HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 12.0s

Alternatives: 16

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

Alternative 2: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

Alternative 3: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. frac-times N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

9. Applied rewrites 98.6%

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

Alternative 4: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/l* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. lower-/.f32 N/A

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

9. Applied rewrites 98.6%

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

Alternative 5: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (costheta_i_m * ((costheta_o / v) * ((((-0.5e0) * ((sintheta_i * sintheta_o) - v)) / (v * v)) / sinh((1.0e0 / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in v around 0

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

10. Applied rewrites 98.6%

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

Alternative 6: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 98.5

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

7. Applied rewrites 98.5%

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

Alternative 7: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (costheta_i_m * ((costheta_o / v) * ((0.5e0 / v) / sinh((1.0e0 / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.1

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

7. Applied rewrites 98.1%

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

Alternative 8: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (costheta_i_m * (costheta_o * ((0.5e0 / v) / (sinh((1.0e0 / v)) * v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.1

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

7. Applied rewrites 98.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-*.f32 98.1

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

9. Applied rewrites 98.1%

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

Alternative 9: 70.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_i\_m \cdot \frac{\frac{\mathsf{fma}\left(-0.5 \cdot \frac{sinTheta\_i}{v \cdot v}, sinTheta\_O, \frac{0.5}{v}\right) \cdot cosTheta\_O}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{\left(-v\right) \cdot v} - -1}{v}}}{v}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_i_m
   (/
    (/
     (* (fma (* -0.5 (/ sinTheta_i (* v v))) sinTheta_O (/ 0.5 v)) cosTheta_O)
     (/
      (-
       (/
        (fma (/ 0.008333333333333333 (* v v)) -1.0 -0.16666666666666666)
        (* (- v) v))
       -1.0)
      v))
    v))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_i_m * (((fmaf((-0.5f * (sinTheta_i / (v * v))), sinTheta_O, (0.5f / v)) * cosTheta_O) / (((fmaf((0.008333333333333333f / (v * v)), -1.0f, -0.16666666666666666f) / (-v * v)) - -1.0f) / v)) / v));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_i_m * Float32(Float32(Float32(fma(Float32(Float32(-0.5) * Float32(sinTheta_i / Float32(v * v))), sinTheta_O, Float32(Float32(0.5) / v)) * cosTheta_O) / Float32(Float32(Float32(fma(Float32(Float32(0.008333333333333333) / Float32(v * v)), Float32(-1.0), Float32(-0.16666666666666666)) / Float32(Float32(-v) * v)) - Float32(-1.0)) / v)) / v)))
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

9. Applied rewrites 98.7%

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

10. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. lower-/.f32 N/A

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

12. Applied rewrites 69.4%

    \[\leadsto cosTheta\_i \cdot \frac{\frac{\mathsf{fma}\left(-0.5 \cdot \frac{sinTheta\_i}{v \cdot v}, sinTheta\_O, \frac{0.5}{v}\right) \cdot cosTheta\_O}{\color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{v \cdot v} - 1}{-v}}}}{v} \]

13. Final simplification 69.4%

    \[\leadsto cosTheta\_i \cdot \frac{\frac{\mathsf{fma}\left(-0.5 \cdot \frac{sinTheta\_i}{v \cdot v}, sinTheta\_O, \frac{0.5}{v}\right) \cdot cosTheta\_O}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{\left(-v\right) \cdot v} - -1}{v}}}{v} \]
14. Add Preprocessing

Alternative 10: 70.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_i\_m \cdot \left(\frac{cosTheta\_O}{v} \cdot \frac{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v \cdot v} \cdot -0.5, \frac{0.5}{v}\right)}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{\left(-v\right) \cdot v} - -1}{v}}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_i_m
   (*
    (/ cosTheta_O v)
    (/
     (fma sinTheta_O (* (/ sinTheta_i (* v v)) -0.5) (/ 0.5 v))
     (/
      (-
       (/
        (fma (/ 0.008333333333333333 (* v v)) -1.0 -0.16666666666666666)
        (* (- v) v))
       -1.0)
      v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_i_m * ((cosTheta_O / v) * (fmaf(sinTheta_O, ((sinTheta_i / (v * v)) * -0.5f), (0.5f / v)) / (((fmaf((0.008333333333333333f / (v * v)), -1.0f, -0.16666666666666666f) / (-v * v)) - -1.0f) / v))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_i_m * Float32(Float32(cosTheta_O / v) * Float32(fma(sinTheta_O, Float32(Float32(sinTheta_i / Float32(v * v)) * Float32(-0.5)), Float32(Float32(0.5) / v)) / Float32(Float32(Float32(fma(Float32(Float32(0.008333333333333333) / Float32(v * v)), Float32(-1.0), Float32(-0.16666666666666666)) / Float32(Float32(-v) * v)) - Float32(-1.0)) / v)))))
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

10. Applied rewrites 69.4%

    \[\leadsto cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v} \cdot \frac{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v \cdot v} \cdot -0.5, \frac{0.5}{v}\right)}{\color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{v \cdot v} - 1}{-v}}}\right) \]

11. Final simplification 69.4%

    \[\leadsto cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v} \cdot \frac{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v \cdot v} \cdot -0.5, \frac{0.5}{v}\right)}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{\left(-v\right) \cdot v} - -1}{v}}\right) \]
12. Add Preprocessing

