HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 23.6s

Alternatives: 22

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.6× speedup

Math

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

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
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* cosTheta_i_m (* cosTheta_O_m (/ 1.0 v))))
    (+ (/ (exp (/ 1.0 v)) (/ 1.0 v)) (/ (exp (/ -1.0 v)) (/ -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 * ((expf(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (1.0f / v)))) / ((expf((1.0f / v)) / (1.0f / v)) + (expf((-1.0f / v)) / (-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 * ((exp(((sintheta_i * sintheta_o) / -v)) * (costheta_i_m * (costheta_o_m * (1.0e0 / v)))) / ((exp((1.0e0 / v)) / (1.0e0 / v)) + (exp(((-1.0e0) / v)) / ((-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(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(cosTheta_i_m * Float32(cosTheta_O_m * Float32(Float32(1.0) / v)))) / Float32(Float32(exp(Float32(Float32(1.0) / v)) / Float32(Float32(1.0) / v)) + Float32(exp(Float32(Float32(-1.0) / v)) / 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 * ((exp(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (single(1.0) / v)))) / ((exp((single(1.0) / v)) / (single(1.0) / v)) + (exp((single(-1.0) / v)) / (single(-1.0) / v)))));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. remove-double-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

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

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 98.6

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sinh.f32 N/A

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

   8. sinh-undef N/A

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

   9. div-sub N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-exp.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. neg-mul-1 N/A

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

   15. lift-/.f32 N/A

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

   16. div-inv N/A

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

   17. lift-/.f32 N/A

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

   18. lower-exp.f32 98.8

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

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
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* cosTheta_i_m (* cosTheta_O_m (/ 1.0 v))))
    (- (/ (exp (/ 1.0 v)) (/ 1.0 v)) (* v (exp (/ -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 * ((expf(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (1.0f / v)))) / ((expf((1.0f / v)) / (1.0f / v)) - (v * expf((-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 * ((exp(((sintheta_i * sintheta_o) / -v)) * (costheta_i_m * (costheta_o_m * (1.0e0 / v)))) / ((exp((1.0e0 / v)) / (1.0e0 / v)) - (v * exp(((-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(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(cosTheta_i_m * Float32(cosTheta_O_m * Float32(Float32(1.0) / v)))) / Float32(Float32(exp(Float32(Float32(1.0) / v)) / Float32(Float32(1.0) / v)) - Float32(v * exp(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 * ((exp(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (single(1.0) / v)))) / ((exp((single(1.0) / v)) / (single(1.0) / v)) - (v * exp((single(-1.0) / v))))));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. remove-double-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

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

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 98.6

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sinh.f32 N/A

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

   8. sinh-undef N/A

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

   9. div-sub N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-exp.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. neg-mul-1 N/A

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

   15. lift-/.f32 N/A

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

   16. div-inv N/A

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

   17. lift-/.f32 N/A

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

   18. lower-exp.f32 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r/ N/A

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

   4. /-rgt-identity N/A

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

   5. lower-*.f32 98.8

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 98.8% accurate, 0.9× speedup

Math

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

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
   (*
    (* cosTheta_i_m cosTheta_O_m)
    (/
     (/ (exp (/ (* sinTheta_i sinTheta_O) (- v))) v)
     (/ (* (sinh (/ 1.0 v)) 2.0) (/ 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 * cosTheta_O_m) * ((expf(((sinTheta_i * sinTheta_O) / -v)) / v) / ((sinhf((1.0f / v)) * 2.0f) / (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 * costheta_o_m) * ((exp(((sintheta_i * sintheta_o) / -v)) / v) / ((sinh((1.0e0 / v)) * 2.0e0) / (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(cosTheta_i_m * cosTheta_O_m) * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) / v) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) / 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 * cosTheta_O_m) * ((exp(((sinTheta_i * sinTheta_O) / -v)) / v) / ((sinh((single(1.0) / v)) * single(2.0)) / (single(1.0) / v)))));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. remove-double-div N/A

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

   7. lift-/.f32 N/A

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

   8. div-inv N/A

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

   9. lower-/.f32 98.8

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

6. Applied rewrites 98.8%

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

Alternative 4: 98.7% accurate, 1.5× speedup

Math

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

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)
     (/
      (/ (fma sinTheta_i (/ sinTheta_O (- v)) 1.0) (* v 2.0))
      (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) * ((fmaf(sinTheta_i, (sinTheta_O / -v), 1.0f) / (v * 2.0f)) / sinhf((1.0f / v))))));
}

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(fma(sinTheta_i, Float32(sinTheta_O / Float32(-v)), Float32(1.0)) / Float32(v * Float32(2.0))) / sinh(Float32(Float32(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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-/.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

