HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 23.6s

Alternatives: 30

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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 98.9

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

6. Applied rewrites 98.9%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_i cosTheta_O)
  (*
   v
   (* (/ (sinh (/ 1.0 v)) (/ 0.5 v)) (exp (/ (* sinTheta_i sinTheta_O) 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)) / (0.5f / v)) * expf(((sinTheta_i * sinTheta_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 * costheta_o) / (v * ((sinh((1.0e0 / v)) / (0.5e0 / v)) * exp(((sintheta_i * sintheta_o) / v))))
end function

Julia

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 98.7

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.7%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* cosTheta_i cosTheta_O)
   (/ 0.5 (* v (exp (/ (* sinTheta_i sinTheta_O) 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 * expf(((sinTheta_i * sinTheta_O) / 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 * exp(((sintheta_i * sintheta_o) / v))))) / (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) * Float32(Float32(0.5) / Float32(v * exp(Float32(Float32(sinTheta_i * sinTheta_O) / v))))) / Float32(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 * exp(((sinTheta_i * sinTheta_O) / v))))) / (v * sinh((single(1.0) / v)));
end

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 6: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 98.9

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

6. Applied rewrites 98.9%

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

8. Applied rewrites 98.6%

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

9. Final simplification 98.6%

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

Alternative 7: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 8: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_i, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_O\right), sinTheta\_O \cdot \left(-v\right)\right)}{v \cdot v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\left(v \cdot v\right) \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (*
   cosTheta_O
   (fma
    sinTheta_i
    (/
     (fma 0.5 (* sinTheta_O (* sinTheta_i sinTheta_O)) (* sinTheta_O (- v)))
     (* v v))
    1.0))
  (/ cosTheta_i (* (* v v) (* (sinh (/ 1.0 v)) 2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * fmaf(sinTheta_i, (fmaf(0.5f, (sinTheta_O * (sinTheta_i * sinTheta_O)), (sinTheta_O * -v)) / (v * v)), 1.0f)) * (cosTheta_i / ((v * v) * (sinhf((1.0f / v)) * 2.0f)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * fma(sinTheta_i, Float32(fma(Float32(0.5), Float32(sinTheta_O * Float32(sinTheta_i * sinTheta_O)), Float32(sinTheta_O * Float32(-v))) / Float32(v * v)), Float32(1.0))) * Float32(cosTheta_i / Float32(Float32(v * v) * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)))))
end

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. neg-mul-1 N/A

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

   15. lower-/.f32 N/A

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

   16. neg-mul-1 N/A

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

   17. lower-neg.f32 98.6

       \[\leadsto \left(\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(0.5, \frac{sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v \cdot v}, \frac{sinTheta\_O}{\color{blue}{-v}}\right), 1\right) \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot v\right)} \]

9. Applied rewrites 98.6%

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(sinTheta\_i, \mathsf{fma}\left(0.5, \frac{sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v \cdot v}, \frac{sinTheta\_O}{-v}\right), 1\right)} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot v\right)} \]

10. Taylor expanded in v around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. mul-1-neg N/A

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

    9. *-commutative N/A

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

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

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

    11. mul-1-neg N/A

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

    12. lower-*.f32 N/A

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

    13. mul-1-neg N/A

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

    14. lower-neg.f32 N/A

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

    15. unpow2 N/A

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

    16. lower-*.f32 98.6

        \[\leadsto \left(\mathsf{fma}\left(sinTheta\_i, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right), v \cdot \left(-sinTheta\_O\right)\right)}{\color{blue}{v \cdot v}}, 1\right) \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot v\right)} \]

12. Applied rewrites 98.6%

    \[\leadsto \left(\mathsf{fma}\left(sinTheta\_i, \color{blue}{\frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right), v \cdot \left(-sinTheta\_O\right)\right)}{v \cdot v}}, 1\right) \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot v\right)} \]

13. Final simplification 98.6%

    \[\leadsto \left(cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_i, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_O\right), sinTheta\_O \cdot \left(-v\right)\right)}{v \cdot v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\left(v \cdot v\right) \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
14. Add Preprocessing

Alternative 9: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 98.6

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

9. Applied rewrites 98.6%

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

10. Final simplification 98.6%

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

Alternative 10: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. associate-/l* N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. frac-2neg N/A

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

   8. associate-/l* N/A

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

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

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

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

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

   11. associate-/l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

7. Applied rewrites 98.3%

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

   1. div-inv N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-sinh.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 98.5

