Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 100.0%

Time: 14.5s

Alternatives: 7

Speedup: 21.8×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (cast (! :precision binary64 (* s (- (log1p (* u -4.0)))))))

C

float code(float s, float u) {
	double tmp = ((double) s) * -log1p((((double) u) * -4.0));
	return (float) tmp;
}

Julia

function code(s, u)
	tmp = Float64(Float64(s) * Float64(-log1p(Float64(Float64(u) * -4.0))))
	return Float32(tmp)
end

Derivation

1. Initial program 100.0%

   \[\langle s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right) \rangle_{\text{binary64}} \]

2. Final simplification 100.0%

   \[\leadsto \langle s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right) \rangle_{\text{binary64}} \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (- (log1p (* u -4.0)))))

C

float code(float s, float u) {
	return s * -log1pf((u * -4.0f));
}

Julia

function code(s, u)
	return Float32(s * Float32(-log1p(Float32(u * Float32(-4.0)))))
end

Derivation

1. Initial program 57.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Step-by-step derivation

   1. log-rec 59.7%

      \[\leadsto s \cdot \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \]

   2. sub-neg 59.7%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + \left(-4 \cdot u\right)\right)}\right) \]

   3. distribute-rgt-neg-out 59.7%

      \[\leadsto s \cdot \left(-\log \left(1 + \color{blue}{4 \cdot \left(-u\right)}\right)\right) \]

   4. log1p-def 99.4%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(4 \cdot \left(-u\right)\right)}\right) \]

   5. distribute-rgt-neg-out 99.4%

      \[\leadsto s \cdot \left(-\mathsf{log1p}\left(\color{blue}{-4 \cdot u}\right)\right) \]

   6. *-commutative 99.4%

      \[\leadsto s \cdot \left(-\mathsf{log1p}\left(-\color{blue}{u \cdot 4}\right)\right) \]

   7. distribute-rgt-neg-in 99.4%

      \[\leadsto s \cdot \left(-\mathsf{log1p}\left(\color{blue}{u \cdot \left(-4\right)}\right)\right) \]

   8. metadata-eval 99.4%

      \[\leadsto s \cdot \left(-\mathsf{log1p}\left(u \cdot \color{blue}{-4}\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right)} \]

4. Final simplification 99.4%

   \[\leadsto s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right) \]

Alternative 3: 92.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \frac{s}{\left(u \cdot -0.3333333333333333 + 0.25 \cdot \frac{1}{u}\right) - 0.5} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (/ s (- (+ (* u -0.3333333333333333) (* 0.25 (/ 1.0 u))) 0.5)))

C

float code(float s, float u) {
	return s / (((u * -0.3333333333333333f) + (0.25f * (1.0f / u))) - 0.5f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s / (((u * (-0.3333333333333333e0)) + (0.25e0 * (1.0e0 / u))) - 0.5e0)
end function

Julia

function code(s, u)
	return Float32(s / Float32(Float32(Float32(u * Float32(-0.3333333333333333)) + Float32(Float32(0.25) * Float32(Float32(1.0) / u))) - Float32(0.5)))
end

MATLAB

function tmp = code(s, u)
	tmp = s / (((u * single(-0.3333333333333333)) + (single(0.25) * (single(1.0) / u))) - single(0.5));
end

Derivation

1. Initial program 57.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Step-by-step derivation

   1. log-rec 59.7%

      \[\leadsto s \cdot \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \]

   2. cancel-sign-sub-inv 59.7%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + \left(-4\right) \cdot u\right)}\right) \]

   3. metadata-eval 59.7%

      \[\leadsto s \cdot \left(-\log \left(1 + \color{blue}{-4} \cdot u\right)\right) \]

   4. *-commutative 59.7%

      \[\leadsto s \cdot \left(-\log \left(1 + \color{blue}{u \cdot -4}\right)\right) \]

   5. log1p-udef 99.4%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   6. neg-sub0 99.4%

      \[\leadsto s \cdot \color{blue}{\left(0 - \mathsf{log1p}\left(u \cdot -4\right)\right)} \]

   7. flip-- 99.2%

      \[\leadsto s \cdot \color{blue}{\frac{0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)}{0 + \mathsf{log1p}\left(u \cdot -4\right)}} \]

   8. +-lft-identity 99.2%

      \[\leadsto s \cdot \frac{0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)}{\color{blue}{\mathsf{log1p}\left(u \cdot -4\right)}} \]

