HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 20.9s

Alternatives: 27

Speedup: 1.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((exp((sinTheta_O * (sinTheta_i / -v))) / v) * (cosTheta_i * cosTheta_O)) / ((sinh((single(1.0) / v)) * single(2.0)) / (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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. lower-*.f32 N/A

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

6. Applied rewrites 98.9%

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

Alternative 2: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((sinTheta_i * -sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) / (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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 3: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) * (single(1.0) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) / (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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. lower-*.f32 N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in sinTheta_O around 0

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

   1. lower-/.f32 98.8

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

9. Applied rewrites 98.8%

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

10. Final simplification 98.8%

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

Alternative 4: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) / ((sinh((single(1.0) / v)) * single(2.0)) / (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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 5: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) * (single(0.5) / (sinh((single(1.0) / v)) / (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 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. *-commutative N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 6: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) * ((single(0.5) / 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 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f32 N/A

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

   5. lower-/.f32 98.6

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

9. Applied rewrites 98.6%

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

10. Final simplification 98.6%

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

Alternative 7: 98.4% 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

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. lower-*.f32 N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in sinTheta_O around 0

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

   1. lower-/.f32 98.8

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

9. Applied rewrites 98.8%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lift-/.f32 N/A

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

    5. lift-sinh.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. lift-/.f32 N/A

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

    8. lift-/.f32 N/A

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

    9. associate-/l* N/A

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

    10. lower-*.f32 N/A

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

    11. lower-/.f32 98.7

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

    12. lift-/.f32 N/A

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

    13. lift-*.f32 N/A

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

    14. associate-/l* N/A

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

    15. lift-/.f32 N/A

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

    16. associate-/r/ N/A

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

    17. metadata-eval N/A

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

    18. *-commutative N/A

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

    19. lift-*.f32 N/A

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

11. Applied rewrites 98.5%

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

Alternative 8: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / (sinh((single(1.0) / v)) * single(2.0))) * (cosTheta_O / (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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-*.f32 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 9: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l/ N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*r* N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f32 N/A

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

9. Applied rewrites 98.5%

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

Alternative 10: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

   1. *-commutative N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-sinh.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. frac-2neg N/A

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

   7. frac-2neg N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. *-rgt-identity N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/l* N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. /-rgt-identity N/A

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

   18. associate-/r/ N/A

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

9. Applied rewrites 98.5%

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

Alternative 11: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) * (single(-0.5) / ((v * 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 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l/ N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

9. Applied rewrites 98.4%

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

10. Final simplification 98.4%

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

Alternative 12: 76.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{2 \cdot \frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}}{\frac{1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* cosTheta_i cosTheta_O) (/ 1.0 v))
  (/
   (*
    2.0
    (/
     (+
      (/
       (+ 0.008333333333333333 (/ 0.0001984126984126984 (* v v)))
       (* v (* v (* v v))))
      (+ 1.0 (/ 0.16666666666666666 (* v v))))
     v))
   (/ 1.0 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) * (1.0f / v)) / ((2.0f * ((((0.008333333333333333f + (0.0001984126984126984f / (v * v))) / (v * (v * (v * v)))) + (1.0f + (0.16666666666666666f / (v * v)))) / v)) / (1.0f / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i * costheta_o) * (1.0e0 / v)) / ((2.0e0 * ((((0.008333333333333333e0 + (0.0001984126984126984e0 / (v * v))) / (v * (v * (v * v)))) + (1.0e0 + (0.16666666666666666e0 / (v * v)))) / v)) / (1.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / v)) / Float32(Float32(Float32(2.0) * Float32(Float32(Float32(Float32(Float32(0.008333333333333333) + Float32(Float32(0.0001984126984126984) / Float32(v * v))) / Float32(v * Float32(v * Float32(v * v)))) + Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v)))) / v)) / Float32(Float32(1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) * (single(1.0) / v)) / ((single(2.0) * ((((single(0.008333333333333333) + (single(0.0001984126984126984) / (v * v))) / (v * (v * (v * v)))) + (single(1.0) + (single(0.16666666666666666) / (v * v)))) / v)) / (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
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. remove-double-div N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. lower-*.f32 N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in sinTheta_O around 0

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

   1. lower-/.f32 98.8

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

9. Applied rewrites 98.8%

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

10. Taylor expanded in v around -inf

    \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)} \cdot 2}{\frac{1}{v}}} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)}{v}} \cdot 2}{\frac{1}{v}}} \]

    2. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}}{v} \cdot 2}{\frac{1}{v}}} \]

    3. sub-neg N/A

       \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v} \cdot 2}{\frac{1}{v}}} \]

    4. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v} \cdot 2}{\frac{1}{v}}} \]