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

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

Alternative 11: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.5

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

7. Applied rewrites 98.5%

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

Alternative 12: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. associate-/l* N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. frac-2neg N/A

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

   8. associate-/l* N/A

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

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

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

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

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

   11. associate-/l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

7. Applied rewrites 98.3%

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

Alternative 13: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. associate-/l* N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

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

Alternative 14: 98.3% accurate, 1.9× speedup

Math

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

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(v * Float32(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 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. associate-/l* N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. div-inv N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 15: 76.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((cosTheta_i / v) / ((v * -2.0f) * (((-1.0f - (0.16666666666666666f / (v * v))) - ((0.008333333333333333f + (0.0001984126984126984f / (v * v))) / (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_o * ((costheta_i / v) / ((v * (-2.0e0)) * ((((-1.0e0) - (0.16666666666666666e0 / (v * v))) - ((0.008333333333333333e0 + (0.0001984126984126984e0 / (v * v))) / (v * (v * (v * v))))) / v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) / ((v * single(-2.0)) * (((single(-1.0) - (single(0.16666666666666666) / (v * v))) - ((single(0.008333333333333333) + (single(0.0001984126984126984) / (v * v))) / (v * (v * (v * v))))) / v)));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. associate-/l* N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. frac-2neg N/A

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

   8. associate-/l* N/A

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

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

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

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

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

   11. associate-/l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 77.5%

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

11. Final simplification 77.5%

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

Alternative 16: 76.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} - \left(-2 + \frac{-0.3333333333333333}{v \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (/
   (/ cosTheta_i v)
   (-
    (/
     (+ 0.016666666666666666 (/ 0.0003968253968253968 (* v v)))
     (* v (* v (* v v))))
    (+ -2.0 (/ -0.3333333333333333 (* v v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((cosTheta_i / v) / (((0.016666666666666666f + (0.0003968253968253968f / (v * v))) / (v * (v * (v * v)))) - (-2.0f + (-0.3333333333333333f / (v * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o * ((costheta_i / v) / (((0.016666666666666666e0 + (0.0003968253968253968e0 / (v * v))) / (v * (v * (v * v)))) - ((-2.0e0) + ((-0.3333333333333333e0) / (v * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(cosTheta_i / v) / Float32(Float32(Float32(Float32(0.016666666666666666) + Float32(Float32(0.0003968253968253968) / Float32(v * v))) / Float32(v * Float32(v * Float32(v * v)))) - Float32(Float32(-2.0) + Float32(Float32(-0.3333333333333333) / Float32(v * v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) / (((single(0.016666666666666666) + (single(0.0003968253968253968) / (v * v))) / (v * (v * (v * v)))) - (single(-2.0) + (single(-0.3333333333333333) / (v * v)))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. associate-/l* N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. frac-2neg N/A

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

   8. associate-/l* N/A

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

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

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

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

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

   11. associate-/l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in v around inf

   \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}} \]

   2. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}} \]

10. Applied rewrites 77.5%

    \[\leadsto \left(-cosTheta\_O\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{\left(-\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)}\right) + \left(-2 + \frac{-0.3333333333333333}{v \cdot v}\right)}} \]

11. Final simplification 77.5%

    \[\leadsto cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} - \left(-2 + \frac{-0.3333333333333333}{v \cdot v}\right)} \]
12. Add Preprocessing

Alternative 17: 70.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / ((v * 2.0f) * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) - -1.0f) / v))) * (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 / ((v * 2.0e0) * ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / v))) * (costheta_o / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 71.1%

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

10. Applied rewrites 71.2%

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

11. Final simplification 71.2%

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

Alternative 18: 70.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 71.1%

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

10. Applied rewrites 71.2%

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

11. Final simplification 71.2%

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

Alternative 19: 70.2% accurate, 4.1× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 71.1%

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

9. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}} \]
10. 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. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. sub-neg N/A

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

    6. metadata-eval N/A

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

    7. distribute-rgt-in N/A

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

    8. lower-fma.f32 N/A

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

11. Applied rewrites 71.1%

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

12. Final simplification 71.1%

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

Alternative 20: 70.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (/
   (/ cosTheta_i v)
   (-
    (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
    -2.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((cosTheta_i / v) / (((0.3333333333333333f + (0.016666666666666666f / (v * v))) / (v * v)) - -2.0f));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o * ((costheta_i / v) / (((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) - (-2.0e0)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(cosTheta_i / v) / Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(-2.0))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) / (((single(0.3333333333333333) + (single(0.016666666666666666) / (v * v))) / (v * v)) - single(-2.0)));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. associate-/l* N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. frac-2neg N/A

