HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.5%

Time: 7.7s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := \sinh \left(\frac{1}{v}\right) \cdot 2\\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \mathsf{fma}\left(\frac{cosTheta\_i\_m}{v \cdot v}, \frac{cosTheta\_O\_m}{t\_0}, -\frac{\frac{\left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot cosTheta\_i\_m\right) \cdot cosTheta\_O\_m}{{v}^{3}}}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* (sinh (/ 1.0 v)) 2.0)))
   (*
    cosTheta_O_s
    (*
     cosTheta_i_s
     (fma
      (/ cosTheta_i_m (* v v))
      (/ cosTheta_O_m t_0)
      (-
       (/
        (/
         (* (* (* sinTheta_O sinTheta_i) cosTheta_i_m) cosTheta_O_m)
         (pow v 3.0))
        t_0)))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = sinhf((1.0f / v)) * 2.0f;
	return cosTheta_O_s * (cosTheta_i_s * fmaf((cosTheta_i_m / (v * v)), (cosTheta_O_m / t_0), -(((((sinTheta_O * sinTheta_i) * cosTheta_i_m) * cosTheta_O_m) / powf(v, 3.0f)) / t_0)));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * fma(Float32(cosTheta_i_m / Float32(v * v)), Float32(cosTheta_O_m / t_0), Float32(-Float32(Float32(Float32(Float32(Float32(sinTheta_O * sinTheta_i) * cosTheta_i_m) * cosTheta_O_m) / (v ^ Float32(3.0))) / t_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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (exp(((sintheta_o * sintheta_i) / -v)) * (((costheta_o_m * costheta_i_m) / v) / (sinh((1.0e0 / v)) * (2.0e0 * v)))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (exp(((sinTheta_O * sinTheta_i) / -v)) * (((cosTheta_O_m * cosTheta_i_m) / v) / (sinh((single(1.0) / v)) * (single(2.0) * 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-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-sinh.f32 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\frac{\frac{cosTheta\_O\_m \cdot v - \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot cosTheta\_O\_m}{2 \cdot \sinh \left(\frac{1}{v}\right)}}{{v}^{3}} \cdot cosTheta\_i\_m\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/
     (/
      (- (* cosTheta_O_m v) (* (* sinTheta_O sinTheta_i) cosTheta_O_m))
      (* 2.0 (sinh (/ 1.0 v))))
     (pow v 3.0))
    cosTheta_i_m))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (((((cosTheta_O_m * v) - ((sinTheta_O * sinTheta_i) * cosTheta_O_m)) / (2.0f * sinhf((1.0f / v)))) / powf(v, 3.0f)) * cosTheta_i_m));
}

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((((costheta_o_m * v) - ((sintheta_o * sintheta_i) * costheta_o_m)) / (2.0e0 * sinh((1.0e0 / v)))) / (v ** 3.0e0)) * costheta_i_m))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(Float32(cosTheta_O_m * v) - Float32(Float32(sinTheta_O * sinTheta_i) * cosTheta_O_m)) / Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v)))) / (v ^ Float32(3.0))) * cosTheta_i_m)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((((cosTheta_O_m * v) - ((sinTheta_O * sinTheta_i) * cosTheta_O_m)) / (single(2.0) * sinh((single(1.0) / v)))) / (v ^ single(3.0))) * cosTheta_i_m));
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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in cosTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in v around 0

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

    1. lower-/.f32 N/A

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

11. Applied rewrites 98.5%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot v - \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot cosTheta\_O}{2 \cdot \sinh \left(\frac{1}{v}\right)}}{{v}^{3}} \cdot cosTheta\_i \]
12. Add Preprocessing

Alternative 4: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(fma(Float32(-1.0), Float32(Float32(sinTheta_O * sinTheta_i) / v), Float32(1.0)) * Float32(Float32(cosTheta_i_m * cosTheta_O_m) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * 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. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 98.5

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

5. Applied rewrites 98.5%

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

Alternative 5: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(fma(Float32(sinTheta_i / Float32(-v)), sinTheta_O, Float32(1.0)) * Float32(Float32(Float32(cosTheta_i_m / v) * cosTheta_O_m) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(Float32(2.0) * 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. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-frac-neg2 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-neg.f32 98.5

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

7. Applied rewrites 98.5%

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

9. Applied rewrites 98.5%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-/.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. lift-/.f32 N/A

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

    8. lift-sinh.f32 N/A

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

    9. associate-/l* N/A

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

    10. lower-*.f32 N/A

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

11. Applied rewrites 98.5%

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

Alternative 6: 98.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_i_m / v) / v) * ((costheta_o_m / sinh((1.0e0 / v))) / 2.0e0)))
end function

