HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 11.7s

Alternatives: 16

Speedup: 1.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / v) * (cosTheta_i * ((powf(expf(sinTheta_O), (-sinTheta_i / v)) / (2.0f * v)) / sinhf((1.0f / 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 = (costheta_o / v) * (costheta_i * (((exp(sintheta_o) ** (-sintheta_i / v)) / (2.0e0 * v)) / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.0%

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

Alternative 2: 98.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (fma
   (/ cosTheta_i v)
   cosTheta_O
   (* (/ (* (- cosTheta_O) (* sinTheta_i cosTheta_i)) (* v v)) sinTheta_O))
  (* (* (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 fmaf((cosTheta_i / v), cosTheta_O, (((-cosTheta_O * (sinTheta_i * cosTheta_i)) / (v * v)) * sinTheta_O)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(fma(Float32(cosTheta_i / v), cosTheta_O, Float32(Float32(Float32(Float32(-cosTheta_O) * Float32(sinTheta_i * cosTheta_i)) / Float32(v * v)) * sinTheta_O)) / 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_O around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Applied rewrites 98.8%

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

Alternative 3: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * (-sinTheta_i * ((sinTheta_O / v) - (1.0f / sinTheta_i)))) * (cosTheta_i / ((v * (2.0f * v)) * sinhf((1.0f / 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 = (costheta_o * (-sintheta_i * ((sintheta_o / v) - (1.0e0 / sintheta_i)))) * (costheta_i / ((v * (2.0e0 * v)) * sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in sinTheta_i around -inf

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

10. Applied rewrites 98.4%

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

Alternative 4: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

Alternative 5: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / v) * (cosTheta_i * ((1.0f / (2.0f * v)) / sinhf((1.0f / 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 = (costheta_o / v) * (costheta_i * ((1.0e0 / (2.0e0 * v)) / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.9%

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

Alternative 6: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * 1.0f) * (cosTheta_i / ((v * (2.0f * v)) * sinhf((1.0f / 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 = (costheta_o * 1.0e0) * (costheta_i / ((v * (2.0e0 * v)) * sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.6%

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

Alternative 7: 70.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\left(v \cdot \left(2 \cdot v\right)\right) \cdot \frac{\frac{\frac{\mathsf{fma}\left(\frac{0.008333333333333333}{v \cdot v}, -1, -0.16666666666666666\right)}{v}}{v} - 1}{-v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_O (* (- sinTheta_i) (- (/ sinTheta_O v) (/ 1.0 sinTheta_i))))
  (/
   cosTheta_i
   (*
    (* v (* 2.0 v))
    (/
     (-
      (/
       (/ (fma (/ 0.008333333333333333 (* v v)) -1.0 -0.16666666666666666) v)
       v)
      1.0)
     (- v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * (-sinTheta_i * ((sinTheta_O / v) - (1.0f / sinTheta_i)))) * (cosTheta_i / ((v * (2.0f * v)) * ((((fmaf((0.008333333333333333f / (v * v)), -1.0f, -0.16666666666666666f) / v) / v) - 1.0f) / -v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(Float32(-sinTheta_i) * Float32(Float32(sinTheta_O / v) - Float32(Float32(1.0) / sinTheta_i)))) * Float32(cosTheta_i / Float32(Float32(v * Float32(Float32(2.0) * v)) * Float32(Float32(Float32(Float32(fma(Float32(Float32(0.008333333333333333) / Float32(v * v)), Float32(-1.0), Float32(-0.16666666666666666)) / v) / v) - Float32(1.0)) / Float32(-v)))))
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in sinTheta_i around -inf

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

10. Applied rewrites 98.4%

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

11. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. lower-/.f32 N/A

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

13. Applied rewrites 71.8%

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

Alternative 8: 70.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\left(-v\right) \cdot \left(\frac{\frac{\mathsf{fma}\left(\frac{0.016666666666666666}{v \cdot v}, -1, -0.3333333333333333\right)}{v}}{v} - 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_O (* (- sinTheta_i) (- (/ sinTheta_O v) (/ 1.0 sinTheta_i))))
  (/
   cosTheta_i
   (*
    (- v)
    (-
     (/
      (/ (fma (/ 0.016666666666666666 (* v v)) -1.0 -0.3333333333333333) v)
      v)
     2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * (-sinTheta_i * ((sinTheta_O / v) - (1.0f / sinTheta_i)))) * (cosTheta_i / (-v * (((fmaf((0.016666666666666666f / (v * v)), -1.0f, -0.3333333333333333f) / v) / v) - 2.0f)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(Float32(-sinTheta_i) * Float32(Float32(sinTheta_O / v) - Float32(Float32(1.0) / sinTheta_i)))) * Float32(cosTheta_i / Float32(Float32(-v) * Float32(Float32(Float32(fma(Float32(Float32(0.016666666666666666) / Float32(v * v)), Float32(-1.0), Float32(-0.3333333333333333)) / v) / v) - Float32(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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in sinTheta_i around -inf