9. Applied rewrites 98.7%

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

Alternative 5: 98.7% accurate, 1.5× speedup

Math

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

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 (/ 1.0 v))
     (/
      (fma sinTheta_O (/ sinTheta_i (- v)) 1.0)
      (* (sinh (/ 1.0 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 * (1.0f / v)) * (fmaf(sinTheta_O, (sinTheta_i / -v), 1.0f) / (sinhf((1.0f / v)) * (v * 2.0f))))));
}

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 * Float32(Float32(1.0) / v)) * Float32(fma(sinTheta_O, Float32(sinTheta_i / Float32(-v)), Float32(1.0)) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(2.0))))))))
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-/.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 6: 98.6% accurate, 1.6× speedup

Math

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

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 (fma sinTheta_i (/ sinTheta_O (- v)) 1.0))
     (* (sinh (/ 1.0 v)) (* 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 * fmaf(sinTheta_i, (sinTheta_O / -v), 1.0f)) / (sinhf((1.0f / v)) * (v * (v * 2.0f))))));
}

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 * fma(sinTheta_i, Float32(sinTheta_O / Float32(-v)), Float32(1.0))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0))))))))
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-/.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.5

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

9. Applied rewrites 98.6%

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

10. Final simplification 98.6%

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

Alternative 7: 98.6% accurate, 1.8× speedup

Math

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

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_i_m cosTheta_O_m)
    (/ (/ 1.0 v) (* (sinh (/ 1.0 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_i_m * cosTheta_O_m) * ((1.0f / v) / (sinhf((1.0f / 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_i_m * costheta_o_m) * ((1.0e0 / v) / (sinh((1.0e0 / 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(Float32(cosTheta_i_m * cosTheta_O_m) * Float32(Float32(Float32(1.0) / v) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(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_i_m * cosTheta_O_m) * ((single(1.0) / v) / (sinh((single(1.0) / v)) * (v * single(2.0))))));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.5%

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

Alternative 8: 98.4% accurate, 1.8× speedup

Math

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

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) (/ 1.0 (* (sinh (/ 1.0 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) * (1.0f / (sinhf((1.0f / 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) * (1.0e0 / (sinh((1.0e0 / 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(Float32(cosTheta_i_m / v) * Float32(Float32(1.0) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(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(1.0) / (sinh((single(1.0) / v)) * (v * single(2.0)))))));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-/.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.3%

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

Alternative 9: 98.4% accurate, 1.8× speedup

Math

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

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 cosTheta_O_m) v) (* v (* (sinh (/ 1.0 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_i_m * cosTheta_O_m) / v) / (v * (sinhf((1.0f / 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_i_m * costheta_o_m) / v) / (v * (sinh((1.0e0 / 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(Float32(Float32(cosTheta_i_m * cosTheta_O_m) / v) / Float32(v * Float32(sinh(Float32(Float32(1.0) / 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_i_m * cosTheta_O_m) / v) / (v * (sinh((single(1.0) / v)) * single(2.0)))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Final simplification 98.3%

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

Alternative 10: 98.4% accurate, 1.8× speedup

Math

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

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_i_m 1.0)
    (/ cosTheta_O_m (* v (* (sinh (/ 1.0 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_i_m * 1.0f) * (cosTheta_O_m / (v * (sinhf((1.0f / 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_i_m * 1.0e0) * (costheta_o_m / (v * (sinh((1.0e0 / 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(Float32(cosTheta_i_m * Float32(1.0)) * Float32(cosTheta_O_m / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(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_i_m * single(1.0)) * (cosTheta_O_m / (v * (sinh((single(1.0) / v)) * (v * single(2.0)))))));
end

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. distribute-neg-frac2 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-neg.f32 N/A

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

   15. associate-/l/ N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 11: 70.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \left(cosTheta\_O\_m \cdot \left(\frac{cosTheta\_i\_m}{v} \cdot \frac{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{-v}, 1\right)}{\left(v \cdot 2\right) \cdot \frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}\right)\right)\right) \end{array} \]

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_O_m
    (*
     (/ cosTheta_i_m v)
     (/
      (fma sinTheta_O (/ sinTheta_i (- v)) 1.0)
      (*
       (* v 2.0)
       (/
        (-
         (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
         -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) * (fmaf(sinTheta_O, (sinTheta_i / -v), 1.0f) / ((v * 2.0f) * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) - -1.0f) / v))))));
}

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(fma(sinTheta_O, Float32(sinTheta_i / Float32(-v)), Float32(1.0)) / Float32(Float32(v * Float32(2.0)) * Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) - Float32(-1.0)) / v)))))))
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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-/.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 67.6%

