HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 20.5s

Alternatives: 22

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: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 99.0

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

6. Applied rewrites 99.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l* N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

6. Applied rewrites 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l* N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. remove-double-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

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

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

   7. mul-1-neg N/A

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

   8. associate-*r* N/A

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

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

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

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

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

   13. lower-*.f32 N/A

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

   14. lower-neg.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 98.7

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 6: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-neg.f32 98.6

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

5. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 98.7

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 7: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 99.0

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 8: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f32 N/A

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

   5. lower-fma.f32 98.7

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

7. Applied rewrites 98.7%

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

Alternative 9: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.5%

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

Alternative 10: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around inf

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

   1. unpow2 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

Alternative 11: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.4%

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

Alternative 12: 70.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-neg.f32 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 71.5%

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

9. Final simplification 71.5%

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

Alternative 13: 64.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l* N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 65.6%

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

8. Final simplification 65.6%

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

Alternative 14: 64.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l* N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 65.6%

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

Alternative 15: 64.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-neg.f32 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 65.6

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

8. Applied rewrites 65.6%

   \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{-v}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]
9. Add Preprocessing

Alternative 16: 59.2% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.0

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

5. Applied rewrites 60.0%

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

7. Applied rewrites 60.6%

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

Alternative 17: 59.1% accurate, 8.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / ((v * 2.0f) / (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 / ((v * 2.0e0) / (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(v * Float32(2.0)) / Float32(cosTheta_i * cosTheta_O)))
end

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.0

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

5. Applied rewrites 60.0%

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

7. Applied rewrites 60.0%

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

9. Applied rewrites 60.5%

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

10. Final simplification 60.5%

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

Alternative 18: 59.1% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.0

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

5. Applied rewrites 60.0%

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

7. Applied rewrites 60.5%

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

8. Final simplification 60.5%

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

Alternative 19: 59.0% 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 98.6%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.0

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

5. Applied rewrites 60.0%

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

7. Applied rewrites 60.2%

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

Alternative 20: 58.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.0

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

5. Applied rewrites 60.0%

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

7. Applied rewrites 60.0%

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

8. Final simplification 60.0%

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

Alternative 21: 58.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.0

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

5. Applied rewrites 60.0%

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

7. Applied rewrites 60.0%

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

9. Applied rewrites 60.0%

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

10. Final simplification 60.0%

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

Alternative 22: 58.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.0

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

5. Applied rewrites 60.0%

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

7. Applied rewrites 60.0%

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

9. Applied rewrites 60.0%

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

10. Final simplification 60.0%

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

Reproduce

herbie shell --seed 2024232 
(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)))