Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.0s

Alternatives: 14

Speedup: 1.1×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \frac{0.125}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}\right) \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ r (* -3.0 s))) (* 18.84955596923828 s))
  (/ 0.75 r)
  (/ 0.125 (* (* PI s) (* (exp (/ r s)) r)))))

C

float code(float s, float r) {
	return fmaf((expf((r / (-3.0f * s))) / (18.84955596923828f * s)), (0.75f / r), (0.125f / ((((float) M_PI) * s) * (expf((r / s)) * r))));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(18.84955596923828) * s)), Float32(Float32(0.75) / r), Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * Float32(exp(Float32(r / s)) * r))))
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\left(6 \cdot \pi\right) \cdot s}, \frac{0.75}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right)} \]

4. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\color{blue}{18.84955596923828} \cdot s}, \frac{0.75}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \color{blue}{\frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}}\right) \]

   2. lift-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\color{blue}{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}}{r}\right) \]

   3. associate-/l/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \color{blue}{\frac{\frac{1}{8}}{\left(\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}\right) \cdot r}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \color{blue}{\frac{\frac{1}{8}}{\left(\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}\right) \cdot r}}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{\color{blue}{\left(\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}\right)} \cdot r}\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{\color{blue}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{\color{blue}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]

   8. lower-*.f32 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \frac{0.125}{\left(\pi \cdot s\right) \cdot \color{blue}{\left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]

6. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \color{blue}{\frac{0.125}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]
7. Add Preprocessing

Alternative 2: 99.6% accurate, 1.2× speedup

Math

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(6.2831854820251465 \cdot s\right) \cdot r} + \frac{e^{\frac{r}{-3 \cdot s}}}{s \cdot r} \cdot 0.039788734167814255 \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* 6.2831854820251465 s) r))
  (* (/ (exp (/ r (* -3.0 s))) (* s r)) 0.039788734167814255)))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / ((6.2831854820251465f * s) * r)) + ((expf((r / (-3.0f * s))) / (s * r)) * 0.039788734167814255f);
}

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(s, r)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: r
    code = ((0.25e0 * exp((-r / s))) / ((6.2831854820251465e0 * s) * r)) + ((exp((r / ((-3.0e0) * s))) / (s * r)) * 0.039788734167814255e0)
end function

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(6.2831854820251465) * s) * r)) + Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(s * r)) * Float32(0.039788734167814255)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / ((single(6.2831854820251465) * s) * r)) + ((exp((r / (single(-3.0) * s))) / (s * r)) * single(0.039788734167814255));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{e^{\frac{-r}{3 \cdot s}} \cdot \frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot \frac{3}{4}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot \frac{3}{4}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right)} \cdot r} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot \frac{3}{4}}{\color{blue}{\left(6 \cdot \pi\right) \cdot \left(s \cdot r\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot \frac{3}{4}}{\color{blue}{\left(s \cdot r\right) \cdot \left(6 \cdot \pi\right)}} \]

   8. times-frac N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{e^{\frac{-r}{3 \cdot s}}}{s \cdot r} \cdot \frac{\frac{3}{4}}{6 \cdot \pi}} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{e^{\frac{-r}{3 \cdot s}}}{s \cdot r} \cdot \frac{\frac{3}{4}}{6 \cdot \pi}} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{e^{\frac{r}{-3 \cdot s}}}{s \cdot r} \cdot \frac{0.75}{6 \cdot \pi}} \]

4. Evaluated real constant 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{e^{\frac{r}{-3 \cdot s}}}{s \cdot r} \cdot \color{blue}{0.039788734167814255} \]

5. Evaluated real constant 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\color{blue}{6.2831854820251465} \cdot s\right) \cdot r} + \frac{e^{\frac{r}{-3 \cdot s}}}{s \cdot r} \cdot 0.039788734167814255 \]
6. Add Preprocessing

Alternative 3: 99.6% accurate, 1.2× speedup

Math

\[\frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{18.84955596923828 \cdot s}, 0.75, \frac{0.125}{\left(e^{\frac{r}{s}} \cdot \pi\right) \cdot s}\right)}{r} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (/ (exp (* (/ r s) -0.3333333333333333)) (* 18.84955596923828 s))
   0.75
   (/ 0.125 (* (* (exp (/ r s)) PI) s)))
  r))