    5. distribute-neg-out N/A

       \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v} \cdot 2}{\frac{1}{v}}} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\frac{\color{blue}{\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}}{v} \cdot 2}{\frac{1}{v}}} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\color{blue}{\frac{\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}} \cdot 2}{\frac{1}{v}}} \]

12. Applied rewrites 73.7%

    \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\frac{\color{blue}{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}} \cdot 2}{\frac{1}{v}}} \]

13. Final simplification 73.7%

    \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{2 \cdot \frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}}{\frac{1}{v}}} \]
14. Add Preprocessing

Alternative 13: 73.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/
   1.0
   (/
    (+
     (/
      (+ 0.008333333333333333 (/ 0.0001984126984126984 (* v v)))
      (* v (* v (* v v))))
     (+ 1.0 (/ 0.16666666666666666 (* v v))))
    v))
  (/ (* v (* 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 / ((((0.008333333333333333f + (0.0001984126984126984f / (v * v))) / (v * (v * (v * v)))) + (1.0f + (0.16666666666666666f / (v * v)))) / v)) / ((v * (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 / ((((0.008333333333333333e0 + (0.0001984126984126984e0 / (v * v))) / (v * (v * (v * v)))) + (1.0e0 + (0.16666666666666666e0 / (v * v)))) / v)) / ((v * (v * 2.0e0)) / (costheta_i * costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.008333333333333333) + Float32(Float32(0.0001984126984126984) / Float32(v * v))) / Float32(v * Float32(v * Float32(v * v)))) + Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v)))) / v)) / Float32(Float32(v * 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) / ((((single(0.008333333333333333) + (single(0.0001984126984126984) / (v * v))) / (v * (v * (v * v)))) + (single(1.0) + (single(0.16666666666666666) / (v * v)))) / v)) / ((v * (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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 95.6%

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

10. Taylor expanded in v around -inf

    \[\leadsto \frac{\frac{1}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)}{v}}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]

    2. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}}{v}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]

    3. sub-neg N/A

       \[\leadsto \frac{\frac{1}{\frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]

    4. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{\frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]

    5. distribute-neg-out N/A

       \[\leadsto \frac{\frac{1}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{\frac{1}{\frac{\color{blue}{\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}}{v}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]

12. Applied rewrites 71.7%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}}}}{\frac{v \cdot \left(v \cdot 2\right)}{cosTheta\_i \cdot cosTheta\_O}} \]
13. Add Preprocessing

Alternative 14: 69.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}} \end{array} \]

FPCore

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * (((costheta_i * costheta_o) / v) / ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / v))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (((cosTheta_i * cosTheta_O) / v) / ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - 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 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 67.4%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{0.5}{v \cdot \color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}}} \]

12. Applied rewrites 67.5%

    \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{-v}} \cdot \frac{0.5}{v}} \]

13. Final simplification 67.5%

    \[\leadsto \frac{0.5}{v} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}} \]
14. Add Preprocessing

Alternative 15: 69.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i * costheta_o) * (0.5e0 * (v / (v * (((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)))))) / v
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) * (single(0.5) * (v / (v * (((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - 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 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 67.4%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{0.5}{v \cdot \color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}}} \]

12. Applied rewrites 67.5%

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

13. Final simplification 67.5%

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

Alternative 16: 69.8% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (/
    v
    (*
     v
     (-
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      -1.0)))
   (* (* 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 ((v / (v * (((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) - -1.0f))) * ((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 = ((v / (v * (((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)))) * ((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(v / Float32(v * Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) - Float32(-1.0)))) * Float32(Float32(cosTheta_i * cosTheta_O) * Float32(0.5))) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((v / (v * (((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - single(-1.0)))) * ((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 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 67.4%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{0.5}{v \cdot \color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}}} \]

12. Applied rewrites 67.4%

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

13. Final simplification 67.4%

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

Alternative 17: 69.8% accurate, 3.4× speedup

Math

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

FPCore

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i * costheta_o) * (-0.5e0)) / (v * ((v * ((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)))) / v))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) * single(-0.5)) / (v * ((v * (single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * 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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 67.4%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{0.5}{v \cdot \color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}}} \]

12. Applied rewrites 67.4%

    \[\leadsto \color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot -0.5}{v \cdot \frac{v \cdot \left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right)}{v}}} \]
13. Add Preprocessing

Alternative 18: 69.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i * costheta_o) / v) * (0.5e0 / ((1.0e0 + (0.16666666666666666e0 / (v * v))) + (0.008333333333333333e0 / (v * (v * (v * v))))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) * (single(0.5) / ((single(1.0) + (single(0.16666666666666666) / (v * v))) + (single(0.008333333333333333) / (v * (v * (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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 67.4%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{0.5}{v \cdot \color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}}} \]