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

10. Applied rewrites 98.4%

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

11. Taylor expanded in v around -inf

    \[\leadsto \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{-1 \cdot \left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)}} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\mathsf{neg}\left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)}} \]

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

       \[\leadsto \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)}} \]

    3. lower-*.f32 N/A

       \[\leadsto \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)}} \]

    4. lower-neg.f32 N/A

       \[\leadsto \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\left(-v\right)} \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)} \]

    5. lower--.f32 N/A

       \[\leadsto \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\left(-v\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)}} \]

13. Applied rewrites 71.8%

    \[\leadsto \left(cosTheta\_O \cdot \left(\left(-sinTheta\_i\right) \cdot \left(\frac{sinTheta\_O}{v} - \frac{1}{sinTheta\_i}\right)\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\left(-v\right) \cdot \left(\frac{\frac{\mathsf{fma}\left(\frac{0.016666666666666666}{v \cdot v}, -1, -0.3333333333333333\right)}{v}}{v} - 2\right)}} \]
14. Add Preprocessing

Alternative 9: 70.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O}{v} \cdot \left(cosTheta\_i \cdot \frac{\frac{1}{v}}{\frac{\frac{\frac{\mathsf{fma}\left(\frac{0.016666666666666666}{v \cdot v}, -1, -0.3333333333333333\right)}{v}}{v} - 2}{-v}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_O v)
  (*
   cosTheta_i
   (/
    (/ 1.0 v)
    (/
     (-
      (/
       (/ (fma (/ 0.016666666666666666 (* v v)) -1.0 -0.3333333333333333) v)
       v)
      2.0)
     (- v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / v) * (cosTheta_i * ((1.0f / v) / ((((fmaf((0.016666666666666666f / (v * v)), -1.0f, -0.3333333333333333f) / v) / v) - 2.0f) / -v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O / v) * Float32(cosTheta_i * Float32(Float32(Float32(1.0) / v) / Float32(Float32(Float32(Float32(fma(Float32(Float32(0.016666666666666666) / Float32(v * v)), Float32(-1.0), Float32(-0.3333333333333333)) / v) / v) - Float32(2.0)) / Float32(-v)))))
end

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rec-exp N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-/.f32 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in v around -inf

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

10. Applied rewrites 72.1%

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

Alternative 10: 70.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O, \frac{sinTheta\_i}{v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\left(-v\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{0.016666666666666666}{v \cdot v}, -1, -0.3333333333333333\right)}{v \cdot v} - 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_O (fma (- sinTheta_O) (/ sinTheta_i v) 1.0))
  (/
   cosTheta_i
   (*
    (- v)
    (-
     (/
      (fma (/ 0.016666666666666666 (* v v)) -1.0 -0.3333333333333333)
      (* v v))
     2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * fmaf(-sinTheta_O, (sinTheta_i / v), 1.0f)) * (cosTheta_i / (-v * ((fmaf((0.016666666666666666f / (v * v)), -1.0f, -0.3333333333333333f) / (v * v)) - 2.0f)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * fma(Float32(-sinTheta_O), Float32(sinTheta_i / v), Float32(1.0))) * Float32(cosTheta_i / Float32(Float32(-v) * Float32(Float32(fma(Float32(Float32(0.016666666666666666) / Float32(v * v)), Float32(-1.0), Float32(-0.3333333333333333)) / Float32(v * v)) - Float32(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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in v around -inf

   \[\leadsto \left(cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O, \frac{sinTheta\_i}{v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{-1 \cdot \left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)}} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O, \frac{sinTheta\_i}{v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\mathsf{neg}\left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)}} \]

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

      \[\leadsto \left(cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O, \frac{sinTheta\_i}{v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \left(cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O, \frac{sinTheta\_i}{v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)}} \]

   4. lower-neg.f32 N/A

      \[\leadsto \left(cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O, \frac{sinTheta\_i}{v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\color{blue}{\left(-v\right)} \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)} \]