   9. div-inv 99.1%

      \[\leadsto s \cdot \color{blue}{\left(\left(0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \]

   10. metadata-eval 99.1%

       \[\leadsto s \cdot \left(\left(\color{blue}{0} - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   11. sub0-neg 99.1%

       \[\leadsto s \cdot \left(\color{blue}{\left(-\mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   12. pow2 99.1%

       \[\leadsto s \cdot \left(\left(-\color{blue}{{\left(\mathsf{log1p}\left(u \cdot -4\right)\right)}^{2}}\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

3. Applied egg-rr 99.1%

   \[\leadsto s \cdot \color{blue}{\left(\left(-{\left(\mathsf{log1p}\left(u \cdot -4\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \]

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{s}{\frac{\frac{s}{-\mathsf{log1p}\left(u \cdot -4\right)}}{s}}} \]

6. Taylor expanded in u around 0 94.0%

   \[\leadsto \frac{s}{\color{blue}{\left(-0.3333333333333333 \cdot u + 0.25 \cdot \frac{1}{u}\right) - 0.5}} \]

7. Final simplification 94.0%

   \[\leadsto \frac{s}{\left(u \cdot -0.3333333333333333 + 0.25 \cdot \frac{1}{u}\right) - 0.5} \]

Alternative 4: 88.4% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \frac{s}{\frac{0.25}{u} + -0.5} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (/ s (+ (/ 0.25 u) -0.5)))

C

float code(float s, float u) {
	return s / ((0.25f / u) + -0.5f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s / ((0.25e0 / u) + (-0.5e0))
end function

Julia

function code(s, u)
	return Float32(s / Float32(Float32(Float32(0.25) / u) + Float32(-0.5)))
end

MATLAB

function tmp = code(s, u)
	tmp = s / ((single(0.25) / u) + single(-0.5));
end

Derivation

1. Initial program 57.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Step-by-step derivation

   1. log-rec 59.7%

      \[\leadsto s \cdot \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \]

   2. cancel-sign-sub-inv 59.7%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + \left(-4\right) \cdot u\right)}\right) \]

   3. metadata-eval 59.7%

      \[\leadsto s \cdot \left(-\log \left(1 + \color{blue}{-4} \cdot u\right)\right) \]

   4. *-commutative 59.7%

      \[\leadsto s \cdot \left(-\log \left(1 + \color{blue}{u \cdot -4}\right)\right) \]

   5. log1p-udef 99.4%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   6. neg-sub0 99.4%

      \[\leadsto s \cdot \color{blue}{\left(0 - \mathsf{log1p}\left(u \cdot -4\right)\right)} \]

   7. flip-- 99.2%

      \[\leadsto s \cdot \color{blue}{\frac{0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)}{0 + \mathsf{log1p}\left(u \cdot -4\right)}} \]

   8. +-lft-identity 99.2%

      \[\leadsto s \cdot \frac{0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)}{\color{blue}{\mathsf{log1p}\left(u \cdot -4\right)}} \]

   9. div-inv 99.1%

      \[\leadsto s \cdot \color{blue}{\left(\left(0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \]

   10. metadata-eval 99.1%

       \[\leadsto s \cdot \left(\left(\color{blue}{0} - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   11. sub0-neg 99.1%

       \[\leadsto s \cdot \left(\color{blue}{\left(-\mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   12. pow2 99.1%

       \[\leadsto s \cdot \left(\left(-\color{blue}{{\left(\mathsf{log1p}\left(u \cdot -4\right)\right)}^{2}}\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

3. Applied egg-rr 99.1%

   \[\leadsto s \cdot \color{blue}{\left(\left(-{\left(\mathsf{log1p}\left(u \cdot -4\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \]

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{s}{\frac{\frac{s}{-\mathsf{log1p}\left(u \cdot -4\right)}}{s}}} \]

6. Taylor expanded in u around 0 90.4%

   \[\leadsto \frac{s}{\color{blue}{0.25 \cdot \frac{1}{u} - 0.5}} \]
7. Step-by-step derivation

   1. sub-neg 90.4%

      \[\leadsto \frac{s}{\color{blue}{0.25 \cdot \frac{1}{u} + \left(-0.5\right)}} \]

   2. associate-*r/ 90.4%

      \[\leadsto \frac{s}{\color{blue}{\frac{0.25 \cdot 1}{u}} + \left(-0.5\right)} \]

   3. metadata-eval 90.4%

      \[\leadsto \frac{s}{\frac{\color{blue}{0.25}}{u} + \left(-0.5\right)} \]

