Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 1.2min

Alternatives: 12

Speedup: 1.0×

• unsoundness detected in the egraph

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\frac{0.25 \cdot e^{\frac{1}{\frac{-s}{r}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(18.84955596923828 \cdot s\right) \cdot r} \]

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ (* 0.25 (exp (/ 1.0 (/ (- s) r)))) (* (* (* 2.0 PI) s) r))
 (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* 18.84955596923828 s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((1.0f / (-s / r)))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / ((18.84955596923828f * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(1.0) / Float32(Float32(-s) / r)))) / 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(18.84955596923828) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((single(1.0) / (-s / r)))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / ((single(18.84955596923828) * s) * r));
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-flip N/A

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

   4. lower-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. remove-double-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{1}{\frac{\color{blue}{-s}}{r}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\color{blue}{\frac{1}{\frac{-s}{r}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Evaluated real constant 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{1}{\frac{-s}{r}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\color{blue}{18.84955596923828} \cdot s\right) \cdot r} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(6.2831854820251465 \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ (* 0.25 (exp (/ (- r) s))) (* (* 6.2831854820251465 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))) / ((6.2831854820251465f * 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(6.2831854820251465) * 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(6.2831854820251465) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.5%

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

2. Evaluated real constant 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\color{blue}{6.2831854820251465} \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Add Preprocessing

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-r}{s}\\ \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \frac{1}{\frac{s}{e^{0.3333333333333333 \cdot t\_0} + e^{t\_0}}} \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ (- r) s)))
  (*
   (/ 1.0 (/ (* (+ PI PI) r) 0.25))
   (/ 1.0 (/ s (+ (exp (* 0.3333333333333333 t_0)) (exp t_0)))))))

C

float code(float s, float r) {
	float t_0 = -r / s;
	return (1.0f / (((((float) M_PI) + ((float) M_PI)) * r) / 0.25f)) * (1.0f / (s / (expf((0.3333333333333333f * t_0)) + expf(t_0))));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(-r) / s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(pi) + Float32(pi)) * r) / Float32(0.25))) * Float32(Float32(1.0) / Float32(s / Float32(exp(Float32(Float32(0.3333333333333333) * t_0)) + exp(t_0)))))
end

MATLAB

function tmp = code(s, r)
	t_0 = -r / s;
	tmp = (single(1.0) / (((single(pi) + single(pi)) * r) / single(0.25))) * (single(1.0) / (s / (exp((single(0.3333333333333333) * t_0)) + exp(t_0))));
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\frac{s}{e^{\frac{-r}{s} \cdot 0.3333333333333333}}}\right)} \]
4. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. div-flip-rev N/A

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

   5. lift-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. div-flip-rev N/A

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

   8. div-add-rev N/A

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

   9. div-flip N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f32 99.4%

       \[\leadsto \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \frac{1}{\frac{s}{\color{blue}{e^{\frac{-r}{s} \cdot 0.3333333333333333} + e^{\frac{-r}{s}}}}} \]

5. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \color{blue}{\frac{1}{\frac{s}{e^{0.3333333333333333 \cdot \frac{-r}{s}} + e^{\frac{-r}{s}}}}} \]
6. Add Preprocessing

Alternative 4: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{-r}{s}\\ t_1 := e^{t\_0}\\ \mathbf{if}\;\frac{0.25 \cdot t\_1}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{1}{\frac{\pi \cdot r}{\left(\left(-e^{0.3333333333333333 \cdot t\_0}\right) - t\_1\right) \cdot -0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{t\_1}} + \frac{1}{s + 0.3333333333333333 \cdot r}\right)\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ (- r) s)) (t_1 (exp t_0)))
  (if (<=
       (+
        (/ (* 0.25 t_1) (* (* (* 2.0 PI) s) r))
        (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
       0.0020000000949949026)
    (/
     1.0
     (/
      (* PI r)
      (* (- (- (exp (* 0.3333333333333333 t_0))) t_1) -0.25)))
    (*
     (/ 1.0 (/ (* (+ PI PI) r) 0.25))
     (+ (/ 1.0 (/ s t_1)) (/ 1.0 (+ s (* 0.3333333333333333 r))))))))

