HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 13.4s

Alternatives: 9

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   (/ cosTheta_O_m v)
   (*
    cosTheta_i
    (/
     (/ (pow (exp sinTheta_O) (/ (- sinTheta_i) v)) (* 2.0 v))
     (sinh (/ 1.0 v)))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * ((cosTheta_O_m / v) * (cosTheta_i * ((powf(expf(sinTheta_O), (-sinTheta_i / v)) / (2.0f * v)) / sinhf((1.0f / v)))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((costheta_o_m / v) * (costheta_i * (((exp(sintheta_o) ** (-sintheta_i / v)) / (2.0e0 * v)) / sinh((1.0e0 / v)))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(cosTheta_O_m / v) * Float32(cosTheta_i * Float32(Float32((exp(sinTheta_O) ^ Float32(Float32(-sinTheta_i) / v)) / Float32(Float32(2.0) * v)) / sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((cosTheta_O_m / v) * (cosTheta_i * (((exp(sinTheta_O) ^ (-sinTheta_i / 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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.1%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_O_m
   (/
    (*
     (/
      (/ (pow (exp sinTheta_O) (/ (- sinTheta_i) v)) (* -2.0 v))
      (sinh (/ -1.0 v)))
     cosTheta_i)
    v))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_O_m * ((((powf(expf(sinTheta_O), (-sinTheta_i / v)) / (-2.0f * v)) / sinhf((-1.0f / v))) * cosTheta_i) / v));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_o_m * (((((exp(sintheta_o) ** (-sintheta_i / v)) / ((-2.0e0) * v)) / sinh(((-1.0e0) / v))) * costheta_i) / v))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_O_m * Float32(Float32(Float32(Float32((exp(sinTheta_O) ^ Float32(Float32(-sinTheta_i) / v)) / Float32(Float32(-2.0) * v)) / sinh(Float32(Float32(-1.0) / v))) * cosTheta_i) / v)))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_O_m * (((((exp(sinTheta_O) ^ (-sinTheta_i / v)) / (single(-2.0) * v)) / sinh((single(-1.0) / v))) * cosTheta_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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.1%

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

6. Applied rewrites 99.1%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* (/ cosTheta_O_m v) (* cosTheta_i (/ (/ 0.5 v) (sinh (/ 1.0 v)))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * ((cosTheta_O_m / v) * (cosTheta_i * ((0.5f / v) / sinhf((1.0f / v)))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((costheta_o_m / v) * (costheta_i * ((0.5e0 / v) / sinh((1.0e0 / v)))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(cosTheta_O_m / v) * Float32(cosTheta_i * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((cosTheta_O_m / v) * (cosTheta_i * ((single(0.5) / 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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 99.0

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

7. Applied rewrites 99.0%

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

Alternative 4: 98.6% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* cosTheta_O_m (/ (* (/ (/ 0.5 v) (sinh (/ 1.0 v))) cosTheta_i) v))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_O_m * ((((0.5f / v) / sinhf((1.0f / v))) * cosTheta_i) / v));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_o_m * ((((0.5e0 / v) / sinh((1.0e0 / v))) * costheta_i) / v))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_O_m * Float32(Float32(Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))) * cosTheta_i) / v)))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_O_m * ((((single(0.5) / v) / sinh((single(1.0) / v))) * cosTheta_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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 99.0

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

7. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 99.0

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.0

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

9. Applied rewrites 99.0%

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

Alternative 5: 98.6% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* cosTheta_O_m (/ (* cosTheta_i (/ 0.5 v)) (* (sinh (/ 1.0 v)) v)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_O_m * ((cosTheta_i * (0.5f / v)) / (sinhf((1.0f / v)) * v)));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_o_m * ((costheta_i * (0.5e0 / v)) / (sinh((1.0e0 / v)) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_O_m * Float32(Float32(cosTheta_i * Float32(Float32(0.5) / v)) / Float32(sinh(Float32(Float32(1.0) / v)) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_O_m * ((cosTheta_i * (single(0.5) / v)) / (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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 99.0

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

7. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 99.0

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.0

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

9. Applied rewrites 99.0%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-/l* N/A

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

    4. lift-/.f32 N/A

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

    5. frac-times N/A

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

    6. lift-*.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 99.0

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

11. Applied rewrites 99.0%

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

Alternative 6: 64.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{\left(1 - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) \cdot \left(\frac{cosTheta\_O\_m}{v} \cdot cosTheta\_i\right)}{\frac{0.3333333333333333}{v \cdot v} + 2} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (*
    (- 1.0 (/ (* sinTheta_i sinTheta_O) v))
    (* (/ cosTheta_O_m v) cosTheta_i))
   (+ (/ 0.3333333333333333 (* v v)) 2.0))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (((1.0f - ((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_O_m / v) * cosTheta_i)) / ((0.3333333333333333f / (v * v)) + 2.0f));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (((1.0e0 - ((sintheta_i * sintheta_o) / v)) * ((costheta_o_m / v) * costheta_i)) / ((0.3333333333333333e0 / (v * v)) + 2.0e0))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(Float32(Float32(1.0) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) * Float32(Float32(cosTheta_O_m / v) * cosTheta_i)) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (((single(1.0) - ((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_O_m / v) * cosTheta_i)) / ((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. 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 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. fp-cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.9

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

10. Applied rewrites 65.9%

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

Alternative 7: 64.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{\frac{0.5}{v} \cdot \left(cosTheta\_O\_m \cdot cosTheta\_i\right)}{\frac{0.16666666666666666}{v \cdot v} + 1} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (* (/ 0.5 v) (* cosTheta_O_m cosTheta_i))
   (+ (/ 0.16666666666666666 (* v v)) 1.0))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (((0.5f / v) * (cosTheta_O_m * cosTheta_i)) / ((0.16666666666666666f / (v * v)) + 1.0f));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (((0.5e0 / v) * (costheta_o_m * costheta_i)) / ((0.16666666666666666e0 / (v * v)) + 1.0e0))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(Float32(Float32(0.5) / v) * Float32(cosTheta_O_m * cosTheta_i)) / Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) + Float32(1.0))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (((single(0.5) / v) * (cosTheta_O_m * cosTheta_i)) / ((single(0.16666666666666666) / (v * v)) + single(1.0)));
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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. associate-*r/ N/A

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

10. Taylor expanded in v around inf

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. lower-/.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 65.9

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

12. Applied rewrites 65.9%

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

Alternative 8: 58.6% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (/ (* (* 0.5 cosTheta_O_m) cosTheta_i) v)))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (((0.5f * cosTheta_O_m) * cosTheta_i) / v);
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (((0.5e0 * costheta_o_m) * costheta_i) / v)
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(Float32(Float32(0.5) * cosTheta_O_m) * cosTheta_i) / v))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (((single(0.5) * cosTheta_O_m) * cosTheta_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. Taylor expanded in v around inf

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

   1. 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 60.4

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

5. Applied rewrites 60.4%

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

7. Applied rewrites 60.4%

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

9. Applied rewrites 60.4%

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

Alternative 9: 58.5% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (/ (* cosTheta_O_m cosTheta_i) v))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (0.5f * ((cosTheta_O_m * cosTheta_i) / v));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (0.5e0 * ((costheta_o_m * costheta_i) / v))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(0.5) * Float32(Float32(cosTheta_O_m * cosTheta_i) / v)))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(0.5) * ((cosTheta_O_m * cosTheta_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. Taylor expanded in v around inf

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

   1. 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 60.4

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

5. Applied rewrites 60.4%

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

Reproduce

herbie shell --seed 2024343 
(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)))