   5. lower--.f32 N/A

      \[\leadsto \left(cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O, \frac{sinTheta\_i}{v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\left(-v\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)}} \]

10. Applied rewrites 72.1%

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

Alternative 11: 64.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{cosTheta\_i}{v}, cosTheta\_O, \frac{\left(-cosTheta\_O\right) \cdot \left(sinTheta\_i \cdot cosTheta\_i\right)}{v \cdot v} \cdot sinTheta\_O\right)}{\frac{0.3333333333333333}{v \cdot v} + 2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (fma
   (/ cosTheta_i v)
   cosTheta_O
   (* (/ (* (- cosTheta_O) (* sinTheta_i cosTheta_i)) (* v v)) sinTheta_O))
  (+ (/ 0.3333333333333333 (* v v)) 2.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return fmaf((cosTheta_i / v), cosTheta_O, (((-cosTheta_O * (sinTheta_i * cosTheta_i)) / (v * v)) * sinTheta_O)) / ((0.3333333333333333f / (v * v)) + 2.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(fma(Float32(cosTheta_i / v), cosTheta_O, Float32(Float32(Float32(Float32(-cosTheta_O) * Float32(sinTheta_i * cosTheta_i)) / Float32(v * v)) * sinTheta_O)) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(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_O around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Applied rewrites 98.8%

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

9. Taylor expanded in v around inf

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. lower-/.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 66.0

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

11. Applied rewrites 66.0%

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

Alternative 12: 64.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O}{v} \cdot \left(cosTheta\_i \cdot \frac{\frac{1}{v}}{\frac{\frac{0.3333333333333333}{v \cdot v} + 2}{v}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_O v)
  (* cosTheta_i (/ (/ 1.0 v) (/ (+ (/ 0.3333333333333333 (* v v)) 2.0) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / v) * (cosTheta_i * ((1.0f / v) / (((0.3333333333333333f / (v * 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 = (costheta_o / v) * (costheta_i * ((1.0e0 / v) / (((0.3333333333333333e0 / (v * v)) + 2.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O / v) * Float32(cosTheta_i * Float32(Float32(Float32(1.0) / v) / Float32(Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / v) * (cosTheta_i * ((single(1.0) / v) / (((single(0.3333333333333333) / (v * 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. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. rec-exp N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-/.f32 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in v around inf

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

10. Applied rewrites 66.0%

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

Alternative 13: 64.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O, \frac{sinTheta\_i}{v}, 1\right)\right) \cdot \frac{cosTheta\_i}{\left(\frac{0.3333333333333333}{v \cdot v} + 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_O (fma (- sinTheta_O) (/ sinTheta_i v) 1.0))
  (/ cosTheta_i (* (+ (/ 0.3333333333333333 (* v v)) 2.0) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * fmaf(-sinTheta_O, (sinTheta_i / v), 1.0f)) * (cosTheta_i / (((0.3333333333333333f / (v * v)) + 2.0f) * v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * fma(Float32(-sinTheta_O), Float32(sinTheta_i / v), Float32(1.0))) * Float32(cosTheta_i / Float32(Float32(Float32(Float32(0.3333333333333333) / Float32(v * 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 66.0

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

10. Applied rewrites 66.0%

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

Alternative 14: 58.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* (* cosTheta_O cosTheta_i) 0.5) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * cosTheta_i) * 0.5f) / 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 = ((costheta_o * costheta_i) * 0.5e0) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_O * cosTheta_i) * Float32(0.5)) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O * cosTheta_i) * 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

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.3

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

5. Applied rewrites 60.3%

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

7. Applied rewrites 60.3%

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

9. Applied rewrites 60.4%

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

Alternative 15: 58.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* (/ cosTheta_i v) cosTheta_O) 0.5))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i / v) * cosTheta_O) * 0.5f;
}

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 = ((costheta_i / v) * costheta_o) * 0.5e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i / v) * cosTheta_O) * Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i / v) * cosTheta_O) * single(0.5);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.3

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

5. Applied rewrites 60.3%

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

7. Applied rewrites 60.3%

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

Alternative 16: 58.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* (/ cosTheta_O v) cosTheta_i)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * ((cosTheta_O / v) * cosTheta_i);
}

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 = 0.5e0 * ((costheta_o / v) * costheta_i)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(Float32(cosTheta_O / v) * cosTheta_i))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((cosTheta_O / v) * cosTheta_i);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.3

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

5. Applied rewrites 60.3%

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

7. Applied rewrites 60.3%

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

Reproduce

herbie shell --seed 2024352 
(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)))