C

float code(float s, float r) {
	return fmaf((expf(((r / s) * -0.3333333333333333f)) / (18.84955596923828f * s)), 0.75f, (0.125f / ((expf((r / s)) * ((float) M_PI)) * s))) / r;
}

Julia

function code(s, r)
	return Float32(fma(Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(18.84955596923828) * s)), Float32(0.75), Float32(Float32(0.125) / Float32(Float32(exp(Float32(r / s)) * Float32(pi)) * s))) / r)
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\left(6 \cdot \pi\right) \cdot s}, \frac{0.75}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right)} \]

4. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\color{blue}{18.84955596923828} \cdot s}, \frac{0.75}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
5. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{\frac{3}{4}}{r} + \frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \color{blue}{\frac{\frac{3}{4}}{r}} + \frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{3}{4}}{r}} + \frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{3}{4}}{r} + \color{blue}{\frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}} \]

   5. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{3}{4} + \frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{3}{4} + \frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{18.84955596923828 \cdot s}, 0.75, \frac{0.125}{\left(e^{\frac{r}{s}} \cdot \pi\right) \cdot s}\right)}{r}} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.2× speedup

Math

\[\frac{0.125}{r} \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi \cdot s} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot s}\right) \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 r)
  (+
   (/ (exp (/ (- r) s)) (* PI s))
   (/ (exp (* (/ r s) -0.3333333333333333)) (* PI s)))))

C

float code(float s, float r) {
	return (0.125f / r) * ((expf((-r / s)) / (((float) M_PI) * s)) + (expf(((r / s) * -0.3333333333333333f)) / (((float) M_PI) * s)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / r) * Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(pi) * s)) + Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(pi) * s))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / r) * ((exp((-r / s)) / (single(pi) * s)) + (exp(((r / s) * single(-0.3333333333333333))) / (single(pi) * s)));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, 0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125\right)}{s}}{r}} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s}}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s}}}{r} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s \cdot r}} \]

   4. lift-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-r}{s}}}{\pi} \cdot \frac{1}{8} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}}}{s \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot \frac{1}{8} + \color{blue}{\frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}}}{s \cdot r} \]

   6. distribute-rgt-out N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}\right)}}{s \cdot r} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}\right)}{\color{blue}{r \cdot s}} \]

   8. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s}} \]

   9. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s}} \]

   10. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r}} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s} \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{0.125}{r} \cdot \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi}}{s}} \]

7. Applied rewrites 99.5%

   \[\leadsto \frac{0.125}{r} \cdot \color{blue}{\left(\frac{e^{\frac{-r}{s}}}{\pi \cdot s} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi \cdot s}\right)} \]
8. Add Preprocessing

Alternative 5: 99.5% accurate, 1.4× speedup

Math

\[\frac{\left(e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right) \cdot 0.125}{\left(s \cdot r\right) \cdot \pi} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (* (+ (exp (/ (- r) s)) (exp (* (/ r s) -0.3333333333333333))) 0.125)
  (* (* s r) PI)))

C

float code(float s, float r) {
	return ((expf((-r / s)) + expf(((r / s) * -0.3333333333333333f))) * 0.125f) / ((s * r) * ((float) M_PI));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) + exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) * Float32(0.125)) / Float32(Float32(s * r) * Float32(pi)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) + exp(((r / s) * single(-0.3333333333333333)))) * single(0.125)) / ((s * r) * single(pi));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, 0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125\right)}{s}}{r}} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s}}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s}}}{r} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s \cdot r}} \]

   4. lift-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-r}{s}}}{\pi} \cdot \frac{1}{8} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}}}{s \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot \frac{1}{8} + \color{blue}{\frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}}}{s \cdot r} \]

   6. distribute-rgt-out N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}\right)}}{s \cdot r} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}\right)}{\color{blue}{r \cdot s}} \]

   8. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s}} \]

   9. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s}} \]

