HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 12.5s

Alternatives: 13

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, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_i_m / v) * Float32(Float32(Float32(Float32(Float32(Float32(0.5) * sinTheta_O) / v) * sinTheta_i) - Float32(0.5)) / v)) * Float32(cosTheta_O_m / 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\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((cosTheta_i_m / v) * (((((single(0.5) * sinTheta_O) / v) * sinTheta_i) - single(0.5)) / v)) * (cosTheta_O_m / sinh((single(-1.0) / v)))));
end

Derivation

1. Initial program 98.5%

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

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. times-frac N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 98.9

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

9. Applied rewrites 98.9%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. associate-*l/ N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f32 N/A

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

11. Applied rewrites 98.9%

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

Alternative 2: 98.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   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. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. times-frac N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 98.9

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

9. Applied rewrites 98.9%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. associate-*r/ N/A

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

    5. associate-/l/ N/A

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

    6. lower-/.f32 N/A

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

11. Applied rewrites 98.8%

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

Alternative 3: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   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. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. metadata-eval N/A

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

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

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 98.7

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

9. Applied rewrites 98.7%

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

Alternative 4: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   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. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

Alternative 5: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_m
    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_i_s * (costheta_o_s * (costheta_o_m * ((costheta_i_m / v) * ((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\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(cosTheta_O_m * Float32(Float32(cosTheta_i_m / v) * 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\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (cosTheta_O_m * ((cosTheta_i_m / v) * ((single(0.5) / v) / sinh((single(1.0) / v))))));
end

Derivation

1. Initial program 98.5%

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

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.6

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

7. Applied rewrites 98.6%

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

Alternative 6: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. rec-exp N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-exp.f32 N/A

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

   13. lower-/.f32 98.4

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 98.5%

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

11. Applied rewrites 98.5%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   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. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. rec-exp N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-exp.f32 N/A

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

   13. lower-/.f32 98.4

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 98.5%

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

Alternative 8: 64.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \left(cosTheta\_O\_m \cdot \frac{cosTheta\_i\_m}{v \cdot \left(\frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v} \cdot \left(2 \cdot v\right)\right)}\right)\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (*
    cosTheta_O_m
    (/
     cosTheta_i_m
     (* v (* (/ (+ (/ 0.16666666666666666 (* v v)) 1.0) v) (* 2.0 v))))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_O_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_O_s * (cosTheta_O_m * (cosTheta_i_m / (v * ((((0.16666666666666666f / (v * v)) + 1.0f) / v) * (2.0f * v))))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_m
    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_i_s * (costheta_o_s * (costheta_o_m * (costheta_i_m / (v * ((((0.16666666666666666e0 / (v * v)) + 1.0e0) / v) * (2.0e0 * v))))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(cosTheta_O_m * Float32(cosTheta_i_m / Float32(v * Float32(Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) + Float32(1.0)) / v) * Float32(Float32(2.0) * v)))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (cosTheta_O_m * (cosTheta_i_m / (v * ((((single(0.16666666666666666) / (v * v)) + 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

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

   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. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. rec-exp N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-exp.f32 N/A

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

   13. lower-/.f32 98.4

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in v around inf

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

12. Applied rewrites 64.2%

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

Alternative 9: 64.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_m
    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_i_s * (costheta_o_s * (costheta_o_m * (costheta_i_m / (v * ((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\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(cosTheta_O_m * Float32(cosTheta_i_m / Float32(v * 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\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (cosTheta_O_m * (cosTheta_i_m / (v * ((single(0.3333333333333333) / (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

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

   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. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. rec-exp N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-exp.f32 N/A

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

   13. lower-/.f32 98.4

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in v around inf

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

12. Applied rewrites 64.2%

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

Alternative 10: 59.2% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   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. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 59.0

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

7. Applied rewrites 59.0%

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

Alternative 11: 59.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. 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 58.9

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

5. Applied rewrites 58.9%

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

7. Applied rewrites 58.9%

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

Alternative 12: 59.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. 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 58.9

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

5. Applied rewrites 58.9%

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

Alternative 13: 59.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. 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 58.9

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

5. Applied rewrites 58.9%

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

7. Applied rewrites 58.8%

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

Reproduce

herbie shell --seed 2024326 
(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)))