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

   8. associate-/l* N/A

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

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

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

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

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

   11. associate-/l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in v around inf

   \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(2\right)\right)}} \]

   2. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(2\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)} + \left(\mathsf{neg}\left(2\right)\right)} \]

   4. lower-neg.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)} + \left(\mathsf{neg}\left(2\right)\right)} \]

   5. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   6. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   7. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \color{blue}{\frac{\frac{1}{60} \cdot 1}{{v}^{2}}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{\color{blue}{\frac{1}{60}}}{{v}^{2}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \color{blue}{\frac{\frac{1}{60}}{{v}^{2}}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   10. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{\frac{1}{60}}{\color{blue}{v \cdot v}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   11. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{\frac{1}{60}}{\color{blue}{v \cdot v}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   12. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{\frac{1}{60}}{v \cdot v}}{\color{blue}{v \cdot v}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   13. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(cosTheta\_O\right)\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{\frac{1}{60}}{v \cdot v}}{\color{blue}{v \cdot v}}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)} \]

   14. metadata-eval 71.2

       \[\leadsto \left(-cosTheta\_O\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\left(-\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v}\right) + \color{blue}{-2}} \]

10. Applied rewrites 71.2%

    \[\leadsto \left(-cosTheta\_O\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{\left(-\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v}\right) + -2}} \]

11. Final simplification 71.2%

    \[\leadsto cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2} \]
12. Add Preprocessing

Alternative 21: 64.0% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

   1. associate-/l* N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. frac-2neg N/A

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

   8. associate-/l* N/A

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

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

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

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

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

   11. associate-/l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in v around inf

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

   1. distribute-lft-in N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 65.2

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

10. Applied rewrites 65.2%

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

11. Final simplification 65.2%

    \[\leadsto cosTheta\_O \cdot \frac{\frac{-cosTheta\_i}{v}}{-2 + \frac{-0.3333333333333333}{v \cdot v}} \]
12. Add Preprocessing

Alternative 22: 64.0% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i * costheta_o) / v) / (2.0e0 + (0.3333333333333333e0 / (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(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))
end

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 65.1

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

8. Applied rewrites 65.1%

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

9. Final simplification 65.1%

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

Alternative 23: 58.9% accurate, 6.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. 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 59.6

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

5. Applied rewrites 59.6%

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

   1. *-commutative N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. associate-/l* N/A

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

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

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. clear-num N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. lower-/.f32 N/A

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

   15. lower-/.f32 60.3

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f32 N/A

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

   21. lower-*.f32 60.3

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

7. Applied rewrites 60.3%

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

8. Final simplification 60.3%

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

Alternative 24: 58.8% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. 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 59.6

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

5. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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. clear-num N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 60.1

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

7. Applied rewrites 60.1%

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

8. Final simplification 60.1%

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

Alternative 25: 58.8% 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 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. 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 59.6

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

5. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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 59.6

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

7. Applied rewrites 59.6%

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

   1. lift-*.f32 N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lower-/.f32 60.1

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

9. Applied rewrites 60.1%

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

Alternative 26: 58.3% 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 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. 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 59.6

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

5. Applied rewrites 59.6%

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

6. Final simplification 59.6%

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

Alternative 27: 58.3% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. 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 59.6

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

5. Applied rewrites 59.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 59.6

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

7. Applied rewrites 59.6%

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

8. Final simplification 59.6%

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

Alternative 28: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. 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 59.6

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

5. Applied rewrites 59.6%

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

   1. *-commutative N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 59.6

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

7. Applied rewrites 59.6%

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

8. Final simplification 59.6%

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

Alternative 29: 58.2% 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 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. 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 59.6

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

5. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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 59.6

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

7. Applied rewrites 59.6%

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

8. Final simplification 59.6%

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

Alternative 30: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. 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 59.6

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

5. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 59.6

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

7. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_O}{v \cdot 2} \cdot cosTheta\_i} \]
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. lift-/.f32 N/A

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

   10. lower-*.f32 59.6

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

9. Applied rewrites 59.6%

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

10. Final simplification 59.6%

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

Reproduce

herbie shell --seed 2024219 
(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)))