   10. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r}} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s} \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{0.125}{r} \cdot \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi}}{s}} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r} \cdot \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi}}{s}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi}}{s} \cdot \frac{\frac{1}{8}}{r}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi}}{s}} \cdot \frac{\frac{1}{8}}{r} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi}}}{s} \cdot \frac{\frac{1}{8}}{r} \]

   5. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi \cdot s}} \cdot \frac{\frac{1}{8}}{r} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\color{blue}{\pi \cdot s}} \cdot \frac{\frac{1}{8}}{r} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi \cdot s} \cdot \color{blue}{\frac{\frac{1}{8}}{r}} \]

   8. frac-times N/A

      \[\leadsto \color{blue}{\frac{\left(e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}\right) \cdot \frac{1}{8}}{\left(\pi \cdot s\right) \cdot r}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\left(e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}\right) \cdot \frac{1}{8}}{\color{blue}{\left(\pi \cdot s\right) \cdot r}} \]

   10. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\left(e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}\right) \cdot \frac{1}{8}}{\left(\pi \cdot s\right) \cdot r}} \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\left(e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right) \cdot 0.125}{\left(s \cdot r\right) \cdot \pi}} \]
8. Add Preprocessing

Alternative 6: 44.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;r \leq 30:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \frac{0.125}{r \cdot \mathsf{fma}\left(r, \pi, s \cdot \pi\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{\pi \cdot r}\right)}^{s}\right)}\\ \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 30.0)
   (fma
    (/ (exp (/ r (* -3.0 s))) (* 18.84955596923828 s))
    (/ 0.75 r)
    (/ 0.125 (* r (fma r PI (* s PI)))))
   (/ 0.25 (log (pow (exp (* PI r)) s)))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 30.0f) {
		tmp = fmaf((expf((r / (-3.0f * s))) / (18.84955596923828f * s)), (0.75f / r), (0.125f / (r * fmaf(r, ((float) M_PI), (s * ((float) M_PI))))));
	} else {
		tmp = 0.25f / logf(powf(expf((((float) M_PI) * r)), s));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(30.0))
		tmp = fma(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(18.84955596923828) * s)), Float32(Float32(0.75) / r), Float32(Float32(0.125) / Float32(r * fma(r, Float32(pi), Float32(s * Float32(pi))))));
	else
		tmp = Float32(Float32(0.25) / log((exp(Float32(Float32(pi) * r)) ^ s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 30

   1. Initial program 99.6%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\left(6 \cdot \pi\right) \cdot s}, \frac{0.75}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right)} \]

   4. Evaluated real constant 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\color{blue}{18.84955596923828} \cdot s}, \frac{0.75}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \color{blue}{\frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}}\right) \]

      2. lift-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\color{blue}{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}}{r}\right) \]

      3. associate-/l/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \color{blue}{\frac{\frac{1}{8}}{\left(\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}\right) \cdot r}}\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \color{blue}{\frac{\frac{1}{8}}{\left(\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}\right) \cdot r}}\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{\color{blue}{\left(\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}\right)} \cdot r}\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{\color{blue}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{\color{blue}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]

      8. lower-*.f32 99.6%

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \frac{0.125}{\left(\pi \cdot s\right) \cdot \color{blue}{\left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]

   6. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \color{blue}{\frac{0.125}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]

   7. Taylor expanded in r around 0

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \frac{0.125}{\color{blue}{r \cdot \left(r \cdot \pi + s \cdot \pi\right)}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(r \cdot \mathsf{PI}\left(\right) + s \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{r \cdot \mathsf{fma}\left(r, \color{blue}{\mathsf{PI}\left(\right)}, s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{r \cdot \mathsf{fma}\left(r, \pi, s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s}, \frac{\frac{3}{4}}{r}, \frac{\frac{1}{8}}{r \cdot \mathsf{fma}\left(r, \pi, s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      5. lower-PI.f32 12.4%

         \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \frac{0.125}{r \cdot \mathsf{fma}\left(r, \pi, s \cdot \pi\right)}\right) \]

