HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 12.1s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{\frac{\left(-sinTheta\_i\right) \cdot sinTheta\_O}{v}} \cdot \left(\left(\frac{1}{v} \cdot cosTheta\_i\right) \cdot cosTheta\_O\right)}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* (- sinTheta_i) sinTheta_O) v))
   (* (* (/ 1.0 v) cosTheta_i) cosTheta_O))
  (/ (* (sinh (/ 1.0 v)) 2.0) (/ 1.0 v))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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)) * (((1.0e0 / v) * costheta_i) * costheta_o)) / ((sinh((1.0e0 / v)) * 2.0e0) / (1.0e0 / v))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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(Float32(Float32(1.0) / v) * cosTheta_i) * cosTheta_O)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) / Float32(Float32(1.0) / v)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((-sinTheta_i * sinTheta_O) / v)) * (((single(1.0) / v) * cosTheta_i) * cosTheta_O)) / ((sinh((single(1.0) / v)) * single(2.0)) / (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. 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 98.7

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

   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 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.8

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

6. Applied rewrites 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}} \]
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 99.0

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

8. Applied rewrites 99.0%

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

9. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{\frac{\left(-sinTheta\_i\right) \cdot sinTheta\_O}{v}} \cdot \left(\left(\frac{1}{v} \cdot cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot v\right) \cdot 2} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* (- sinTheta_i) sinTheta_O) v))
   (* (* (/ 1.0 v) cosTheta_i) cosTheta_O))
  (* (* (sinh (/ 1.0 v)) v) 2.0)))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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)) * (((1.0e0 / v) * costheta_i) * costheta_o)) / ((sinh((1.0e0 / v)) * v) * 2.0e0)
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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(Float32(Float32(1.0) / v) * cosTheta_i) * cosTheta_O)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * v) * Float32(2.0)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((-sinTheta_i * sinTheta_O) / v)) * (((single(1.0) / v) * cosTheta_i) * cosTheta_O)) / ((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. 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 98.7

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

   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 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.8

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

6. Applied rewrites 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}} \]
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 99.0

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

8. Applied rewrites 99.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. remove-double-div N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 98.9

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

10. Applied rewrites 98.9%

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

11. Final simplification 98.9%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{\frac{\left(-sinTheta\_i\right) \cdot sinTheta\_O}{v}} \cdot \left(cosTheta\_i \cdot \left(\frac{1}{v} \cdot cosTheta\_O\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* (- sinTheta_i) sinTheta_O) v))
   (* cosTheta_i (* (/ 1.0 v) cosTheta_O)))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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 * ((1.0f / v) * cosTheta_O))) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 * ((1.0e0 / v) * costheta_o))) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((-sinTheta_i * sinTheta_O) / v)) * (cosTheta_i * ((single(1.0) / v) * cosTheta_O))) / ((sinh((single(1.0) / v)) * single(2.0)) * 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. 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. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0 - cosTheta\_i \cdot cosTheta\_O}}{\mathsf{neg}\left(v\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-sub N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(v\right)} - \frac{cosTheta\_i \cdot cosTheta\_O}{\mathsf{neg}\left(v\right)}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. distribute-frac-neg2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{0}{\mathsf{neg}\left(v\right)} - \color{blue}{\left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. distribute-frac-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{0}{\mathsf{neg}\left(v\right)} - \color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. frac-sub N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 \cdot v - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 \cdot v - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. mul0-lft N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0} - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. lower--.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0 - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. lower-neg.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \color{blue}{\left(-v\right)} \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\mathsf{neg}\left(\color{blue}{cosTheta\_i \cdot cosTheta\_O}\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

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

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i\right)\right) \cdot cosTheta\_O\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i\right)\right) \cdot cosTheta\_O\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   16. lower-neg.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\color{blue}{\left(-cosTheta\_i\right)} \cdot cosTheta\_O\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   18. lower-neg.f32 98.6

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\color{blue}{\left(-v\right)} \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}}{\left(-v\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. sub0-neg N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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

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

   8. lift-neg.f32 N/A

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

   9. remove-double-neg N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. *-inverses N/A

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

   11. *-rgt-identity N/A

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

   12. div-inv N/A

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

   13. lift-/.f32 N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 98.8

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{\frac{\left(-sinTheta\_i\right) \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* (- sinTheta_i) sinTheta_O) v)) (* (/ cosTheta_i v) cosTheta_O))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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 / v) * cosTheta_O)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 / v) * costheta_o)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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 / v) * cosTheta_O)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((-sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i / v) * cosTheta_O)) / ((sinh((single(1.0) / v)) * single(2.0)) * 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{\frac{\left(-sinTheta\_i\right) \cdot sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* (- sinTheta_i) sinTheta_O) v)) (* (/ cosTheta_O v) cosTheta_i))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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_o / v) * costheta_i)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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_O / v) * cosTheta_i)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((-sinTheta_i * sinTheta_O) / v)) * ((cosTheta_O / v) * cosTheta_i)) / ((sinh((single(1.0) / v)) * single(2.0)) * 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. 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 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 6: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{-1}{\left(-2 \cdot v\right) \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ -1.0 (* (* -2.0 v) (sinh (/ 1.0 v))))
  (/
   (* cosTheta_O (- cosTheta_i (* cosTheta_i (/ (* sinTheta_O sinTheta_i) v))))
   v)))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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) * sinh((1.0e0 / v)))) * ((costheta_o * (costheta_i - (costheta_i * ((sintheta_o * sintheta_i) / v)))) / v)
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(-1.0) / ((single(-2.0) * v) * sinh((single(1.0) / v)))) * ((cosTheta_O * (cosTheta_i - (cosTheta_i * ((sinTheta_O * sinTheta_i) / 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

   7. lower-/.f32 N/A

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

   8. associate-/l* N/A

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

   9. distribute-lft-out-- N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-*.f32 98.4

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

9. Applied rewrites 98.4%

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

Alternative 7: 93.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{1}{\left(\left(2 \cdot v\right) \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot \frac{\frac{v}{cosTheta\_i}}{cosTheta\_O}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (* (* (* 2.0 v) (sinh (/ 1.0 v))) (/ (/ v cosTheta_i) cosTheta_O))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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) * sinh((1.0e0 / v))) * ((v / costheta_i) / costheta_o))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (((single(2.0) * v) * sinh((single(1.0) / v))) * ((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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

6. Applied rewrites 92.3%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Applied rewrites 92.1%

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

Alternative 8: 59.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{1}{\frac{v}{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ v (* 0.5 (* cosTheta_i cosTheta_O)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
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 98.5%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.2

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

8. Applied rewrites 58.2%

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

10. Applied rewrites 58.7%

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

Alternative 9: 59.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (/ v (* cosTheta_i cosTheta_O))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
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

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.2

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

8. Applied rewrites 58.2%

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

10. Applied rewrites 58.7%

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

Alternative 10: 58.7% accurate, 12.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (* cosTheta_i cosTheta_O)) v))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
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

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.2

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

8. Applied rewrites 58.2%

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

10. Applied rewrites 58.3%

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

Alternative 11: 58.7% accurate, 12.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (* cosTheta_O cosTheta_i) v)))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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_o * costheta_i) / v)
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((cosTheta_O * cosTheta_i) / 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

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.2

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

8. Applied rewrites 58.2%

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

Reproduce

herbie shell --seed 2024303 
(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)))