C

float code(float s, float r) {
	float t_0 = -r / s;
	float t_1 = expf(t_0);
	float tmp;
	if ((((0.25f * t_1) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r))) <= 0.0020000000949949026f) {
		tmp = 1.0f / ((((float) M_PI) * r) / ((-expf((0.3333333333333333f * t_0)) - t_1) * -0.25f));
	} else {
		tmp = (1.0f / (((((float) M_PI) + ((float) M_PI)) * r) / 0.25f)) * ((1.0f / (s / t_1)) + (1.0f / (s + (0.3333333333333333f * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(-r) / s)
	t_1 = exp(t_0)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(Float32(0.25) * t_1) / 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))) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(pi) * r) / Float32(Float32(Float32(-exp(Float32(Float32(0.3333333333333333) * t_0))) - t_1) * Float32(-0.25))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(pi) + Float32(pi)) * r) / Float32(0.25))) * Float32(Float32(Float32(1.0) / Float32(s / t_1)) + Float32(Float32(1.0) / Float32(s + Float32(Float32(0.3333333333333333) * r)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = -r / s;
	t_1 = exp(t_0);
	tmp = single(0.0);
	if ((((single(0.25) * t_1) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r))) <= single(0.0020000000949949026))
		tmp = single(1.0) / ((single(pi) * r) / ((-exp((single(0.3333333333333333) * t_0)) - t_1) * single(-0.25)));
	else
		tmp = (single(1.0) / (((single(pi) + single(pi)) * r) / single(0.25))) * ((single(1.0) / (s / t_1)) + (single(1.0) / (s + (single(0.3333333333333333) * r))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.00200000009

   1. Initial program 99.5%

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

   3. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi + \pi}{-0.25}} \cdot \left(\frac{1}{\frac{r}{1}} \cdot \left(\left(-e^{\frac{-r}{s} \cdot 0.3333333333333333}\right) - e^{\frac{-r}{s}}\right)\right)} \]

   5. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi \cdot r}{\left(\left(-e^{0.3333333333333333 \cdot \frac{-r}{s}}\right) - e^{\frac{-r}{s}}\right) \cdot -0.25}}} \]

   if 0.00200000009 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\frac{s}{e^{\frac{-r}{s} \cdot 0.3333333333333333}}}\right)} \]

   4. Taylor expanded in r around 0

      \[\leadsto \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\color{blue}{s + \frac{1}{3} \cdot r}}\right) \]
   5. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 13.0%

         \[\leadsto \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{s + 0.3333333333333333 \cdot \color{blue}{r}}\right) \]

   6. Applied rewrites 13.0%

      \[\leadsto \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\color{blue}{s + 0.3333333333333333 \cdot r}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{-r}{s}\\ t_1 := e^{t\_0}\\ \mathbf{if}\;\frac{0.25 \cdot t\_1}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0.0020000000949949026:\\ \;\;\;\;\left(\frac{1}{\pi \cdot r} \cdot \left(e^{0.3333333333333333 \cdot t\_0} + t\_1\right)\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{t\_1}} + \frac{1}{s + 0.3333333333333333 \cdot r}\right)\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ (- r) s)) (t_1 (exp t_0)))
  (if (<=
       (+
        (/ (* 0.25 t_1) (* (* (* 2.0 PI) s) r))
        (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
       0.0020000000949949026)
    (*
     (* (/ 1.0 (* PI r)) (+ (exp (* 0.3333333333333333 t_0)) t_1))
     0.25)
    (*
     (/ 1.0 (/ (* (+ PI PI) r) 0.25))
     (+ (/ 1.0 (/ s t_1)) (/ 1.0 (+ s (* 0.3333333333333333 r))))))))