   9. Applied rewrites 12.4%

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{18.84955596923828 \cdot s}, \frac{0.75}{r}, \frac{0.125}{\color{blue}{r \cdot \mathsf{fma}\left(r, \pi, s \cdot \pi\right)}}\right) \]

   if 30 < r

   1. Initial program 99.6%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

      4. lower-PI.f32 9.0%

         \[\leadsto \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]

   4. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \pi\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\pi}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot s\right) \cdot \color{blue}{\pi}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

      6. lift-PI.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \]

      7. add-log-exp N/A

         \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

      8. log-pow-rev N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

      9. lower-log.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

      10. lift-PI.f32 N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

      11. pow-exp N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot \left(s \cdot r\right)}\right)} \]

      12. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot \left(s \cdot r\right)}\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

      14. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

      15. lift-*.f32 N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

      16. lower-exp.f32 9.9%

          \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

   6. Applied rewrites 9.9%

      \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]
   7. Step-by-step derivation

      1. lift-exp.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(\pi \cdot s\right)}\right)} \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(\pi \cdot s\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(r \cdot \pi\right) \cdot s}\right)} \]

      6. lift-PI.f32 N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(r \cdot \mathsf{PI}\left(\right)\right) \cdot s}\right)} \]

      7. add-log-exp N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(r \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right) \cdot s}\right)} \]

      8. log-pow-rev N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s}\right)} \]

      9. exp-to-pow N/A

         \[\leadsto \frac{\frac{1}{4}}{\log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right)}^{s}\right)} \]

      10. lower-pow.f32 N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right)}^{s}\right)} \]

      11. lift-PI.f32 N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left({\left({\left(e^{\pi}\right)}^{r}\right)}^{s}\right)} \]

      12. pow-exp N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi \cdot r}\right)}^{s}\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{r \cdot \pi}\right)}^{s}\right)} \]

      14. lower-exp.f32 N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{r \cdot \pi}\right)}^{s}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi \cdot r}\right)}^{s}\right)} \]

      16. lower-*.f32 40.6%

          \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi \cdot r}\right)}^{s}\right)} \]

   8. Applied rewrites 40.6%

      \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi \cdot r}\right)}^{s}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 41.9% accurate, 2.5× speedup

Math

\[\frac{0.25}{\log \left(e^{\pi \cdot r}\right) \cdot s} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (* (log (exp (* PI r))) s)))

C

float code(float s, float r) {
	return 0.25f / (logf(expf((((float) M_PI) * r))) * s);
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(log(exp(Float32(Float32(pi) * r))) * s))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (log(exp((single(pi) * r))) * s);
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f32 9.0%

      \[\leadsto \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \pi\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\pi}\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(\pi \cdot \color{blue}{s}\right)} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(\pi \cdot \color{blue}{s}\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot s\right) \cdot \color{blue}{r}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot s\right) \cdot r} \]

   7. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4}}{\pi \cdot \color{blue}{\left(s \cdot r\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\pi \cdot \left(r \cdot \color{blue}{s}\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot r\right) \cdot \color{blue}{s}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot r\right) \cdot \color{blue}{s}} \]

   11. lower-*.f32 9.0%

       \[\leadsto \frac{0.25}{\left(\pi \cdot r\right) \cdot s} \]

6. Applied rewrites 9.0%

   \[\leadsto \frac{0.25}{\left(\pi \cdot r\right) \cdot \color{blue}{s}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(\pi \cdot r\right) \cdot s} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot \pi\right) \cdot s} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot \mathsf{PI}\left(\right)\right) \cdot s} \]

   4. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right) \cdot s} \]

   5. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s} \]

   6. lower-log.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{r}\right) \cdot s} \]

   8. pow-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot r}\right) \cdot s} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot r}\right) \cdot s} \]

   10. lower-exp.f32 41.9%

       \[\leadsto \frac{0.25}{\log \left(e^{\pi \cdot r}\right) \cdot s} \]

8. Applied rewrites 41.9%

   \[\leadsto \frac{0.25}{\log \left(e^{\pi \cdot r}\right) \cdot s} \]
9. Add Preprocessing

Alternative 8: 9.9% accurate, 2.5× speedup

Math

\[\frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (log (exp (* (* PI s) r)))))