   4. metadata-eval 90.4%

      \[\leadsto \frac{s}{\frac{0.25}{u} + \color{blue}{-0.5}} \]

8. Simplified 90.4%

   \[\leadsto \frac{s}{\color{blue}{\frac{0.25}{u} + -0.5}} \]

9. Final simplification 90.4%

   \[\leadsto \frac{s}{\frac{0.25}{u} + -0.5} \]

Alternative 5: 73.4% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(s \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* 4.0 (* s u)))

C

float code(float s, float u) {
	return 4.0f * (s * u);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 4.0e0 * (s * u)
end function

Julia

function code(s, u)
	return Float32(Float32(4.0) * Float32(s * u))
end

MATLAB

function tmp = code(s, u)
	tmp = single(4.0) * (s * u);
end

Derivation

1. Initial program 57.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 75.9%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]

3. Final simplification 75.9%

   \[\leadsto 4 \cdot \left(s \cdot u\right) \]

Alternative 6: 73.6% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* u 4.0)))

C

float code(float s, float u) {
	return s * (u * 4.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * 4.0e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(4.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * single(4.0));
end

Derivation

1. Initial program 57.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 76.1%

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

   1. *-commutative 76.1%

      \[\leadsto s \cdot \color{blue}{\left(u \cdot 4\right)} \]

4. Simplified 76.1%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot 4\right)} \]

5. Final simplification 76.1%

   \[\leadsto s \cdot \left(u \cdot 4\right) \]

Alternative 7: 8.1% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ \frac{s}{-0.5} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (/ s -0.5))

C

float code(float s, float u) {
	return s / -0.5f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s / (-0.5e0)
end function

Julia

function code(s, u)
	return Float32(s / Float32(-0.5))
end

MATLAB

function tmp = code(s, u)
	tmp = s / single(-0.5);
end

Derivation

1. Initial program 57.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Step-by-step derivation

   1. log-rec 59.7%

      \[\leadsto s \cdot \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \]

   2. cancel-sign-sub-inv 59.7%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + \left(-4\right) \cdot u\right)}\right) \]

   3. metadata-eval 59.7%

      \[\leadsto s \cdot \left(-\log \left(1 + \color{blue}{-4} \cdot u\right)\right) \]

   4. *-commutative 59.7%

      \[\leadsto s \cdot \left(-\log \left(1 + \color{blue}{u \cdot -4}\right)\right) \]

   5. log1p-udef 99.4%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   6. neg-sub0 99.4%

      \[\leadsto s \cdot \color{blue}{\left(0 - \mathsf{log1p}\left(u \cdot -4\right)\right)} \]

   7. flip-- 99.2%

      \[\leadsto s \cdot \color{blue}{\frac{0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)}{0 + \mathsf{log1p}\left(u \cdot -4\right)}} \]

   8. +-lft-identity 99.2%

      \[\leadsto s \cdot \frac{0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)}{\color{blue}{\mathsf{log1p}\left(u \cdot -4\right)}} \]

   9. div-inv 99.1%

      \[\leadsto s \cdot \color{blue}{\left(\left(0 \cdot 0 - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \]

   10. metadata-eval 99.1%

       \[\leadsto s \cdot \left(\left(\color{blue}{0} - \mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   11. sub0-neg 99.1%

       \[\leadsto s \cdot \left(\color{blue}{\left(-\mathsf{log1p}\left(u \cdot -4\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

   12. pow2 99.1%

       \[\leadsto s \cdot \left(\left(-\color{blue}{{\left(\mathsf{log1p}\left(u \cdot -4\right)\right)}^{2}}\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right) \]

3. Applied egg-rr 99.1%

   \[\leadsto s \cdot \color{blue}{\left(\left(-{\left(\mathsf{log1p}\left(u \cdot -4\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \]

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{s}{\frac{\frac{s}{-\mathsf{log1p}\left(u \cdot -4\right)}}{s}}} \]

6. Taylor expanded in u around 0 90.3%

   \[\leadsto \frac{s}{\frac{\color{blue}{-0.5 \cdot s + 0.25 \cdot \frac{s}{u}}}{s}} \]

7. Taylor expanded in u around inf 7.8%

   \[\leadsto \frac{s}{\color{blue}{-0.5}} \]

8. Final simplification 7.8%

   \[\leadsto \frac{s}{-0.5} \]

Reproduce

herbie shell --seed 2023297 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))