C

float code(float s, float r) {
	float t_0 = -r / s;
	float t_1 = expf(t_0);
	float tmp;
	if ((((0.25f * t_1) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r))) <= 0.0020000000949949026f) {
		tmp = ((1.0f / (((float) M_PI) * r)) * (expf((0.3333333333333333f * t_0)) + t_1)) * 0.25f;
	} else {
		tmp = (1.0f / (((((float) M_PI) + ((float) M_PI)) * r) / 0.25f)) * ((1.0f / (s / t_1)) + (1.0f / (s + (0.3333333333333333f * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(-r) / s)
	t_1 = exp(t_0)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(Float32(0.25) * t_1) / 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))) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(Float32(pi) * r)) * Float32(exp(Float32(Float32(0.3333333333333333) * t_0)) + t_1)) * Float32(0.25));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(pi) + Float32(pi)) * r) / Float32(0.25))) * Float32(Float32(Float32(1.0) / Float32(s / t_1)) + Float32(Float32(1.0) / Float32(s + Float32(Float32(0.3333333333333333) * r)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = -r / s;
	t_1 = exp(t_0);
	tmp = single(0.0);
	if ((((single(0.25) * t_1) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r))) <= single(0.0020000000949949026))
		tmp = ((single(1.0) / (single(pi) * r)) * (exp((single(0.3333333333333333) * t_0)) + t_1)) * single(0.25);
	else
		tmp = (single(1.0) / (((single(pi) + single(pi)) * r) / single(0.25))) * ((single(1.0) / (s / t_1)) + (single(1.0) / (s + (single(0.3333333333333333) * r))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.00200000009

   1. Initial program 99.5%

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

   3. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi + \pi}{-0.25}} \cdot \left(\frac{1}{\frac{r}{1}} \cdot \left(\left(-e^{\frac{-r}{s} \cdot 0.3333333333333333}\right) - e^{\frac{-r}{s}}\right)\right)} \]

   5. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\left(\frac{1}{\pi \cdot r} \cdot \left(e^{0.3333333333333333 \cdot \frac{-r}{s}} + e^{\frac{-r}{s}}\right)\right) \cdot 0.25} \]

   if 0.00200000009 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 99.5%

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

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\frac{s}{e^{\frac{-r}{s} \cdot 0.3333333333333333}}}\right)} \]

   4. Taylor expanded in r around 0

      \[\leadsto \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\color{blue}{s + \frac{1}{3} \cdot r}}\right) \]
   5. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 13.0%

         \[\leadsto \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{s + 0.3333333333333333 \cdot \color{blue}{r}}\right) \]

   6. Applied rewrites 13.0%

      \[\leadsto \frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\color{blue}{s + 0.3333333333333333 \cdot r}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\\ \mathbf{if}\;t\_0 + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{0.75}{\left(18.84955596923828 \cdot s\right) \cdot r}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))))
  (if (<=
       (+
        t_0
        (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
       0.0)
    0.0
    (+ t_0 (/ 0.75 (* (* 18.84955596923828 s) r))))))

C

float code(float s, float r) {
	float t_0 = (0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r);
	float tmp;
	if ((t_0 + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r))) <= 0.0f) {
		tmp = 0.0f;
	} else {
		tmp = t_0 + (0.75f / ((18.84955596923828f * s) * r));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r))
	tmp = Float32(0.0)
	if (Float32(t_0 + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r))) <= Float32(0.0))
		tmp = Float32(0.0);
	else
		tmp = Float32(t_0 + Float32(Float32(0.75) / Float32(Float32(Float32(18.84955596923828) * s) * r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = (single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r);
	tmp = single(0.0);
	if ((t_0 + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r))) <= single(0.0))
		tmp = single(0.0);
	else
		tmp = t_0 + (single(0.75) / ((single(18.84955596923828) * s) * r));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   1. Initial program 99.5%

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

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\frac{s}{e^{\frac{-r}{s} \cdot 0.3333333333333333}}}\right)} \]