C

float code(float s, float r) {
	return 0.25f / logf(expf(((((float) M_PI) * s) * r)));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / log(exp(Float32(Float32(Float32(pi) * s) * r))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / log(exp(((single(pi) * s) * r)));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f32 9.0%

      \[\leadsto \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \pi\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\pi}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot s\right) \cdot \color{blue}{\pi}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   6. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \]

   7. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

   8. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   9. lower-log.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   10. lift-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

   11. pow-exp N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot \left(s \cdot r\right)}\right)} \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot \left(s \cdot r\right)}\right)} \]

   13. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

   16. lower-exp.f32 9.9%

       \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

6. Applied rewrites 9.9%

   \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]
7. Add Preprocessing

Alternative 9: 9.1% accurate, 2.6× speedup

Math

\[\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (- (* 0.25 (/ 1.0 (* r PI))) (* 0.16666666666666666 (/ 1.0 (* s PI)))) s))

C

float code(float s, float r) {
	return ((0.25f * (1.0f / (r * ((float) M_PI)))) - (0.16666666666666666f * (1.0f / (s * ((float) M_PI))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi)))) - Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(s * Float32(pi))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) / (r * single(pi)))) - (single(0.16666666666666666) * (single(1.0) / (s * single(pi))))) / s;
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \pi}}{s}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{s}} \]

   2. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

   6. lower-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s} \]

   10. lower-PI.f32 9.1%

       \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s} \]

4. Applied rewrites 9.1%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
5. Add Preprocessing

Alternative 10: 9.1% accurate, 2.8× speedup

Math

\[\frac{0.125}{r} \cdot \frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{s}}{\pi}}{s} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 0.125 r) (/ (/ (+ 2.0 (* -1.3333333333333333 (/ r s))) PI) s)))

C

float code(float s, float r) {
	return (0.125f / r) * (((2.0f + (-1.3333333333333333f * (r / s))) / ((float) M_PI)) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / r) * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(-1.3333333333333333) * Float32(r / s))) / Float32(pi)) / s))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / r) * (((single(2.0) + (single(-1.3333333333333333) * (r / s))) / single(pi)) / s);
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, 0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125\right)}{s}}{r}} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s}}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s}}}{r} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, \frac{1}{8}, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}\right)}{s \cdot r}} \]

   4. lift-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{\frac{-r}{s}}}{\pi} \cdot \frac{1}{8} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}}}{s \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}}}{\pi} \cdot \frac{1}{8} + \color{blue}{\frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot \frac{1}{8}}}{s \cdot r} \]

   6. distribute-rgt-out N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}\right)}}{s \cdot r} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}\right)}{\color{blue}{r \cdot s}} \]

   8. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s}} \]

   9. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s}} \]

   10. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{8}}{r}} \cdot \frac{\frac{e^{\frac{-r}{s}}}{\pi} + \frac{e^{\frac{r}{-3 \cdot s}}}{\pi}}{s} \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{0.125}{r} \cdot \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s \cdot -3}}}{\pi}}{s}} \]

6. Taylor expanded in r around 0

   \[\leadsto \frac{0.125}{r} \cdot \frac{\frac{\color{blue}{2 + \frac{-4}{3} \cdot \frac{r}{s}}}{\pi}}{s} \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{r} \cdot \frac{\frac{2 + \color{blue}{\frac{-4}{3} \cdot \frac{r}{s}}}{\pi}}{s} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{r} \cdot \frac{\frac{2 + \frac{-4}{3} \cdot \color{blue}{\frac{r}{s}}}{\pi}}{s} \]

   3. lower-/.f32 9.1%

      \[\leadsto \frac{0.125}{r} \cdot \frac{\frac{2 + -1.3333333333333333 \cdot \frac{r}{\color{blue}{s}}}{\pi}}{s} \]

8. Applied rewrites 9.1%

   \[\leadsto \frac{0.125}{r} \cdot \frac{\frac{\color{blue}{2 + -1.3333333333333333 \cdot \frac{r}{s}}}{\pi}}{s} \]
9. Add Preprocessing

