HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 6.3s

Alternatives: 15

Speedup: 1.7×

Specification

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

Math

\[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\frac{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\frac{1}{v}\right)}} \cdot 0.5}{0.500023603439331 \cdot v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (/
    1.0
    (pow
     (exp (- (fma sinTheta_O sinTheta_i 1.0) (* cosTheta_O cosTheta_i)))
     (/ 1.0 v)))
   0.5)
  (* 0.500023603439331 v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((1.0f / powf(expf((fmaf(sinTheta_O, sinTheta_i, 1.0f) - (cosTheta_O * cosTheta_i))), (1.0f / v))) * 0.5f) / (0.500023603439331f * v);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(1.0) / (exp(Float32(fma(sinTheta_O, sinTheta_i, Float32(1.0)) - Float32(cosTheta_O * cosTheta_i))) ^ Float32(Float32(1.0) / v))) * Float32(0.5)) / Float32(Float32(0.500023603439331) * v))
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   4. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   5. add-flip N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) - \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   6. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   8. rem-exp-log N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \frac{1}{\color{blue}{2 \cdot v}} \]

   11. associate-/r* N/A

       \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   12. frac-times N/A

       \[\leadsto \color{blue}{\frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}} \cdot \frac{1}{2}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)} \cdot v}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \cdot 0.5}{e^{-0.6931} \cdot v}} \]
4. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   3. frac-2neg N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)}{\mathsf{neg}\left(v\right)}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   4. mult-flip N/A

      \[\leadsto \frac{e^{\color{blue}{\left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(v\right)}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \frac{e^{\left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   6. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   7. lower-pow.f32 N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   8. lower-exp.f32 N/A

      \[\leadsto \frac{{\color{blue}{\left(e^{\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   9. lift--.f32 N/A

      \[\leadsto \frac{{\left(e^{\mathsf{neg}\left(\color{blue}{\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   10. sub-negate-rev N/A

       \[\leadsto \frac{{\left(e^{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - cosTheta\_O \cdot cosTheta\_i}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   11. lower--.f32 N/A

       \[\leadsto \frac{{\left(e^{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - cosTheta\_O \cdot cosTheta\_i}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   12. lift-fma.f32 N/A

       \[\leadsto \frac{{\left(e^{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right)} - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   13. *-commutative N/A

       \[\leadsto \frac{{\left(e^{\left(\color{blue}{sinTheta\_i \cdot sinTheta\_O} + 1\right) - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   14. lower-fma.f32 N/A

       \[\leadsto \frac{{\left(e^{\color{blue}{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)} - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - \color{blue}{cosTheta\_O \cdot cosTheta\_i}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   16. *-commutative N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - \color{blue}{cosTheta\_i \cdot cosTheta\_O}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - \color{blue}{cosTheta\_i \cdot cosTheta\_O}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   18. distribute-neg-frac N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   19. metadata-eval N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{\color{blue}{-1}}{v}\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   20. lower-/.f32 99.7%

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\color{blue}{\left(\frac{-1}{v}\right)}} \cdot 0.5}{e^{-0.6931} \cdot v} \]

5. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{-1}{v}\right)}} \cdot 0.5}{e^{-0.6931} \cdot v} \]
6. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{-1}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\color{blue}{\left(\frac{-1}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   3. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   4. distribute-neg-frac N/A

      \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   5. pow-neg N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{1}{v}\right)}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   6. lower-unsound-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{1}{v}\right)}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   7. lower-unsound-pow.f32 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{1}{v}\right)}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   8. lift-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{{\left(e^{\color{blue}{\left(sinTheta\_i \cdot sinTheta\_O + 1\right)} - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{1}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{1}{{\left(e^{\left(\color{blue}{sinTheta\_O \cdot sinTheta\_i} + 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{1}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   10. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{{\left(e^{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)} - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{1}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - \color{blue}{cosTheta\_i \cdot cosTheta\_O}}\right)}^{\left(\frac{1}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   12. *-commutative N/A

       \[\leadsto \frac{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - \color{blue}{cosTheta\_O \cdot cosTheta\_i}}\right)}^{\left(\frac{1}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - \color{blue}{cosTheta\_O \cdot cosTheta\_i}}\right)}^{\left(\frac{1}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   14. lower-/.f32 99.7%

       \[\leadsto \frac{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - cosTheta\_O \cdot cosTheta\_i}\right)}^{\color{blue}{\left(\frac{1}{v}\right)}}} \cdot 0.5}{e^{-0.6931} \cdot v} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\frac{1}{v}\right)}}} \cdot 0.5}{e^{-0.6931} \cdot v} \]

