Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 13.8s

Alternatives: 11

Speedup: N/A×

Specification

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

Math

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

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

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

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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{1}{s \cdot \left(\pi \cdot e^{\frac{r}{s}}\right)}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* 0.125 (/ (exp (* -0.3333333333333333 (/ r s))) (* s PI)))
   (* 0.125 (/ 1.0 (* s (* PI (exp (/ r s)))))))
  r))

C

float code(float s, float r) {
	return ((0.125f * (expf((-0.3333333333333333f * (r / s))) / (s * ((float) M_PI)))) + (0.125f * (1.0f / (s * (((float) M_PI) * expf((r / s))))))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / Float32(s * Float32(pi)))) + Float32(Float32(0.125) * Float32(Float32(1.0) / Float32(s * Float32(Float32(pi) * exp(Float32(r / s))))))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * (exp((single(-0.3333333333333333) * (r / s))) / (s * single(pi)))) + (single(0.125) * (single(1.0) / (s * (single(pi) * exp((r / s))))))) / r;
end

Derivation

1. Initial program 99.3%

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

   1. times-frac 99.3%

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

   2. *-commutative 99.3%

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

   3. distribute-frac-neg 99.3%

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

   4. associate-/l* 99.3%

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

   5. *-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. associate-*l* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.3%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.3%

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

   2. exp-neg 99.3%

      \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 99.3%

      \[\leadsto \frac{\color{blue}{\frac{0.125 \cdot 1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in r around 0 99.3%

   \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \left(\pi \cdot r\right)} + 0.75 \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \left(\pi \cdot r\right)} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \left(\pi \cdot r\right)} + 0.75 \cdot \frac{e^{\frac{r}{s} \cdot \color{blue}{\frac{1}{-3}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. times-frac 99.3%

      \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \left(\pi \cdot r\right)} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot 1}{s \cdot -3}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. *-rgt-identity 99.3%

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

10. Simplified 99.3%

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

11. Taylor expanded in r around inf 99.4%

    \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{1}{s \cdot \left(\pi \cdot e^{\frac{r}{s}}\right)}}{r}} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{e^{\frac{r}{-s}}}{s \cdot \pi} + 0.125 \cdot \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{s \cdot \pi}}{r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.2%

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

5. Taylor expanded in r around inf 99.3%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi}}{r}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.3%

      \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{s \cdot \pi}}{r} \]

7. Applied egg-rr 99.3%

   \[\leadsto \frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{s \cdot \pi}}{r} \]

8. Final simplification 99.3%

   \[\leadsto \frac{0.125 \cdot \frac{e^{\frac{r}{-s}}}{s \cdot \pi} + 0.125 \cdot \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{s \cdot \pi}}{r} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi} + \frac{\frac{\frac{0.125}{\pi}}{s}}{e^{\frac{r}{s}}}}{r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. times-frac 99.3%

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

   2. *-commutative 99.3%

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

   3. distribute-frac-neg 99.3%

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

   4. associate-/l* 99.3%

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

   5. *-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. associate-*l* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.3%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.3%

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

   2. exp-neg 99.3%

      \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 99.3%

      \[\leadsto \frac{\color{blue}{\frac{0.125 \cdot 1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in r around 0 99.3%

   \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \left(\pi \cdot r\right)} + 0.75 \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \left(\pi \cdot r\right)} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \left(\pi \cdot r\right)} + 0.75 \cdot \frac{e^{\frac{r}{s} \cdot \color{blue}{\frac{1}{-3}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. times-frac 99.3%

      \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \left(\pi \cdot r\right)} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot 1}{s \cdot -3}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. *-rgt-identity 99.3%

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

10. Simplified 99.3%

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

11. Taylor expanded in r around inf 99.4%

    \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi} + 0.125 \cdot \frac{1}{s \cdot \left(\pi \cdot e^{\frac{r}{s}}\right)}}{r}} \]

