HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 14.5s

Alternatives: 11

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 99.0

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 99.0

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

6. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 98.9

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 98.8

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 6: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_o / v) / ((2.0e0 * v) * sinh((1.0e0 / v)))) * (costheta_i / (1.0e0 - ((((-0.5e0) * ((((sintheta_i * sintheta_i) * sintheta_o) * sintheta_o) / v)) - (sintheta_o * sintheta_i)) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-sinh.f32 N/A

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

   5. sinh-undef N/A

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

   6. flip-- N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 7: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-sinh.f32 N/A

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

   5. sinh-undef N/A

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

   6. flip-- N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 98.6

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

9. Applied rewrites 98.6%

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

10. Final simplification 98.6%

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

Alternative 8: 64.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 66.2

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

5. Applied rewrites 66.2%

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

6. Final simplification 66.2%

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

Alternative 9: 58.8% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 61.3%

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

8. Taylor expanded in sinTheta_i around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 61.3

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

10. Applied rewrites 61.3%

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

11. Final simplification 61.3%

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

Alternative 10: 58.8% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-sinh.f32 N/A

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

   5. sinh-undef N/A

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

   6. flip-- N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. log-E N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{2}{\color{blue}{4}} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]

   7. metadata-eval N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 60.6

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

7. Applied rewrites 60.6%

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

9. Applied rewrites 61.3%

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

10. Final simplification 61.3%

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

Alternative 11: 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 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-sinh.f32 N/A

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

   5. sinh-undef N/A

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

   6. flip-- N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. log-E N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{2}{\color{blue}{4}} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]

   7. metadata-eval N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 60.6

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

7. Applied rewrites 60.6%

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

Reproduce

herbie shell --seed 2024288 
(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)))