   5. Applied rewrites 87.4%

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

   7. Applied rewrites 88.2%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 99.5%

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

   2. Taylor expanded in s around inf

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   3. Step-by-step derivation

   4. Applied rewrites 9.7%

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

   5. Evaluated real constant 9.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75}{\left(\color{blue}{18.84955596923828} \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r\\ \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{t\_0} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\pi \cdot \left(2 \cdot \left(s \cdot r\right)\right)} + \frac{0.75}{t\_0}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (* (* (* 6.0 PI) s) r)))
  (if (<=
       (+
        (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
        (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) t_0))
       0.0)
    0.0
    (+ (/ 0.25 (* PI (* 2.0 (* s r)))) (/ 0.75 t_0)))))

C

float code(float s, float r) {
	float t_0 = ((6.0f * ((float) M_PI)) * s) * r;
	float tmp;
	if ((((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / t_0)) <= 0.0f) {
		tmp = 0.0f;
	} else {
		tmp = (0.25f / (((float) M_PI) * (2.0f * (s * r)))) + (0.75f / t_0);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)
	tmp = Float32(0.0)
	if (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)))) / t_0)) <= Float32(0.0))
		tmp = Float32(0.0);
	else
		tmp = Float32(Float32(Float32(0.25) / Float32(Float32(pi) * Float32(Float32(2.0) * Float32(s * r)))) + Float32(Float32(0.75) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = ((single(6.0) * single(pi)) * s) * r;
	tmp = single(0.0);
	if ((((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / t_0)) <= single(0.0))
		tmp = single(0.0);
	else
		tmp = (single(0.25) / (single(pi) * (single(2.0) * (s * r)))) + (single(0.75) / t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   1. Initial program 99.5%

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

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\frac{s}{e^{\frac{-r}{s} \cdot 0.3333333333333333}}}\right)} \]

   5. Applied rewrites 87.4%

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

   7. Applied rewrites 88.2%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 99.5%

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

   2. Taylor expanded in s around inf

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   3. Step-by-step derivation

   4. Applied rewrites 9.7%

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

   5. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{4}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   6. Step-by-step derivation

   7. Applied rewrites 9.2%

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

   9. Applied rewrites 9.2%

      \[\leadsto \frac{0.25}{\color{blue}{\pi \cdot \left(2 \cdot \left(s \cdot r\right)\right)}} + \frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r\\ \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{t\_0} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{t\_0} + \frac{0.75}{\left(18.84955596923828 \cdot s\right) \cdot r}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (* (* (* 2.0 PI) s) r)))
  (if (<=
       (+
        (/ (* 0.25 (exp (/ (- r) s))) t_0)
        (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r)))
       0.0)
    0.0
    (+ (/ 0.25 t_0) (/ 0.75 (* (* 18.84955596923828 s) r))))))

C

float code(float s, float r) {
	float t_0 = ((2.0f * ((float) M_PI)) * s) * r;
	float tmp;
	if ((((0.25f * expf((-r / s))) / t_0) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r))) <= 0.0f) {
		tmp = 0.0f;
	} else {
		tmp = (0.25f / t_0) + (0.75f / ((18.84955596923828f * s) * r));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / t_0) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r))) <= Float32(0.0))
		tmp = Float32(0.0);
	else
		tmp = Float32(Float32(Float32(0.25) / t_0) + Float32(Float32(0.75) / Float32(Float32(Float32(18.84955596923828) * s) * r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = ((single(2.0) * single(pi)) * s) * r;
	tmp = single(0.0);
	if ((((single(0.25) * exp((-r / s))) / t_0) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r))) <= single(0.0))
		tmp = single(0.0);
	else
		tmp = (single(0.25) / t_0) + (single(0.75) / ((single(18.84955596923828) * s) * r));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   1. Initial program 99.5%

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

   3. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\frac{s}{e^{\frac{-r}{s} \cdot 0.3333333333333333}}}\right)} \]

   5. Applied rewrites 87.4%

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

   7. Applied rewrites 88.2%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 99.5%

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

   2. Taylor expanded in s around inf

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   3. Step-by-step derivation

