Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 14.0s

Alternatives: 12

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32((Float32(exp(1)) ^ Float32(r * Float32(Float32(-0.3333333333333333) / s))) + exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((single(2.71828182845904523536) ^ (r * (single(-0.3333333333333333) / s))) + exp((r / -s))) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around inf 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. associate-*r/ 99.7%

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

7. Applied egg-rr 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-1-e 99.7%

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

9. Applied egg-rr 99.7%

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

    1. *-lft-identity 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in r around 0 99.7%

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

    1. *-commutative 99.7%

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

    2. associate-*l/ 99.7%

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

    3. associate-*r/ 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(exp(Float32(r / Float32(-s))) + (Float32(exp(1)) ^ Float32(Float32(r * Float32(-0.3333333333333333)) / s))) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + (single(2.71828182845904523536) ^ ((r * single(-0.3333333333333333)) / s))) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around inf 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. associate-*r/ 99.7%

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

7. Applied egg-rr 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-1-e 99.7%

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

9. Applied egg-rr 99.7%

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

    1. *-lft-identity 99.7%

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

11. Simplified 99.7%

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

    1. mul-1-neg 99.7%

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

    2. distribute-frac-neg 99.7%

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

    3. add-log-exp 99.7%

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

    4. *-un-lft-identity 99.7%

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

    5. log-prod 99.7%

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

    6. metadata-eval 99.7%

       \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{0} + \log \left(e^{\frac{-r}{s}}\right)} + {e}^{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

    7. add-log-exp 99.7%

       \[\leadsto 0.125 \cdot \frac{e^{0 + \color{blue}{\frac{-r}{s}}} + {e}^{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

13. Applied egg-rr 99.7%

    \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{0 + \frac{-r}{s}}} + {e}^{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]
14. Step-by-step derivation

    1. +-lft-identity 99.7%

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

15. Simplified 99.7%

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

16. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. add-log-exp 99.7%

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

   4. *-un-lft-identity 99.7%

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

   5. log-prod 99.7%

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

   6. metadata-eval 99.7%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{0} + \log \left(e^{\frac{-r}{s}}\right)} + {e}^{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

   7. add-log-exp 99.7%

      \[\leadsto 0.125 \cdot \frac{e^{0 + \color{blue}{\frac{-r}{s}}} + {e}^{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.7%

   \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{0 + \frac{-r}{s}}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. +-lft-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 4: 47.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 20:\\ \;\;\;\;\frac{\frac{0.125}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 20.0)
   (+
    (/ (/ 0.125 (+ (/ r s) 1.0)) (* r (* s PI)))
    (* 0.75 (/ (exp (/ r (* 3.0 (- s)))) (* r (* (* s PI) 6.0)))))
   (/ (/ -0.25 (log1p (expm1 (* r PI)))) s)))

C

float code(float s, float r) {
	float tmp;
	if (r <= 20.0f) {
		tmp = ((0.125f / ((r / s) + 1.0f)) / (r * (s * ((float) M_PI)))) + (0.75f * (expf((r / (3.0f * -s))) / (r * ((s * ((float) M_PI)) * 6.0f))));
	} else {
		tmp = (-0.25f / log1pf(expm1f((r * ((float) M_PI))))) / s;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(20.0))
		tmp = Float32(Float32(Float32(Float32(0.125) / Float32(Float32(r / s) + Float32(1.0))) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(0.75) * Float32(exp(Float32(r / Float32(Float32(3.0) * Float32(-s)))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(6.0))))));
	else
		tmp = Float32(Float32(Float32(-0.25) / log1p(expm1(Float32(r * Float32(pi))))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 20

   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. times-frac 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-frac-neg 99.6%

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

      4. associate-/l* 99.6%

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

      5. *-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in s around 0 99.6%

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

      1. associate-*r/ 99.6%

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

      2. rec-exp 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in r around 0 12.2%

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

   if 20 < r

   1. Initial program 99.7%

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

      1. add-exp-log 99.6%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in r around 0 5.6%

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

      1. associate-*r* 5.6%

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

      2. *-commutative 5.6%

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

   7. Simplified 5.6%

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

      1. frac-2neg 5.6%

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

      2. *-commutative 5.6%

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

      3. rem-exp-log 5.6%

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

      4. div-inv 5.6%

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

      5. metadata-eval 5.6%

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

      6. rem-exp-log 5.6%

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

      7. associate-*r* 5.6%

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

      8. distribute-lft-neg-in 5.6%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 5.0%

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

      11. sqr-neg 5.0%

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

      12. sqrt-unprod 5.0%

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

      13. add-sqr-sqrt 5.0%

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

   9. Applied egg-rr 5.0%

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

       1. metadata-eval 5.0%

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

       2. distribute-lft-neg-in 5.0%

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

       3. associate-*r/ 5.0%

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

       4. metadata-eval 5.0%

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

       5. associate-*r* 5.0%

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

       6. associate-/l/ 5.0%

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

       7. associate-/r* 5.0%

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

       8. distribute-neg-frac 5.0%

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

       9. associate-/l/ 5.0%

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

       10. distribute-neg-frac 5.0%

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

       11. metadata-eval 5.0%

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

   11. Simplified 5.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.25}{r \cdot \pi}}{s}} \]
   12. Step-by-step derivation

       1. log1p-expm1-u 95.8%

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

   13. Applied egg-rr 95.8%

       \[\leadsto \frac{\frac{-0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 20:\\ \;\;\;\;\frac{\frac{0.125}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 15.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r}{3 \cdot \left(-s\right)}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.125 (+ (/ r s) 1.0)) (* r (* s PI)))
  (* 0.75 (/ (exp (/ r (* 3.0 (- s)))) (* r (* (* s PI) 6.0))))))