12. Taylor expanded in s around 0 99.4%

    \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi} + \color{blue}{\frac{0.125}{s \cdot \left(\pi \cdot e^{\frac{r}{s}}\right)}}}{r} \]
13. Step-by-step derivation

    1. associate-*r* 99.3%

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

    2. associate-/r* 99.3%

       \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi} + \color{blue}{\frac{\frac{0.125}{s \cdot \pi}}{e^{\frac{r}{s}}}}}{r} \]

    3. associate-/l/ 99.3%

       \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi} + \frac{\color{blue}{\frac{\frac{0.125}{\pi}}{s}}}{e^{\frac{r}{s}}}}{r} \]

14. Simplified 99.3%

    \[\leadsto \frac{0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi} + \color{blue}{\frac{\frac{\frac{0.125}{\pi}}{s}}{e^{\frac{r}{s}}}}}{r} \]
15. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around 0 99.3%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]

6. Final simplification 99.3%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi} \]
7. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around inf 99.3%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]

6. Final simplification 99.3%

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

Alternative 6: 12.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(r \cdot \pi\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (log1p (expm1 (* s (* r PI))))))

C

float code(float s, float r) {
	return 0.25f / log1pf(expm1f((s * (r * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(s * Float32(r * Float32(pi))))))
end

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.2%

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

5. Taylor expanded in s around inf 9.1%

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

   1. log1p-expm1-u 8.9%

      \[\leadsto \frac{0.25}{r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \pi\right)\right)}} \]

7. Applied egg-rr 8.9%

   \[\leadsto \frac{0.25}{r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \pi\right)\right)}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 9.1%

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

   2. *-commutative 9.1%

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

   3. associate-*r* 9.1%

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

   4. log1p-expm1-u 11.4%

      \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(\pi \cdot r\right)\right)\right)}} \]

9. Applied egg-rr 11.4%

   \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(\pi \cdot r\right)\right)\right)}} \]

10. Final simplification 11.4%

    \[\leadsto \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(r \cdot \pi\right)\right)\right)} \]
11. Add Preprocessing

Alternative 7: 10.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s} + \frac{0.125}{r \cdot \pi}}{s} + 0.75 \cdot \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (+ (/ (- (* 0.0625 (/ r (* s PI))) (/ 0.125 PI)) s) (/ 0.125 (* r PI))) s)
  (* 0.75 (/ (exp (/ (* -0.3333333333333333 r) s)) (* r (* (* s PI) 6.0))))))

C

float code(float s, float r) {
	return (((((0.0625f * (r / (s * ((float) M_PI)))) - (0.125f / ((float) M_PI))) / s) + (0.125f / (r * ((float) M_PI)))) / s) + (0.75f * (expf(((-0.3333333333333333f * r) / s)) / (r * ((s * ((float) M_PI)) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.0625) * Float32(r / Float32(s * Float32(pi)))) - Float32(Float32(0.125) / Float32(pi))) / s) + Float32(Float32(0.125) / Float32(r * Float32(pi)))) / s) + Float32(Float32(0.75) * Float32(exp(Float32(Float32(Float32(-0.3333333333333333) * r) / s)) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (((((single(0.0625) * (r / (s * single(pi)))) - (single(0.125) / single(pi))) / s) + (single(0.125) / (r * single(pi)))) / s) + (single(0.75) * (exp(((single(-0.3333333333333333) * r) / s)) / (r * ((s * single(pi)) * single(6.0)))));
end

Derivation

1. Initial program 99.3%

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

   1. times-frac 99.3%

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

   2. *-commutative 99.3%

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

   3. distribute-frac-neg 99.3%

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

   4. associate-/l* 99.3%

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

   5. *-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. associate-*l* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around -inf 9.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 9.8%

      \[\leadsto \color{blue}{\left(-\frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   2. mul-1-neg 9.8%

      \[\leadsto \left(-\frac{\color{blue}{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s}\right)} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 9.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \color{blue}{\frac{0.125 \cdot 1}{\pi}}}{s}\right) - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 9.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{\color{blue}{0.125}}{\pi}}{s}\right) - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   5. associate-*r/ 9.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \color{blue}{\frac{0.125 \cdot 1}{r \cdot \pi}}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   6. metadata-eval 9.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{\color{blue}{0.125}}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   7. *-commutative 9.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{0.125}{\color{blue}{\pi \cdot r}}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