Alternative 11: 9.0% accurate, 4.5× speedup

Math

\[\frac{1}{\frac{\pi \cdot s}{\frac{0.25}{r}}} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.0 (/ (* PI s) (/ 0.25 r))))

C

float code(float s, float r) {
	return 1.0f / ((((float) M_PI) * s) / (0.25f / r));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(Float32(Float32(pi) * s) / Float32(Float32(0.25) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / ((single(pi) * s) / (single(0.25) / r));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f32 9.0%

      \[\leadsto \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \pi\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \pi\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\color{blue}{s \cdot \pi}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{s \cdot \color{blue}{\pi}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\pi \cdot \color{blue}{s}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\pi \cdot \color{blue}{s}} \]

   7. div-flip N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\pi \cdot s}{\frac{\frac{1}{4}}{r}}}} \]

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

      \[\leadsto \frac{1}{\color{blue}{\frac{\pi \cdot s}{\frac{\frac{1}{4}}{r}}}} \]

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

      \[\leadsto \frac{1}{\frac{\pi \cdot s}{\color{blue}{\frac{\frac{1}{4}}{r}}}} \]

   10. lower-/.f32 9.0%

       \[\leadsto \frac{1}{\frac{\pi \cdot s}{\frac{0.25}{\color{blue}{r}}}} \]

6. Applied rewrites 9.0%

   \[\leadsto \frac{1}{\color{blue}{\frac{\pi \cdot s}{\frac{0.25}{r}}}} \]
7. Add Preprocessing

Alternative 12: 9.0% accurate, 4.8× speedup

Math

\[\frac{0.25}{r} \cdot \frac{1}{\pi \cdot s} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 0.25 r) (/ 1.0 (* PI s))))

C

float code(float s, float r) {
	return (0.25f / r) * (1.0f / (((float) M_PI) * s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / r) * Float32(Float32(1.0) / Float32(Float32(pi) * s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / r) * (single(1.0) / (single(pi) * s));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f32 9.0%

      \[\leadsto \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \pi\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \pi\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\color{blue}{s \cdot \pi}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{s \cdot \color{blue}{\pi}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\pi \cdot \color{blue}{s}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\pi \cdot \color{blue}{s}} \]

   7. mult-flip N/A

      \[\leadsto \frac{\frac{1}{4}}{r} \cdot \color{blue}{\frac{1}{\pi \cdot s}} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r} \cdot \color{blue}{\frac{1}{\pi \cdot s}} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r} \cdot \frac{\color{blue}{1}}{\pi \cdot s} \]

   10. lower-/.f32 9.0%

       \[\leadsto \frac{0.25}{r} \cdot \frac{1}{\color{blue}{\pi \cdot s}} \]

6. Applied rewrites 9.0%

   \[\leadsto \frac{0.25}{r} \cdot \color{blue}{\frac{1}{\pi \cdot s}} \]
7. Add Preprocessing

Alternative 13: 9.0% accurate, 6.0× speedup

Math

\[\frac{\frac{0.25}{s \cdot \pi}}{r} \]

FPCore

(FPCore (s r) :precision binary32 (/ (/ 0.25 (* s PI)) r))

C

float code(float s, float r) {
	return (0.25f / (s * ((float) M_PI))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / Float32(s * Float32(pi))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (s * single(pi))) / r;
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, 0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125\right)}{s}}{r}} \]

4. Taylor expanded in s around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \pi}}}{r} \]
5. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s \cdot \color{blue}{\mathsf{PI}\left(\right)}}}{r} \]

   3. lower-PI.f32 9.0%

      \[\leadsto \frac{\frac{0.25}{s \cdot \pi}}{r} \]

6. Applied rewrites 9.0%

   \[\leadsto \frac{\color{blue}{\frac{0.25}{s \cdot \pi}}}{r} \]
7. Add Preprocessing

Alternative 14: 9.0% accurate, 6.4× speedup

Math

\[\frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (* r (* s PI))))

C

float code(float s, float r) {
	return 0.25f / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(r * Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (r * (s * single(pi)));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

   4. lower-PI.f32 9.0%

      \[\leadsto \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025182 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))