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

11. Final simplification 67.6%

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

Alternative 12: 64.3% accurate, 3.8× speedup

Math

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

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 v)
     (/
      (fma sinTheta_O (/ sinTheta_i (- v)) 1.0)
      (+ 2.0 (/ 0.3333333333333333 (* 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 / v) * (fmaf(sinTheta_O, (sinTheta_i / -v), 1.0f) / (2.0f + (0.3333333333333333f / (v * v)))))));
}

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(fma(sinTheta_O, Float32(sinTheta_i / Float32(-v)), Float32(1.0)) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-/.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 61.5

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

10. Applied rewrites 61.5%

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

Alternative 13: 59.1% accurate, 6.2× speedup

Math

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

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
   (/ (* (/ -1.0 (/ -1.0 (* cosTheta_i_m cosTheta_O_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 * (((-1.0f / (-1.0f / (cosTheta_i_m * cosTheta_O_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 * ((((-1.0e0) / ((-1.0e0) / (costheta_i_m * costheta_o_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(Float32(-1.0) / Float32(Float32(-1.0) / Float32(cosTheta_i_m * cosTheta_O_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 * (((single(-1.0) / (single(-1.0) / (cosTheta_i_m * cosTheta_O_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. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 56.6%

   \[\leadsto \frac{\frac{1}{\frac{1}{cosTheta\_i \cdot cosTheta\_O}} \cdot 0.5}{v} \]

8. Final simplification 56.6%

   \[\leadsto \frac{\frac{-1}{\frac{-1}{cosTheta\_i \cdot cosTheta\_O}} \cdot 0.5}{v} \]
9. Add Preprocessing

Alternative 14: 59.0% accurate, 8.2× speedup

Math

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

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 (/ 1.0 (/ v (* cosTheta_O_m (* cosTheta_i_m 0.5)))))))

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 * (1.0f / (v / (cosTheta_O_m * (cosTheta_i_m * 0.5f)))));
}

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 * (1.0e0 / (v / (costheta_o_m * (costheta_i_m * 0.5e0)))))
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(1.0) / Float32(v / Float32(cosTheta_O_m * Float32(cosTheta_i_m * Float32(0.5)))))))
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(1.0) / (v / (cosTheta_O_m * (cosTheta_i_m * single(0.5))))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 56.3%

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

Alternative 15: 59.0% accurate, 9.7× speedup

Math

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

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 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 * 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 * 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(0.5) / Float32(v / Float32(cosTheta_i_m * 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 * 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. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 56.3%

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

Alternative 16: 58.6% accurate, 12.4× speedup

Math

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

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 (* (* cosTheta_i_m cosTheta_O_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_i_m * cosTheta_O_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_i_m * costheta_o_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(cosTheta_i_m * cosTheta_O_m) * Float32(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_i_m * cosTheta_O_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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around inf

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

   1. lower-/.f32 55.7

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

7. Applied rewrites 55.7%

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

Alternative 17: 58.6% accurate, 12.4× speedup

Math

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

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(cosTheta_O_m * Float32(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. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 55.7%

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

8. Final simplification 55.7%

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

Alternative 18: 58.6% accurate, 12.4× speedup

Math

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

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_i_m cosTheta_O_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_i_m * cosTheta_O_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_i_m * costheta_o_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_i_m * cosTheta_O_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_i_m * cosTheta_O_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. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

6. Final simplification 55.7%

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

Alternative 19: 58.6% accurate, 12.4× speedup

Math

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

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_i_m cosTheta_O_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_i_m * cosTheta_O_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_i_m * costheta_o_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_i_m * cosTheta_O_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_i_m * cosTheta_O_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. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 55.7%

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

8. Final simplification 55.7%

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

Alternative 20: 58.6% accurate, 12.4× speedup

Math

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

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 (/ cosTheta_O_m (* 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_i_m * (cosTheta_O_m / (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_i_m * (costheta_o_m / (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_i_m * Float32(cosTheta_O_m / Float32(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_i_m * (cosTheta_O_m / (v * single(2.0)))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 55.7%

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

8. Final simplification 55.7%

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

Alternative 21: 58.6% accurate, 12.4× speedup

Math

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

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 (* cosTheta_O_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_i_m * (cosTheta_O_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_i_m * (costheta_o_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(cosTheta_i_m * Float32(cosTheta_O_m * Float32(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_i_m * (cosTheta_O_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. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 55.7%

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

9. Applied rewrites 55.7%

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

10. Final simplification 55.7%

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

Alternative 22: 58.6% accurate, 12.4× speedup

Math

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

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 (/ 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(cosTheta_O_m * Float32(cosTheta_i_m * Float32(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. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 55.7%

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

9. Applied rewrites 55.7%

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

10. Final simplification 55.7%

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

Reproduce

herbie shell --seed 2024237 
(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)))