11. Taylor expanded in v around inf

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

    1. +-commutative N/A

       \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{\frac{1}{2}}{\color{blue}{\left(\frac{\frac{1}{120}}{{v}^{4}} + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right) + 1}} \]

    2. associate-+l+ N/A

       \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{120}}{{v}^{4}} + \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}} + 1\right)}} \]

    3. +-commutative N/A

       \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{\frac{1}{2}}{\frac{\frac{1}{120}}{{v}^{4}} + \color{blue}{\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}} \]

    4. lower-+.f32 N/A

       \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{120}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}} \]

    5. lower-/.f32 N/A

       \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{\frac{1}{2}}{\color{blue}{\frac{\frac{1}{120}}{{v}^{4}}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)} \]

    6. metadata-eval N/A

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

    7. pow-sqr N/A

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

    8. unpow2 N/A

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

    9. associate-*l* N/A

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

    10. unpow2 N/A

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

    11. cube-mult N/A

        \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{\frac{1}{2}}{\frac{\frac{1}{120}}{v \cdot \color{blue}{{v}^{3}}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)} \]

    12. lower-*.f32 N/A

        \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{\frac{1}{2}}{\frac{\frac{1}{120}}{\color{blue}{v \cdot {v}^{3}}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)} \]

    13. cube-mult N/A

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

    14. unpow2 N/A

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

    15. lower-*.f32 N/A

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

    16. unpow2 N/A

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

    17. lower-*.f32 N/A

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

    18. lower-+.f32 N/A

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

    19. associate-*r/ N/A

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

    20. metadata-eval N/A

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

    21. lower-/.f32 N/A

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

    22. unpow2 N/A

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

    23. lower-*.f32 67.4

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

13. Applied rewrites 67.4%

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

14. Final simplification 67.4%

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

Alternative 19: 69.8% accurate, 4.2× speedup

Math

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

FPCore

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * (costheta_o * (-0.5e0))) / (v * ((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * (cosTheta_O * single(-0.5))) / (v * (single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 67.4%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{0.5}{v \cdot \color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}}} \]

11. Taylor expanded in cosTheta_O around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. sub-neg N/A

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

    10. metadata-eval N/A

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

    11. +-commutative N/A

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

    12. mul-1-neg N/A

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

    13. unsub-neg N/A

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

    14. lower--.f32 N/A

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

    15. lower-/.f32 N/A

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

13. Applied rewrites 67.4%

    \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot -0.5\right)}{v \cdot \left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right)}} \]
14. Add Preprocessing

Alternative 20: 63.5% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i * costheta_o) / v) * (0.5e0 / (1.0e0 + (0.16666666666666666e0 / (v * v))))
end function

Julia

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

MATLAB

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 61.9

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

10. Applied rewrites 61.9%

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

11. Final simplification 61.9%

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

Alternative 21: 58.4% accurate, 8.2× speedup

Math

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

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(v * Float32(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 56.7

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

5. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. associate-/l* N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. associate-/r/ N/A

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

   15. lift-/.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. associate-/r/ N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f32 56.7

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

7. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. clear-num N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 57.4

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

9. Applied rewrites 57.4%

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

Alternative 22: 58.3% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 56.7

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

5. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 57.1

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

7. Applied rewrites 57.1%

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

Alternative 23: 57.8% accurate, 12.4× speedup

Math

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

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(Float32(Float32(cosTheta_i * cosTheta_O) * 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 56.7

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

5. Applied rewrites 56.7%

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

6. Final simplification 56.7%

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

Alternative 24: 57.8% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) * single(0.5);
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 56.7

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

5. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lift-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 56.7

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

7. Applied rewrites 56.7%

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

Alternative 25: 57.8% accurate, 12.4× speedup

Math

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

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(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 = (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 56.7

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

5. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. associate-/l* N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 56.7

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

7. Applied rewrites 56.7%

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

8. Final simplification 56.7%

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

Alternative 26: 57.8% 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 56.7

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

5. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. associate-/l* N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. associate-/r/ N/A

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

   15. lift-/.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. associate-/r/ N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f32 56.7

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

7. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 56.7

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

9. Applied rewrites 56.7%

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

10. Final simplification 56.7%

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

Alternative 27: 57.8% 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 56.7

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

5. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. associate-/l* N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. associate-/r/ N/A

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

   15. lift-/.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. associate-/r/ N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f32 56.7

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

7. Applied rewrites 56.7%

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

   1. lift-*.f32 N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 56.7

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

9. Applied rewrites 56.7%

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

10. Final simplification 56.7%

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

Reproduce

herbie shell --seed 2024216 
(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)))