C

float code(float s, float r) {
	return ((0.125f / ((r / s) + 1.0f)) / (r * (s * ((float) M_PI)))) + (0.75f * (expf((r / (3.0f * -s))) / (r * ((s * ((float) M_PI)) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(Float32(r / s) + Float32(1.0))) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(0.75) * Float32(exp(Float32(r / Float32(Float32(3.0) * Float32(-s)))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / ((r / s) + single(1.0))) / (r * (s * single(pi)))) + (single(0.75) * (exp((r / (single(3.0) * -s))) / (r * ((s * single(pi)) * single(6.0)))));
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. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

   1. associate-*r/ 99.6%

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

   2. rec-exp 99.6%

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

   3. associate-*r/ 99.6%

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

   4. metadata-eval 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in r around 0 13.0%

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

9. Final simplification 13.0%

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

Alternative 6: 9.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{e^{\frac{r}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.25 (exp (/ r s))) (* r (* s (* PI 2.0))))
  (/ 0.125 (* PI (* r s)))))

C

float code(float s, float r) {
	return ((0.25f / expf((r / s))) / (r * (s * (((float) M_PI) * 2.0f)))) + (0.125f / (((float) M_PI) * (r * s)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / exp(Float32(r / s))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(0.125) / Float32(Float32(pi) * Float32(r * s))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / exp((r / s))) / (r * (s * (single(pi) * single(2.0))))) + (single(0.125) / (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. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in r around 0 7.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{0.125}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

   1. associate-*r* 7.9%

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

   2. *-commutative 7.9%

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

7. Simplified 7.9%

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

8. Taylor expanded in r around inf 7.9%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 7.9%

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

   2. rec-exp 7.9%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]

   3. associate-*r/ 7.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]

   4. metadata-eval 7.9%

      \[\leadsto \frac{\frac{\color{blue}{0.25}}{e^{\frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]

10. Simplified 7.9%

    \[\leadsto \frac{\color{blue}{\frac{0.25}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]

11. Final simplification 7.9%

    \[\leadsto \frac{\frac{0.25}{e^{\frac{r}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]
12. Add Preprocessing

Alternative 7: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) / exp((r / s))) / (r * single(pi))) + (single(0.125) / (r * 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. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in r around 0 7.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{0.125}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

   1. associate-*r* 7.9%

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

   2. *-commutative 7.9%

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

7. Simplified 7.9%

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

8. Taylor expanded in s around 0 7.9%

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

   1. associate-*r/ 7.9%

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

   2. rem-exp-log 7.9%

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

   3. exp-sum 7.9%

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

   4. mul-1-neg 7.9%

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

   5. sub-neg 7.9%

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

   6. exp-diff 7.9%

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

   7. rem-exp-log 7.9%

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

   8. associate-*r/ 7.9%

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

   9. metadata-eval 7.9%

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

10. Simplified 7.9%

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

Alternative 8: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around inf 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. associate-*r/ 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in r around 0 7.9%

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

9. Final simplification 7.9%

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

Alternative 9: 10.2% 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 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.6%

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

   4. associate-*l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. *-commutative 99.6%

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

   8. neg-mul-1 99.6%

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

   9. times-frac 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around -inf 7.8%

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

   1. mul-1-neg 7.8%

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

7. Simplified 7.8%

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

8. Final simplification 7.8%

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

Alternative 10: 9.1% 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 99.6%

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

   1. add-exp-log 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in r around 0 7.6%

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

   1. associate-*r* 7.6%

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

   2. *-commutative 7.6%

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

7. Simplified 7.6%

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

Alternative 11: 9.1% 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 99.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 7.6%

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

Alternative 12: 3.5% accurate, 77.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{0} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 0.0))

C

float code(float s, float r) {
	return 0.25f / 0.0f;
}

Fortran

real(4) function code(s, r)
    real(4), intent (in) :: s
    real(4), intent (in) :: r
    code = 0.25e0 / 0.0e0
end function

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(0.0))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / single(0.0);
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. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 99.5%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in r around 0 7.6%

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

   1. associate-*r* 7.6%

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

   2. *-commutative 7.6%

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

7. Simplified 7.6%

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

   1. *-commutative 7.6%

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

   2. rem-exp-log 7.6%

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

   3. expm1-log1p-u 7.6%

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

   4. expm1-undefine 6.5%

      \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(e^{\log \left(\left(r \cdot s\right) \cdot \pi\right)}\right)} - 1}} \]

   5. rem-exp-log 6.5%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(r \cdot s\right) \cdot \pi}\right)} - 1} \]

   6. associate-*r* 6.5%

      \[\leadsto \frac{0.25}{e^{\mathsf{log1p}\left(\color{blue}{r \cdot \left(s \cdot \pi\right)}\right)} - 1} \]

9. Applied egg-rr 6.5%

   \[\leadsto \frac{0.25}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]
10. Step-by-step derivation

    1. log1p-undefine 6.5%

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

    2. rem-exp-log 6.5%

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

    3. associate-+r- 6.6%

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

    4. fmm-def 6.6%

       \[\leadsto \frac{0.25}{1 + \color{blue}{\mathsf{fma}\left(r, s \cdot \pi, -1\right)}} \]

    5. metadata-eval 6.6%

       \[\leadsto \frac{0.25}{1 + \mathsf{fma}\left(r, s \cdot \pi, \color{blue}{-1}\right)} \]

11. Simplified 6.6%

    \[\leadsto \frac{0.25}{\color{blue}{1 + \mathsf{fma}\left(r, s \cdot \pi, -1\right)}} \]

12. Taylor expanded in r around 0 3.4%

    \[\leadsto \frac{0.25}{1 + \color{blue}{-1}} \]

13. Final simplification 3.4%

    \[\leadsto \frac{0.25}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024153 
(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))))