HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 13.6s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o * costheta_i) * (((exp(sintheta_o) ** (-sintheta_i / v)) / v) * ((0.5e0 / 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(0.5) / v) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

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

Derivation

1. Initial program 98.4%

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

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f32 N/A

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f32 N/A

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

   10. associate-*l/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (* cosTheta_O cosTheta_i) (/ 1.0 v))
   (exp (/ (* (- sinTheta_i) sinTheta_O) v)))
  (* (* 2.0 (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) * (1.0f / v)) * expf(((-sinTheta_i * sinTheta_O) / v))) / ((2.0f * 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) * (1.0e0 / v)) * exp(((-sintheta_i * sintheta_o) / v))) / ((2.0e0 * sinh((1.0e0 / v))) * v)
end function

Julia

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

Derivation

1. Initial program 98.4%

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

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* (/ cosTheta_O v) cosTheta_i) (exp (/ (* (- sinTheta_i) sinTheta_O) v)))
  (* (* 2.0 (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 / v) * cosTheta_i) * expf(((-sinTheta_i * sinTheta_O) / v))) / ((2.0f * 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 / v) * costheta_i) * exp(((-sintheta_i * sintheta_o) / v))) / ((2.0e0 * sinh((1.0e0 / v))) * v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. rec-exp N/A

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

   14. lower-exp.f32 N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f32 98.3

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

8. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
9. Add Preprocessing

Alternative 6: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f32 59.8

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 59.7%

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

11. Taylor expanded in sinTheta_i around 0

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

    1. times-frac N/A

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

    2. associate-*r/ N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-/.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 N/A

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

    8. lower--.f32 N/A

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

    9. lower-exp.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. rec-exp N/A

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

    12. lower-exp.f32 N/A

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

    13. distribute-neg-frac N/A

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

    14. metadata-eval N/A

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

    15. lower-/.f32 98.3

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

13. Applied rewrites 98.3%

    \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
14. Add Preprocessing

Alternative 7: 70.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_O cosTheta_i) v) (exp (/ (* (- sinTheta_i) sinTheta_O) v)))
  (*
   (*
    (/
     (-
      (/ (/ (- (/ 0.008333333333333333 (* v v)) -0.16666666666666666) v) v)
      -1.0)
     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) / v) * expf(((-sinTheta_i * sinTheta_O) / v))) / ((((((((0.008333333333333333f / (v * v)) - -0.16666666666666666f) / v) / v) - -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 = (((costheta_o * costheta_i) / v) * exp(((-sintheta_i * sintheta_o) / v))) / ((((((((0.008333333333333333e0 / (v * v)) - (-0.16666666666666666e0)) / v) / v) - (-1.0e0)) / v) * 2.0e0) * v)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_O * cosTheta_i) / v) * exp(((-sinTheta_i * sinTheta_O) / v))) / ((((((((single(0.008333333333333333) / (v * v)) - single(-0.16666666666666666)) / v) / v) - single(-1.0)) / v) * single(2.0)) * v);
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around -inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft1-in N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-rgt-identity N/A

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

   10. lower-/.f32 N/A

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

5. Applied rewrites 70.8%

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

6. Final simplification 70.8%

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

Alternative 8: 64.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 64.9

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

7. Applied rewrites 64.9%

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

8. Final simplification 64.9%

   \[\leadsto \frac{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}} \cdot e^{\frac{\left(-sinTheta\_i\right) \cdot sinTheta\_O}{v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \]
9. Add Preprocessing

Alternative 9: 64.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

7. Applied rewrites 65.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]

8. Final simplification 65.0%

   \[\leadsto \frac{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{\left(-sinTheta\_i\right) \cdot sinTheta\_O}{v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \]
9. Add Preprocessing

Alternative 10: 64.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.0

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

5. Applied rewrites 65.0%

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

6. Final simplification 65.0%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot e^{\frac{\left(-sinTheta\_i\right) \cdot sinTheta\_O}{v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \]
7. Add Preprocessing

Alternative 11: 59.1% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f32 59.8

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 60.3%

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

11. Final simplification 60.3%

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

Alternative 12: 59.0% accurate, 8.0× speedup

Math

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

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(Float32(v / 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 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f32 59.8

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 60.1%

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

Alternative 13: 59.0% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f32 59.8

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 60.0%

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

11. Final simplification 60.0%

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

Alternative 14: 58.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f32 59.8

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 59.8%

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

11. Final simplification 59.8%

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

Alternative 15: 58.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f32 59.8

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 59.7%

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

12. Applied rewrites 59.8%

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

Alternative 16: 58.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f32 59.8

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 59.8%

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

11. Final simplification 59.8%

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

Alternative 17: 58.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f32 59.8

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

8. Applied rewrites 59.8%

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

9. Final simplification 59.8%

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

Reproduce

herbie shell --seed 2024285 
(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)))