   4. Applied rewrites 9.7%

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

   5. Taylor expanded in s around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{4}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   6. Step-by-step derivation

   7. Applied rewrites 9.2%

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

   8. Evaluated real constant 9.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75}{\left(\color{blue}{18.84955596923828} \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{-r}{s}\\ \left(\frac{1}{\pi \cdot r} \cdot \left(e^{0.3333333333333333 \cdot t\_0} + e^{t\_0}\right)\right) \cdot 0.25 \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ (- r) s)))
  (*
   (* (/ 1.0 (* PI r)) (+ (exp (* 0.3333333333333333 t_0)) (exp t_0)))
   0.25)))

C

float code(float s, float r) {
	float t_0 = -r / s;
	return ((1.0f / (((float) M_PI) * r)) * (expf((0.3333333333333333f * t_0)) + expf(t_0))) * 0.25f;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(-r) / s)
	return Float32(Float32(Float32(Float32(1.0) / Float32(Float32(pi) * r)) * Float32(exp(Float32(Float32(0.3333333333333333) * t_0)) + exp(t_0))) * Float32(0.25))
end

MATLAB

function tmp = code(s, r)
	t_0 = -r / s;
	tmp = ((single(1.0) / (single(pi) * r)) * (exp((single(0.3333333333333333) * t_0)) + exp(t_0))) * single(0.25);
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 90.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\pi + \pi}{-0.25}} \cdot \left(\frac{1}{\frac{r}{1}} \cdot \left(\left(-e^{\frac{-r}{s} \cdot 0.3333333333333333}\right) - e^{\frac{-r}{s}}\right)\right)} \]

5. Applied rewrites 90.0%

   \[\leadsto \color{blue}{\left(\frac{1}{\pi \cdot r} \cdot \left(e^{0.3333333333333333 \cdot \frac{-r}{s}} + e^{\frac{-r}{s}}\right)\right) \cdot 0.25} \]
6. Add Preprocessing

Alternative 10: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-r}{s}\\ \frac{1}{\pi \cdot r} \cdot \left(e^{0.3333333333333333 \cdot t\_0} + e^{t\_0}\right) \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ (- r) s)))
  (* (/ 1.0 (* PI r)) (+ (exp (* 0.3333333333333333 t_0)) (exp t_0)))))

C

float code(float s, float r) {
	float t_0 = -r / s;
	return (1.0f / (((float) M_PI) * r)) * (expf((0.3333333333333333f * t_0)) + expf(t_0));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(-r) / s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(pi) * r)) * Float32(exp(Float32(Float32(0.3333333333333333) * t_0)) + exp(t_0)))
end

MATLAB

function tmp = code(s, r)
	t_0 = -r / s;
	tmp = (single(1.0) / (single(pi) * r)) * (exp((single(0.3333333333333333) * t_0)) + exp(t_0));
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 90.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\pi + \pi}{-0.25}} \cdot \left(\frac{1}{\frac{r}{1}} \cdot \left(\left(-e^{\frac{-r}{s} \cdot 0.3333333333333333}\right) - e^{\frac{-r}{s}}\right)\right)} \]

5. Applied rewrites 90.0%

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

Alternative 11: 89.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-r}{s}\\ \frac{1}{r} \cdot \left(\left(e^{t\_0} - e^{0.3333333333333333 \cdot t\_0}\right) \cdot -0.039788734167814255\right) \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ (- r) s)))
  (*
   (/ 1.0 r)
   (*
    (- (exp t_0) (exp (* 0.3333333333333333 t_0)))
    -0.039788734167814255))))

C

float code(float s, float r) {
	float t_0 = -r / s;
	return (1.0f / r) * ((expf(t_0) - expf((0.3333333333333333f * t_0))) * -0.039788734167814255f);
}

Fortran

real(4) function code(s, r)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: r
    real(4) :: t_0
    t_0 = -r / s
    code = (1.0e0 / r) * ((exp(t_0) - exp((0.3333333333333333e0 * t_0))) * (-0.039788734167814255e0))
end function