8. Evaluated real constant 99.7%

   \[\leadsto \frac{\frac{1}{{\left(e^{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\frac{1}{v}\right)}} \cdot 0.5}{\color{blue}{0.500023603439331} \cdot v} \]
9. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{-1}{v}\right)} \cdot 0.5}{0.500023603439331 \cdot v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (pow
    (exp (- (fma sinTheta_i sinTheta_O 1.0) (* cosTheta_i cosTheta_O)))
    (/ -1.0 v))
   0.5)
  (* 0.500023603439331 v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (powf(expf((fmaf(sinTheta_i, sinTheta_O, 1.0f) - (cosTheta_i * cosTheta_O))), (-1.0f / v)) * 0.5f) / (0.500023603439331f * v);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32((exp(Float32(fma(sinTheta_i, sinTheta_O, Float32(1.0)) - Float32(cosTheta_i * cosTheta_O))) ^ Float32(Float32(-1.0) / v)) * Float32(0.5)) / Float32(Float32(0.500023603439331) * v))
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   4. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   5. add-flip N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) - \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   6. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   8. rem-exp-log N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \frac{1}{\color{blue}{2 \cdot v}} \]

   11. associate-/r* N/A

       \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   12. frac-times N/A

       \[\leadsto \color{blue}{\frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}} \cdot \frac{1}{2}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)} \cdot v}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \cdot 0.5}{e^{-0.6931} \cdot v}} \]
4. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   3. frac-2neg N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)}{\mathsf{neg}\left(v\right)}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   4. mult-flip N/A

      \[\leadsto \frac{e^{\color{blue}{\left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(v\right)}}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \frac{e^{\left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   6. exp-prod N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   7. lower-pow.f32 N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   8. lower-exp.f32 N/A

      \[\leadsto \frac{{\color{blue}{\left(e^{\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)\right)}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   9. lift--.f32 N/A

      \[\leadsto \frac{{\left(e^{\mathsf{neg}\left(\color{blue}{\left(cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   10. sub-negate-rev N/A

       \[\leadsto \frac{{\left(e^{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - cosTheta\_O \cdot cosTheta\_i}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   11. lower--.f32 N/A

       \[\leadsto \frac{{\left(e^{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) - cosTheta\_O \cdot cosTheta\_i}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   12. lift-fma.f32 N/A

       \[\leadsto \frac{{\left(e^{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right)} - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   13. *-commutative N/A

       \[\leadsto \frac{{\left(e^{\left(\color{blue}{sinTheta\_i \cdot sinTheta\_O} + 1\right) - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   14. lower-fma.f32 N/A

       \[\leadsto \frac{{\left(e^{\color{blue}{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)} - cosTheta\_O \cdot cosTheta\_i}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - \color{blue}{cosTheta\_O \cdot cosTheta\_i}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   16. *-commutative N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - \color{blue}{cosTheta\_i \cdot cosTheta\_O}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - \color{blue}{cosTheta\_i \cdot cosTheta\_O}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   18. distribute-neg-frac N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{v}\right)}} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   19. metadata-eval N/A

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{\color{blue}{-1}}{v}\right)} \cdot \frac{1}{2}}{e^{\frac{-6931}{10000}} \cdot v} \]

   20. lower-/.f32 99.7%

       \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\color{blue}{\left(\frac{-1}{v}\right)}} \cdot 0.5}{e^{-0.6931} \cdot v} \]

5. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{-1}{v}\right)}} \cdot 0.5}{e^{-0.6931} \cdot v} \]

6. Evaluated real constant 99.7%

   \[\leadsto \frac{{\left(e^{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right) - cosTheta\_i \cdot cosTheta\_O}\right)}^{\left(\frac{-1}{v}\right)} \cdot 0.5}{\color{blue}{0.500023603439331} \cdot v} \]
7. Add Preprocessing

Alternative 3: 99.7% accurate, 1.5× speedup

Math

\[\frac{1}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ (+ v v) (exp (- (/ (- (* cosTheta_O cosTheta_i) 1.0) v) -0.6931)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / ((v + v) / expf(((((cosTheta_O * cosTheta_i) - 1.0f) / v) - -0.6931f)));
}

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 = 1.0e0 / ((v + v) / exp(((((costheta_o * costheta_i) - 1.0e0) / v) - (-0.6931e0))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(Float32(v + v) / exp(Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(1.0)) / v) - Float32(-0.6931)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / ((v + v) / exp(((((cosTheta_O * cosTheta_i) - single(1.0)) / v) - single(-0.6931))));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. add-flip N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \left(\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)\right)}} \]

   4. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{e^{\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{e^{\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)}}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931}}{v + v}} \]

4. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{1}}{v} - -0.6931}}{v + v} \]
5. Step-by-step derivation

6. Applied rewrites 99.7%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{1}}{v} - -0.6931}}{v + v} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}{v + v}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   3. lower-unsound-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   4. lower-unsound-/.f32 99.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}}}} \]

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}}}} \]
9. Add Preprocessing

Alternative 4: 99.7% accurate, 1.7× speedup

Math

\[\frac{\frac{0.5}{v}}{e^{-0.6931 - \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (/ 0.5 v) (exp (- -0.6931 (/ (- (* cosTheta_i cosTheta_O) 1.0) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) / expf((-0.6931f - (((cosTheta_i * cosTheta_O) - 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 = (0.5e0 / v) / exp(((-0.6931e0) - (((costheta_i * costheta_o) - 1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) / exp(Float32(Float32(-0.6931) - Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(1.0)) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) / exp((single(-0.6931) - (((cosTheta_i * cosTheta_O) - single(1.0)) / v)));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. add-flip N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \left(\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)\right)}} \]

   4. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{e^{\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{e^{\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)}}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931}}{v + v}} \]

4. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{1}}{v} - -0.6931}}{v + v} \]
5. Step-by-step derivation

6. Applied rewrites 99.7%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{1}}{v} - -0.6931}}{v + v} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}{v + v}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   3. lower-unsound-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   4. lower-unsound-/.f32 99.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}}}} \]

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}}}} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{v + v}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   3. mult-flip N/A

      \[\leadsto \frac{1}{\color{blue}{\left(v + v\right) \cdot \frac{1}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{v + v}}{\frac{1}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   5. lift-+.f32 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{v + v}}}{\frac{1}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}} \]

   6. count-2 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot v}}}{\frac{1}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{\frac{1}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2}}}{v}}{\frac{1}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{\frac{1}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}} \]

   10. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{v}}{\frac{1}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   11. lift-exp.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\frac{1}{\color{blue}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}}} \]

   12. rec-exp N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{e^{\mathsf{neg}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}\right)\right)}}} \]

   13. lift--.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{e^{\mathsf{neg}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}\right)}\right)}} \]

10. Applied rewrites 99.6%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{e^{-0.6931 - \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}}}} \]
11. Add Preprocessing

Alternative 5: 99.6% accurate, 1.7× speedup

Math

\[\frac{0.5}{v} \cdot e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (- (/ (- (* cosTheta_O cosTheta_i) 1.0) v) -0.6931))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf(((((cosTheta_O * cosTheta_i) - 1.0f) / v) - -0.6931f));
}

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 / v) * exp(((((costheta_o * costheta_i) - 1.0e0) / v) - (-0.6931e0)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(1.0)) / v) - Float32(-0.6931))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp(((((cosTheta_O * cosTheta_i) - single(1.0)) / v) - single(-0.6931)));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. add-flip N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \left(\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)\right)}} \]

   4. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{e^{\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{e^{\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)}}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931}}{v + v}} \]

4. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{1}}{v} - -0.6931}}{v + v} \]
5. Step-by-step derivation

6. Applied rewrites 99.7%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{1}}{v} - -0.6931}}{v + v} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}}{v + v}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}} \cdot \frac{1}{v + v}} \]

   3. lift-+.f32 N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}} \cdot \frac{1}{\color{blue}{v + v}} \]

   4. count-2 N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}} \cdot \frac{1}{\color{blue}{2 \cdot v}} \]

   5. associate-/r* N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   6. metadata-eval N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}} \cdot \frac{\color{blue}{\frac{1}{2}}}{v} \]

   7. lift-/.f32 N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v} \cdot e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - \frac{-6931}{10000}}} \]

   9. lower-*.f32 99.6%

      \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}} \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}} \]
9. Add Preprocessing

Alternative 6: 99.6% accurate, 1.7× speedup

Math

\[\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}}{v + v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (exp (- (/ (- (* cosTheta_O cosTheta_i) 1.0) v) -0.6931)) (+ v v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((cosTheta_O * cosTheta_i) - 1.0f) / v) - -0.6931f)) / (v + 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(((((costheta_o * costheta_i) - 1.0e0) / v) - (-0.6931e0))) / (v + v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(1.0)) / v) - Float32(-0.6931))) / Float32(v + v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((cosTheta_O * cosTheta_i) - single(1.0)) / v) - single(-0.6931))) / (v + v);
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. add-flip N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \left(\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)\right)}} \]

   4. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{e^{\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{e^{\mathsf{neg}\left(\log \left(\frac{1}{2 \cdot v}\right)\right)}}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931}}{v + v}} \]

4. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{1}}{v} - -0.6931}}{v + v} \]
5. Step-by-step derivation

6. Applied rewrites 99.7%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{1}}{v} - -0.6931}}{v + v} \]
7. Add Preprocessing

Alternative 7: 17.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;\frac{0.5}{e^{\left(-\mathsf{min}\left(cosTheta\_i, cosTheta\_O\right)\right) \cdot \frac{\mathsf{max}\left(cosTheta\_i, cosTheta\_O\right)}{v}} \cdot v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{v}}{e^{\frac{1}{v} \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 2.0000000063421537e-30)
   (/
    0.5
    (*
     (exp
      (* (- (fmin cosTheta_i cosTheta_O)) (/ (fmax cosTheta_i cosTheta_O) v)))
     v))
   (/ (/ 0.5 v) (exp (* (/ 1.0 v) (* sinTheta_O sinTheta_i))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 2.0000000063421537e-30f) {
		tmp = 0.5f / (expf((-fminf(cosTheta_i, cosTheta_O) * (fmaxf(cosTheta_i, cosTheta_O) / v))) * v);
	} else {
		tmp = (0.5f / v) / expf(((1.0f / v) * (sinTheta_O * sinTheta_i)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 2.0000000063421537e-30) then
        tmp = 0.5e0 / (exp((-fmin(costheta_i, costheta_o) * (fmax(costheta_i, costheta_o) / v))) * v)
    else
        tmp = (0.5e0 / v) / exp(((1.0e0 / v) * (sintheta_o * sintheta_i)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(2.0000000063421537e-30))
		tmp = Float32(Float32(0.5) / Float32(exp(Float32(Float32(-fmin(cosTheta_i, cosTheta_O)) * Float32(fmax(cosTheta_i, cosTheta_O) / v))) * v));
	else
		tmp = Float32(Float32(Float32(0.5) / v) / exp(Float32(Float32(Float32(1.0) / v) * Float32(sinTheta_O * sinTheta_i))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(2.0000000063421537e-30))
		tmp = single(0.5) / (exp((-min(cosTheta_i, cosTheta_O) * (max(cosTheta_i, cosTheta_O) / v))) * v);
	else
		tmp = (single(0.5) / v) / exp(((single(1.0) / v) * (sinTheta_O * sinTheta_i)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 2.00000001e-30

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. +-commutative N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

      4. add-flip N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) - \left(\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)\right)}} \]

      5. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{1}{2 \cdot v}\right)}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      6. lift-log.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      7. rem-exp-log N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      9. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2}}}{v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      14. lower-exp.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{e^{-\left(\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931\right)}}} \]

   4. Taylor expanded in cosTheta_i around inf

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{-1 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}}}} \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 12.0%

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

   6. Applied rewrites 12.0%

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{-1 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}}}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 12.1%

         \[\leadsto \frac{0.5}{\color{blue}{e^{-1 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \cdot v}} \]

   8. Applied rewrites 12.1%

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

   if 2.00000001e-30 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. +-commutative N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

      4. add-flip N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) - \left(\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)\right)}} \]

      5. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{1}{2 \cdot v}\right)}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      6. lift-log.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      7. rem-exp-log N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      9. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2}}}{v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      14. lower-exp.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{e^{-\left(\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931\right)}}} \]

   4. Taylor expanded in sinTheta_i around inf

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
   5. Step-by-step derivation

      1. lower-/.f32 N/A

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

      2. lower-*.f32 11.3%

         \[\leadsto \frac{\frac{0.5}{v}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   6. Applied rewrites 11.3%

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 11.3%

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

   8. Applied rewrites 11.3%

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\frac{1}{v} \cdot \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 17.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;\frac{0.5}{e^{\left(-\mathsf{min}\left(cosTheta\_i, cosTheta\_O\right)\right) \cdot \frac{\mathsf{max}\left(cosTheta\_i, cosTheta\_O\right)}{v}} \cdot v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{v}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 2.0000000063421537e-30)
   (/
    0.5
    (*
     (exp
      (* (- (fmin cosTheta_i cosTheta_O)) (/ (fmax cosTheta_i cosTheta_O) v)))
     v))
   (/ (/ 0.5 v) (exp (/ (* sinTheta_O sinTheta_i) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 2.0000000063421537e-30f) {
		tmp = 0.5f / (expf((-fminf(cosTheta_i, cosTheta_O) * (fmaxf(cosTheta_i, cosTheta_O) / v))) * v);
	} else {
		tmp = (0.5f / v) / expf(((sinTheta_O * sinTheta_i) / v));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 2.0000000063421537e-30) then
        tmp = 0.5e0 / (exp((-fmin(costheta_i, costheta_o) * (fmax(costheta_i, costheta_o) / v))) * v)
    else
        tmp = (0.5e0 / v) / exp(((sintheta_o * sintheta_i) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(2.0000000063421537e-30))
		tmp = Float32(Float32(0.5) / Float32(exp(Float32(Float32(-fmin(cosTheta_i, cosTheta_O)) * Float32(fmax(cosTheta_i, cosTheta_O) / v))) * v));
	else
		tmp = Float32(Float32(Float32(0.5) / v) / exp(Float32(Float32(sinTheta_O * sinTheta_i) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(2.0000000063421537e-30))
		tmp = single(0.5) / (exp((-min(cosTheta_i, cosTheta_O) * (max(cosTheta_i, cosTheta_O) / v))) * v);
	else
		tmp = (single(0.5) / v) / exp(((sinTheta_O * sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 2.00000001e-30