Julia

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

MATLAB

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. *-commutative N/A

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

   5. sinh-undef N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift--.f32 98.5

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

   12. lift-neg.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. pow2 N/A

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

   3. frac-times N/A

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

   4. lower-/.f32 N/A

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

   5. frac-times N/A

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

   6. pow2 N/A

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

   7. associate-/l* N/A

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

   8. pow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lift-/.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. rec-exp N/A

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

   14. sinh-undef-rev N/A

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

   15. lower-*.f32 N/A

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

   16. lift-sinh.f32 N/A

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

   17. lift-/.f32 98.3

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

10. Applied rewrites 98.3%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-/.f32 N/A

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

    5. lift-sinh.f32 N/A

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

    6. *-commutative N/A

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

    7. lift-/.f32 N/A

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

    8. lift-/.f32 N/A

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

    9. associate-/r* N/A

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

    10. *-commutative N/A

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

    11. associate-*r/ N/A

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

    12. associate-/r* N/A

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

    13. lift-/.f32 N/A

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

    14. lift-/.f32 N/A

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

    15. lower-*.f32 N/A

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

    16. associate-/r* N/A

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

    17. lower-/.f32 N/A

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

    18. lower-/.f32 N/A

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

    19. lift-sinh.f32 N/A

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

    20. lift-/.f32 98.3

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

12. Applied rewrites 98.3%

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

Alternative 7: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_o_m * costheta_i_m) / v) / ((sinh((1.0e0 / v)) * 2.0e0) * v)))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((cosTheta_O_m * cosTheta_i_m) / v) / ((sinh((single(1.0) / v)) * single(2.0)) * 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. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

Alternative 8: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_i_m / v) * costheta_o_m) / ((sinh((1.0e0 / v)) * 2.0e0) * v)))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((cosTheta_i_m / v) * cosTheta_O_m) / ((sinh((single(1.0) / v)) * single(2.0)) * 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. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-frac-neg2 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-neg.f32 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. lift-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 98.3

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

10. Applied rewrites 98.3%

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

Alternative 9: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_o_m * costheta_i_m) / ((2.0e0 * sinh((1.0e0 / v))) * (v * v))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * cosTheta_i_m) / ((single(2.0) * sinh((single(1.0) / v))) * (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. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sinh.f32 N/A

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

   4. *-commutative N/A

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

   5. sinh-undef N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift--.f32 98.5

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

   12. lift-neg.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. pow2 N/A

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

   3. frac-times N/A

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

   4. lower-/.f32 N/A

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

   5. frac-times N/A

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

   6. pow2 N/A

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

   7. associate-/l* N/A

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

   8. pow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lift-/.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. rec-exp N/A

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

   14. sinh-undef-rev N/A

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

   15. lower-*.f32 N/A

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

   16. lift-sinh.f32 N/A

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

   17. lift-/.f32 98.3

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

10. Applied rewrites 98.3%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-/.f32 N/A

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

    5. lift-sinh.f32 N/A

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

    6. associate-/l* N/A

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

    7. lift-/.f32 N/A

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

    8. lift-/.f32 N/A

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

    9. associate-/r* N/A

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

    10. pow2 N/A

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

    11. sinh-undef-rev N/A

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

    12. rec-exp N/A

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

    13. associate-/r* N/A

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

    14. associate-/l* N/A

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

    15. lower-/.f32 N/A

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

    16. lower-*.f32 N/A

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

    17. *-commutative N/A

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

12. Applied rewrites 98.3%

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

Alternative 10: 58.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 58.4

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

5. Applied rewrites 58.4%

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

Alternative 11: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_o_m * costheta_i_m) * 0.5e0) / v))
end function

Julia

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

MATLAB

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

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.4

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

5. Applied rewrites 58.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-*.f32 58.4

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

7. Applied rewrites 58.4%

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

Alternative 12: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (0.5e0 * ((costheta_o_m * costheta_i_m) / v)))
end function

Julia

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

MATLAB

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

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.4

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

5. Applied rewrites 58.4%

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

Alternative 13: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i\_m =     private
cosTheta_i\_s =     private
cosTheta_O\_m =     private
cosTheta_O\_s =     private
NOTE: cosTheta_i_m, cosTheta_O_m, 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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (0.5e0 * (costheta_o_m * (costheta_i_m / v))))
end function

Julia

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

MATLAB

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

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 58.4

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

5. Applied rewrites 58.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-/.f32 58.4

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

7. Applied rewrites 58.4%

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

Reproduce

herbie shell --seed 2025091 
(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)))