / v) / ((((2.0e0 + t_0) + ((0.5e0 / v) + t_0)) - (0.5e0 / v)) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(0.16666666666666666) / Float32(v * v))
	return Float32(Float32(cosTheta_O * cosTheta_i) * Float32(Float32(Float32(Float32(1.0) / v) / v) / Float32(Float32(Float32(Float32(Float32(2.0) + t_0) + Float32(Float32(Float32(0.5) / v) + t_0)) - Float32(Float32(0.5) / v)) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(0.16666666666666666) / (v * v);
	tmp = (cosTheta_O * cosTheta_i) * (((single(1.0) / v) / v) / ((((single(2.0) + t_0) + ((single(0.5) / v) + t_0)) - (single(0.5) / v)) / v));
end

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 97.9%

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

10. Taylor expanded in v around inf

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

12. Applied rewrites 61.9%

    \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\frac{\frac{1}{v}}{v}}{\frac{\left(\left(2 + \frac{0.16666666666666666}{v \cdot v}\right) + \left(\frac{0.5}{v} + \frac{0.16666666666666666}{v \cdot v}\right)\right) - \frac{0.5}{v}}{\color{blue}{v}}} \]
13. Add Preprocessing

Alternative 13: 64.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 98.3

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in v around inf

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

10. Applied rewrites 61.9%

    \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\frac{\frac{1}{v}}{v}}{\frac{\frac{0.3333333333333333}{v \cdot v} + 2}{\color{blue}{v}}} \]
11. Add Preprocessing

Alternative 14: 59.4% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. 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. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 56.1%

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

Alternative 15: 59.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. 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. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 56.0%

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

Alternative 16: 58.9% accurate, 12.4× speedup

Math

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

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(Float32(Float32(0.5) * 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 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. 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. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 55.9%

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

9. Applied rewrites 55.9%

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

10. Final simplification 55.9%

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

Alternative 17: 58.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. 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. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 55.9%

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

Alternative 18: 58.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. 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. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 55.9%

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

Reproduce

herbie shell --seed 2024308 
(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)))