7. Simplified 9.8%

   \[\leadsto \color{blue}{\left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{0.125}{\pi \cdot r}}{s}\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

8. Taylor expanded in r around 0 9.8%

   \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{0.125}{\pi \cdot r}}{s}\right) + 0.75 \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 9.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{0.125}{\pi \cdot r}}{s}\right) + 0.75 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   2. associate-*l/ 9.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{0.125}{\pi \cdot r}}{s}\right) + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

10. Simplified 9.8%

    \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{0.125}{\pi \cdot r}}{s}\right) + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

11. Final simplification 9.8%

    \[\leadsto \frac{\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s} + \frac{0.125}{r \cdot \pi}}{s} + 0.75 \cdot \frac{e^{\frac{-0.3333333333333333 \cdot r}{s}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} \]
12. Add Preprocessing

Alternative 8: 10.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* r PI))
   (/
    (+ (/ (* (/ r PI) -0.06944444444444445) s) (/ 0.16666666666666666 PI))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) - (((((r / ((float) M_PI)) * -0.06944444444444445f) / s) + (0.16666666666666666f / ((float) M_PI))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) - Float32(Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s) + Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (((((r / single(pi)) * single(-0.06944444444444445)) / s) + (single(0.16666666666666666) / single(pi))) / s)) / s;
end

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.2%

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

5. Taylor expanded in s around -inf 9.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. mul-1-neg 9.7%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

7. Simplified 9.7%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

8. Final simplification 9.7%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \]
9. Add Preprocessing

Alternative 9: 9.3% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\pi} \cdot \frac{\frac{0.25}{r}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.2%

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

5. Taylor expanded in s around inf 9.1%

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

   1. log1p-expm1-u 8.9%

      \[\leadsto \frac{0.25}{r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \pi\right)\right)}} \]

7. Applied egg-rr 8.9%

   \[\leadsto \frac{0.25}{r \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \pi\right)\right)}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 9.1%

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

   2. associate-/r* 9.1%

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

   3. *-un-lft-identity 9.1%

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

   4. *-commutative 9.1%

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

   5. times-frac 9.1%

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

9. Applied egg-rr 9.1%

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

Alternative 10: 9.3% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\pi \cdot \left(r \cdot s\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.2%

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

5. Taylor expanded in r around 0 8.8%

   \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \frac{r}{{s}^{2} \cdot \pi} + 0.25 \cdot \frac{1}{s \cdot \pi}}{r}} \]

6. Taylor expanded in r around inf 8.7%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
7. Step-by-step derivation

   1. associate-*r/ 8.7%

      \[\leadsto \color{blue}{\frac{0.25 \cdot 1}{r \cdot \left(s \cdot \pi\right)}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   2. metadata-eval 8.7%

      \[\leadsto \frac{\color{blue}{0.25}}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   3. associate-/r* 8.7%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   4. associate-*r/ 8.7%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{{s}^{2} \cdot \pi}} \]

   5. metadata-eval 8.7%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{{s}^{2} \cdot \pi} \]

   6. *-commutative 8.7%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{0.16666666666666666}{\color{blue}{\pi \cdot {s}^{2}}} \]

8. Simplified 8.7%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{0.16666666666666666}{\pi \cdot {s}^{2}}} \]

9. Taylor expanded in r around 0 9.1%

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

    1. associate-*r* 9.1%

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

11. Simplified 9.1%

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

12. Final simplification 9.1%

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

Alternative 11: 9.3% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

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 99.3%

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

   1. +-commutative 99.3%

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

   2. times-frac 99.3%

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

   3. fma-define 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.2%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.2%

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

5. Taylor expanded in s around inf 9.1%

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

Reproduce

herbie shell --seed 2024111 
(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))))