Julia

function code(s, r)
	t_0 = Float32(Float32(-r) / s)
	return Float32(Float32(Float32(1.0) / r) * Float32(Float32(exp(t_0) - exp(Float32(Float32(0.3333333333333333) * t_0))) * Float32(-0.039788734167814255)))
end

MATLAB

function tmp = code(s, r)
	t_0 = -r / s;
	tmp = (single(1.0) / r) * ((exp(t_0) - exp((single(0.3333333333333333) * t_0))) * single(-0.039788734167814255));
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 89.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{\pi + \pi}{-0.25}} \cdot \left(\frac{1}{\frac{r}{1}} \cdot \left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot 0.3333333333333333}\right)\right)} \]

4. Evaluated real constant 89.7%

   \[\leadsto \color{blue}{-0.039788734167814255} \cdot \left(\frac{1}{\frac{r}{1}} \cdot \left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot 0.3333333333333333}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{-10680707}{268435456} \cdot \left(\frac{1}{\frac{r}{1}} \cdot \left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot \frac{1}{3}}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\frac{r}{1}} \cdot \left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot \frac{1}{3}}\right)\right) \cdot \frac{-10680707}{268435456}} \]

   3. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\frac{r}{1}} \cdot \left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot \frac{1}{3}}\right)\right)} \cdot \frac{-10680707}{268435456} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{r}{1}} \cdot \left(\left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot \frac{1}{3}}\right) \cdot \frac{-10680707}{268435456}\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{r}{1}} \cdot \left(\left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot \frac{1}{3}}\right) \cdot \frac{-10680707}{268435456}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{r}{1}}} \cdot \left(\left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot \frac{1}{3}}\right) \cdot \frac{-10680707}{268435456}\right) \]

   7. /-rgt-identity N/A

      \[\leadsto \frac{1}{\color{blue}{r}} \cdot \left(\left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot \frac{1}{3}}\right) \cdot \frac{-10680707}{268435456}\right) \]

   8. lower-*.f32 89.7%

      \[\leadsto \frac{1}{r} \cdot \color{blue}{\left(\left(e^{\frac{-r}{s}} - e^{\frac{-r}{s} \cdot 0.3333333333333333}\right) \cdot -0.039788734167814255\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{1}{r} \cdot \left(\left(e^{\frac{-r}{s}} - e^{\color{blue}{\frac{-r}{s} \cdot \frac{1}{3}}}\right) \cdot \frac{-10680707}{268435456}\right) \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{r} \cdot \left(\left(e^{\frac{-r}{s}} - e^{\color{blue}{\frac{1}{3} \cdot \frac{-r}{s}}}\right) \cdot \frac{-10680707}{268435456}\right) \]

   11. lower-*.f32 89.7%

       \[\leadsto \frac{1}{r} \cdot \left(\left(e^{\frac{-r}{s}} - e^{\color{blue}{0.3333333333333333 \cdot \frac{-r}{s}}}\right) \cdot -0.039788734167814255\right) \]

6. Applied rewrites 89.7%

   \[\leadsto \color{blue}{\frac{1}{r} \cdot \left(\left(e^{\frac{-r}{s}} - e^{0.3333333333333333 \cdot \frac{-r}{s}}\right) \cdot -0.039788734167814255\right)} \]
7. Add Preprocessing

Alternative 12: 88.2% accurate, 69.8× speedup

Math

\[0 \]

FPCore

(FPCore (s r)
  :precision binary32
  0.0)

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\left(\pi + \pi\right) \cdot r}{0.25}} \cdot \left(\frac{1}{\frac{s}{e^{\frac{-r}{s}}}} + \frac{1}{\frac{s}{e^{\frac{-r}{s} \cdot 0.3333333333333333}}}\right)} \]

5. Applied rewrites 87.4%

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

7. Applied rewrites 88.2%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(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))))