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. +-commutative N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

      4. add-flip N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) - \left(\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)\right)}} \]

      5. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{1}{2 \cdot v}\right)}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      6. lift-log.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      7. rem-exp-log N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      9. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2}}}{v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      14. lower-exp.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{e^{-\left(\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931\right)}}} \]

   4. Taylor expanded in cosTheta_i around inf

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{-1 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}}}} \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 12.0%

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

   6. Applied rewrites 12.0%

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{-1 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}}}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 12.1%

         \[\leadsto \frac{0.5}{\color{blue}{e^{-1 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \cdot v}} \]

   8. Applied rewrites 12.1%

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

   if 2.00000001e-30 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. +-commutative N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

      4. add-flip N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) - \left(\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)\right)}} \]

      5. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{1}{2 \cdot v}\right)}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      6. lift-log.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      7. rem-exp-log N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      9. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2}}}{v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      14. lower-exp.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{e^{-\left(\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931\right)}}} \]

   4. Taylor expanded in sinTheta_i around inf

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
   5. Step-by-step derivation

      1. lower-/.f32 N/A

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

      2. lower-*.f32 11.3%

         \[\leadsto \frac{\frac{0.5}{v}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   6. Applied rewrites 11.3%

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 17.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{\mathsf{max}\left(cosTheta\_i, cosTheta\_O\right)}{v} \cdot \mathsf{min}\left(cosTheta\_i, cosTheta\_O\right)}}{v + v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{v}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 2.0000000063421537e-30)
   (/
    (exp (* (/ (fmax cosTheta_i cosTheta_O) v) (fmin cosTheta_i cosTheta_O)))
    (+ v v))
   (/ (/ 0.5 v) (exp (/ (* sinTheta_O sinTheta_i) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 2.0000000063421537e-30f) {
		tmp = expf(((fmaxf(cosTheta_i, cosTheta_O) / v) * fminf(cosTheta_i, cosTheta_O))) / (v + v);
	} else {
		tmp = (0.5f / v) / expf(((sinTheta_O * sinTheta_i) / v));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 2.0000000063421537e-30) then
        tmp = exp(((fmax(costheta_i, costheta_o) / v) * fmin(costheta_i, costheta_o))) / (v + v)
    else
        tmp = (0.5e0 / v) / exp(((sintheta_o * sintheta_i) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(2.0000000063421537e-30))
		tmp = Float32(exp(Float32(Float32(fmax(cosTheta_i, cosTheta_O) / v) * fmin(cosTheta_i, cosTheta_O))) / Float32(v + v));
	else
		tmp = Float32(Float32(Float32(0.5) / v) / exp(Float32(Float32(sinTheta_O * sinTheta_i) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(2.0000000063421537e-30))
		tmp = exp(((max(cosTheta_i, cosTheta_O) / v) * min(cosTheta_i, cosTheta_O))) / (v + v);
	else
		tmp = (single(0.5) / v) / exp(((sinTheta_O * sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 2.00000001e-30

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   2. Taylor expanded in v around inf

      \[\leadsto e^{\color{blue}{\frac{6931}{10000}} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 4.6%

      \[\leadsto e^{\color{blue}{0.6931} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   5. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. lift-log.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

      4. lift-/.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

      5. log-rec N/A

         \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

      6. lift-*.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(2 \cdot v\right)}\right)\right)} \]

      7. count-2-rev N/A

         \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

      8. lift-+.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

      9. lift-log.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\color{blue}{\log \left(v + v\right)}\right)\right)} \]

      10. sub-flip-reverse N/A

          \[\leadsto e^{\color{blue}{\frac{6931}{10000} - \log \left(v + v\right)}} \]

      11. div-exp N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{e^{\log \left(v + v\right)}}} \]