Alternative 11: 70.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_i\_m \cdot \left(\frac{cosTheta\_O}{v} \cdot \frac{\frac{0.5}{v}}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{\left(-v\right) \cdot v} - -1}{v}}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_i_m
   (*
    (/ cosTheta_O v)
    (/
     (/ 0.5 v)
     (/
      (-
       (/
        (fma (/ 0.008333333333333333 (* v v)) -1.0 -0.16666666666666666)
        (* (- v) v))
       -1.0)
      v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_i_m * ((cosTheta_O / v) * ((0.5f / v) / (((fmaf((0.008333333333333333f / (v * v)), -1.0f, -0.16666666666666666f) / (-v * v)) - -1.0f) / v))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_i_m * Float32(Float32(cosTheta_O / v) * Float32(Float32(Float32(0.5) / v) / Float32(Float32(Float32(fma(Float32(Float32(0.008333333333333333) / Float32(v * v)), Float32(-1.0), Float32(-0.16666666666666666)) / Float32(Float32(-v) * v)) - Float32(-1.0)) / v)))))
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.1

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

10. Applied rewrites 69.4%

    \[\leadsto cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v} \cdot \frac{\frac{0.5}{v}}{\color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{v \cdot v} - 1}{-v}}}\right) \]

11. Final simplification 69.4%

    \[\leadsto cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v} \cdot \frac{\frac{0.5}{v}}{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{\left(-v\right) \cdot v} - -1}{v}}\right) \]
12. Add Preprocessing

Alternative 12: 70.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(\frac{\frac{1}{v}}{\frac{\frac{-0.3333333333333333 - \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - 2}{-v}} \cdot \frac{cosTheta\_O \cdot cosTheta\_i\_m}{v}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (/
    (/ 1.0 v)
    (/
     (-
      (/ (- -0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
      2.0)
     (- v)))
   (/ (* cosTheta_O cosTheta_i_m) v))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((1.0f / v) / ((((-0.3333333333333333f - (0.016666666666666666f / (v * v))) / (v * v)) - 2.0f) / -v)) * ((cosTheta_O * cosTheta_i_m) / v));
}

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (((1.0e0 / v) / (((((-0.3333333333333333e0) - (0.016666666666666666e0 / (v * v))) / (v * v)) - 2.0e0) / -v)) * ((costheta_o * costheta_i_m) / v))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(1.0) / v) / Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) - Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(2.0)) / Float32(-v))) * Float32(Float32(cosTheta_O * cosTheta_i_m) / v)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((single(1.0) / v) / ((((single(-0.3333333333333333) - (single(0.016666666666666666) / (v * v))) / (v * v)) - single(2.0)) / -v)) * ((cosTheta_O * cosTheta_i_m) / v));
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rec-exp N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-/.f32 98.0

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

7. Applied rewrites 98.0%

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

8. Taylor expanded in v around -inf

   \[\leadsto \frac{\frac{1}{v}}{-1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{v}}} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
9. Step-by-step derivation

10. Applied rewrites 69.4%

    \[\leadsto \frac{\frac{1}{v}}{\frac{\frac{\mathsf{fma}\left(\frac{0.016666666666666666}{v \cdot v}, -1, -0.3333333333333333\right)}{v \cdot v} - 2}{\color{blue}{-v}}} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
11. Step-by-step derivation

12. Applied rewrites 69.4%

    \[\leadsto \frac{\frac{1}{v}}{\frac{\frac{-0.3333333333333333 - \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - 2}{-v}} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
13. Add Preprocessing

Alternative 13: 63.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (costheta_i_m * ((costheta_o / v) * ((0.5e0 / v) / (((0.16666666666666666e0 / (v * v)) + 1.0e0) / v))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_i_m * ((cosTheta_O / v) * ((single(0.5) / v) / (((single(0.16666666666666666) / (v * v)) + single(1.0)) / v))));
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.1

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 62.9

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

10. Applied rewrites 62.9%

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

Alternative 14: 63.9% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * ((1.0e0 * ((costheta_i_m * costheta_o) / v)) / ((0.3333333333333333e0 / (v * v)) + 2.0e0))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((single(1.0) * ((cosTheta_i_m * cosTheta_O) / v)) / ((single(0.3333333333333333) / (v * v)) + single(2.0)));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in sinTheta_i around 0

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 62.9

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, sinTheta\_i, \frac{\left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot sinTheta\_O\right) \cdot 0.5}{v}\right)}{v}, sinTheta\_O, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

8. Applied rewrites 62.9%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1, sinTheta\_i, \frac{\left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot sinTheta\_O\right) \cdot 0.5}{v}\right)}{v}, sinTheta\_O, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]

9. Taylor expanded in sinTheta_i around 0

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

11. Applied rewrites 62.9%

    \[\leadsto \frac{\color{blue}{1} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{v \cdot v} + 2} \]
12. Add Preprocessing

Alternative 15: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (((costheta_o * 0.5e0) * costheta_i_m) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 56.8

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

5. Applied rewrites 56.8%

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

7. Applied rewrites 56.8%

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

9. Applied rewrites 56.8%

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

Alternative 16: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 * ((costheta_o * costheta_i_m) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 56.8

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

5. Applied rewrites 56.8%

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

Reproduce

herbie shell --seed 2024363 
(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)))