      12. lift-log.f32 N/A

          \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\color{blue}{\log \left(v + v\right)}}} \]

      13. lift-+.f32 N/A

          \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(v + v\right)}}} \]

      14. count-2-rev N/A

          \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

   6. Applied rewrites 4.6%

      \[\leadsto \color{blue}{\frac{e^{0.6931}}{v + v}} \]

   7. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 12.0%

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

   9. Applied rewrites 12.0%

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

       1. lift-/.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 N/A

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

       7. lower-/.f32 12.0%

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

   11. Applied rewrites 12.0%

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

   if 2.00000001e-30 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. +-commutative N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

      4. add-flip N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) - \left(\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)\right)}} \]

      5. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{1}{2 \cdot v}\right)}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      6. lift-log.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      7. rem-exp-log N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      9. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2}}}{v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      14. lower-exp.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{e^{-\left(\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931\right)}}} \]

   4. Taylor expanded in sinTheta_i around inf

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
   5. Step-by-step derivation

      1. lower-/.f32 N/A

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

      2. lower-*.f32 11.3%

         \[\leadsto \frac{\frac{0.5}{v}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   6. Applied rewrites 11.3%

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 17.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{\mathsf{max}\left(cosTheta\_i, cosTheta\_O\right)}{v} \cdot \mathsf{min}\left(cosTheta\_i, cosTheta\_O\right)}}{v + v}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \cdot v}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 2.0000000063421537e-30)
   (/
    (exp (* (/ (fmax cosTheta_i cosTheta_O) v) (fmin cosTheta_i cosTheta_O)))
    (+ v v))
   (/ 0.5 (* (exp (/ (* sinTheta_O sinTheta_i) v)) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 2.0000000063421537e-30f) {
		tmp = expf(((fmaxf(cosTheta_i, cosTheta_O) / v) * fminf(cosTheta_i, cosTheta_O))) / (v + v);
	} else {
		tmp = 0.5f / (expf(((sinTheta_O * sinTheta_i) / v)) * v);
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 2.0000000063421537e-30) then
        tmp = exp(((fmax(costheta_i, costheta_o) / v) * fmin(costheta_i, costheta_o))) / (v + v)
    else
        tmp = 0.5e0 / (exp(((sintheta_o * sintheta_i) / v)) * v)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(2.0000000063421537e-30))
		tmp = Float32(exp(Float32(Float32(fmax(cosTheta_i, cosTheta_O) / v) * fmin(cosTheta_i, cosTheta_O))) / Float32(v + v));
	else
		tmp = Float32(Float32(0.5) / Float32(exp(Float32(Float32(sinTheta_O * sinTheta_i) / v)) * v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(2.0000000063421537e-30))
		tmp = exp(((max(cosTheta_i, cosTheta_O) / v) * min(cosTheta_i, cosTheta_O))) / (v + v);
	else
		tmp = single(0.5) / (exp(((sinTheta_O * sinTheta_i) / v)) * v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 2.00000001e-30

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   2. Taylor expanded in v around inf

      \[\leadsto e^{\color{blue}{\frac{6931}{10000}} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 4.6%

      \[\leadsto e^{\color{blue}{0.6931} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   5. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. lift-log.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

      4. lift-/.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

      5. log-rec N/A

         \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

      6. lift-*.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(2 \cdot v\right)}\right)\right)} \]

      7. count-2-rev N/A

         \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

      8. lift-+.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

      9. lift-log.f32 N/A

         \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\color{blue}{\log \left(v + v\right)}\right)\right)} \]

      10. sub-flip-reverse N/A

          \[\leadsto e^{\color{blue}{\frac{6931}{10000} - \log \left(v + v\right)}} \]

      11. div-exp N/A

          \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{e^{\log \left(v + v\right)}}} \]

      12. lift-log.f32 N/A

          \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\color{blue}{\log \left(v + v\right)}}} \]

      13. lift-+.f32 N/A

          \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(v + v\right)}}} \]

      14. count-2-rev N/A

          \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

   6. Applied rewrites 4.6%

      \[\leadsto \color{blue}{\frac{e^{0.6931}}{v + v}} \]

   7. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 12.0%

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

   9. Applied rewrites 12.0%

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

       1. lift-/.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 N/A

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

       7. lower-/.f32 12.0%

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

   11. Applied rewrites 12.0%

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

   if 2.00000001e-30 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. +-commutative N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

      4. add-flip N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) - \left(\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)\right)}} \]

      5. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{1}{2 \cdot v}\right)}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      6. lift-log.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      7. rem-exp-log N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

      9. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2}}}{v}}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}} \]

      14. lower-exp.f32 N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{v}}{\color{blue}{e^{\mathsf{neg}\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)\right)}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{e^{-\left(\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} - -0.6931\right)}}} \]

   4. Taylor expanded in sinTheta_i around inf

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
   5. Step-by-step derivation

      1. lower-/.f32 N/A

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

      2. lower-*.f32 11.3%

         \[\leadsto \frac{\frac{0.5}{v}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   6. Applied rewrites 11.3%

      \[\leadsto \frac{\frac{0.5}{v}}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
   7. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 11.3%

         \[\leadsto \frac{0.5}{\color{blue}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \cdot v}} \]

   8. Applied rewrites 11.3%

      \[\leadsto \color{blue}{\frac{0.5}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \cdot v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 12.0% accurate, 1.7× speedup

Math

\[\frac{e^{\frac{\mathsf{max}\left(cosTheta\_i, cosTheta\_O\right)}{v} \cdot \mathsf{min}\left(cosTheta\_i, cosTheta\_O\right)}}{v + v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (exp (* (/ (fmax cosTheta_i cosTheta_O) v) (fmin cosTheta_i cosTheta_O)))
  (+ v v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((fmaxf(cosTheta_i, cosTheta_O) / v) * fminf(cosTheta_i, cosTheta_O))) / (v + 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(((fmax(costheta_i, costheta_o) / v) * fmin(costheta_i, costheta_o))) / (v + v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(Float32(fmax(cosTheta_i, cosTheta_O) / v) * fmin(cosTheta_i, cosTheta_O))) / Float32(v + v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((max(cosTheta_i, cosTheta_O) / v) * min(cosTheta_i, cosTheta_O))) / (v + v);
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in v around inf

   \[\leadsto e^{\color{blue}{\frac{6931}{10000}} + \log \left(\frac{1}{2 \cdot v}\right)} \]
3. Step-by-step derivation

4. Applied rewrites 4.6%

   \[\leadsto e^{\color{blue}{0.6931} + \log \left(\frac{1}{2 \cdot v}\right)} \]
5. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   4. lift-/.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

   5. log-rec N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

   6. lift-*.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(2 \cdot v\right)}\right)\right)} \]

   7. count-2-rev N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

   8. lift-+.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

   9. lift-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\color{blue}{\log \left(v + v\right)}\right)\right)} \]

   10. sub-flip-reverse N/A

       \[\leadsto e^{\color{blue}{\frac{6931}{10000} - \log \left(v + v\right)}} \]

   11. div-exp N/A

       \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{e^{\log \left(v + v\right)}}} \]

   12. lift-log.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\color{blue}{\log \left(v + v\right)}}} \]

   13. lift-+.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(v + v\right)}}} \]

   14. count-2-rev N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

6. Applied rewrites 4.6%

   \[\leadsto \color{blue}{\frac{e^{0.6931}}{v + v}} \]

7. Taylor expanded in cosTheta_i around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 12.0%

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

9. Applied rewrites 12.0%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. associate-/l* N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. lower-/.f32 12.0%

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

11. Applied rewrites 12.0%

    \[\leadsto \frac{e^{\frac{cosTheta\_O}{v} \cdot \color{blue}{cosTheta\_i}}}{v + v} \]
12. Add Preprocessing

Alternative 12: 4.6% accurate, 2.3× speedup

Math

\[e^{0.6931 - \log \left(v + v\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (- 0.6931 (log (+ v v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((0.6931f - logf((v + 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((0.6931e0 - log((v + v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(0.6931) - log(Float32(v + v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) - log((v + v))));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in v around inf

   \[\leadsto e^{\color{blue}{\frac{6931}{10000}} + \log \left(\frac{1}{2 \cdot v}\right)} \]
3. Step-by-step derivation

4. Applied rewrites 4.6%

   \[\leadsto e^{\color{blue}{0.6931} + \log \left(\frac{1}{2 \cdot v}\right)} \]
5. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-/.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

   4. log-rec N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

   5. lift-*.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(2 \cdot v\right)}\right)\right)} \]

   6. count-2-rev N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

   7. lift-+.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\color{blue}{\log \left(v + v\right)}\right)\right)} \]

   9. sub-flip-reverse N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} - \log \left(v + v\right)}} \]

   10. lower--.f32 4.6%

       \[\leadsto e^{\color{blue}{0.6931 - \log \left(v + v\right)}} \]

6. Applied rewrites 4.6%

   \[\leadsto e^{\color{blue}{0.6931 - \log \left(v + v\right)}} \]
7. Add Preprocessing

Alternative 13: 4.6% accurate, 2.4× speedup

Math

\[\frac{1}{\frac{v + v}{e^{0.6931}}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ (+ v v) (exp 0.6931))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / ((v + v) / expf(0.6931f));
}

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 = 1.0e0 / ((v + v) / exp(0.6931e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(Float32(v + v) / exp(Float32(0.6931))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / ((v + v) / exp(single(0.6931)));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in v around inf

   \[\leadsto e^{\color{blue}{\frac{6931}{10000}} + \log \left(\frac{1}{2 \cdot v}\right)} \]
3. Step-by-step derivation

4. Applied rewrites 4.6%

   \[\leadsto e^{\color{blue}{0.6931} + \log \left(\frac{1}{2 \cdot v}\right)} \]
5. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   4. lift-/.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

   5. log-rec N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

   6. lift-*.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(2 \cdot v\right)}\right)\right)} \]

   7. count-2-rev N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

   8. lift-+.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

   9. lift-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\color{blue}{\log \left(v + v\right)}\right)\right)} \]

   10. sub-flip-reverse N/A

       \[\leadsto e^{\color{blue}{\frac{6931}{10000} - \log \left(v + v\right)}} \]

   11. div-exp N/A

       \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{e^{\log \left(v + v\right)}}} \]

   12. lift-log.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\color{blue}{\log \left(v + v\right)}}} \]

   13. lift-+.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(v + v\right)}}} \]

   14. count-2-rev N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

6. Applied rewrites 4.6%

   \[\leadsto \color{blue}{\frac{e^{0.6931}}{v + v}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{v + v}} \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{6931}{10000}}}}} \]

   3. lower-unsound-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{\frac{6931}{10000}}}}} \]

   4. lower-unsound-/.f32 4.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{v + v}{e^{0.6931}}}} \]

8. Applied rewrites 4.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{v + v}{e^{0.6931}}}} \]
9. Add Preprocessing

Alternative 14: 4.6% accurate, 2.8× speedup

Math

\[\frac{0.5}{e^{-0.6931} \cdot v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (* (exp -0.6931) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / (expf(-0.6931f) * 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 = 0.5e0 / (exp((-0.6931e0)) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / Float32(exp(Float32(-0.6931)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (exp(single(-0.6931)) * v);
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   4. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   5. add-flip N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) - \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   6. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   8. rem-exp-log N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \frac{1}{\color{blue}{2 \cdot v}} \]

   11. associate-/r* N/A

       \[\leadsto \frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)}} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   12. frac-times N/A

       \[\leadsto \color{blue}{\frac{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}} \cdot \frac{1}{2}}{e^{\mathsf{neg}\left(\frac{6931}{10000}\right)} \cdot v}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \cdot 0.5}{e^{-0.6931} \cdot v}} \]

4. Taylor expanded in v around inf

   \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{e^{-0.6931} \cdot v} \]
5. Step-by-step derivation

6. Applied rewrites 4.6%

   \[\leadsto \frac{\color{blue}{0.5}}{e^{-0.6931} \cdot v} \]
7. Add Preprocessing

Alternative 15: 4.6% accurate, 2.9× speedup

Math

\[\frac{e^{0.6931}}{v + v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (exp 0.6931) (+ v v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(0.6931f) / (v + 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(0.6931e0) / (v + v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(0.6931)) / Float32(v + v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(single(0.6931)) / (v + v);
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in v around inf

   \[\leadsto e^{\color{blue}{\frac{6931}{10000}} + \log \left(\frac{1}{2 \cdot v}\right)} \]
3. Step-by-step derivation

4. Applied rewrites 4.6%

   \[\leadsto e^{\color{blue}{0.6931} + \log \left(\frac{1}{2 \cdot v}\right)} \]
5. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   4. lift-/.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

   5. log-rec N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

   6. lift-*.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(2 \cdot v\right)}\right)\right)} \]

   7. count-2-rev N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

   8. lift-+.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\log \color{blue}{\left(v + v\right)}\right)\right)} \]

   9. lift-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\mathsf{neg}\left(\color{blue}{\log \left(v + v\right)}\right)\right)} \]

   10. sub-flip-reverse N/A

       \[\leadsto e^{\color{blue}{\frac{6931}{10000} - \log \left(v + v\right)}} \]

   11. div-exp N/A

       \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{e^{\log \left(v + v\right)}}} \]

   12. lift-log.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\color{blue}{\log \left(v + v\right)}}} \]

   13. lift-+.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(v + v\right)}}} \]

   14. count-2-rev N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{e^{\log \color{blue}{\left(2 \cdot v\right)}}} \]

6. Applied rewrites 4.6%

   \[\leadsto \color{blue}{\frac{e^{0.6931}}{v + v}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025207 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (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))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))