Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 100.0%

Time: 17.5s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Initial Program: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (* s (* -3.0 (log1p (+ (* -1.3333333333333333 u) 0.3333333333333333)))))))

C

float code(float s, float u) {
	double tmp = ((double) s) * (-3.0 * log1p(((-1.3333333333333333 * ((double) u)) + 0.3333333333333333)));
	return (float) tmp;
}

Julia

function code(s, u)
	tmp = Float64(Float64(s) * Float64(-3.0 * log1p(Float64(Float64(-1.3333333333333333 * Float64(u)) + 0.3333333333333333))))
	return Float32(tmp)
end

Derivation

1. Initial program 100.0%

   \[\langle s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right) \rangle_{\text{binary64}} \]

2. Final simplification 100.0%

   \[\leadsto \langle s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right) \rangle_{\text{binary64}} \]

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \langle \left( \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (cast
   (!
    :precision
    binary64
    (* -3.0 (log1p (+ (* -1.3333333333333333 u) 0.3333333333333333)))))))

C

float code(float s, float u) {
	double tmp = -3.0 * log1p(((-1.3333333333333333 * ((double) u)) + 0.3333333333333333));
	return s * ((float) tmp);
}

Julia

function code(s, u)
	tmp = Float64(-3.0 * log1p(Float64(Float64(-1.3333333333333333 * Float64(u)) + 0.3333333333333333)))
	return Float32(s * Float32(tmp))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. *-commutative 95.9%

      \[\leadsto \color{blue}{\left(s \cdot 3\right)} \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

   2. associate-*l* 95.9%

      \[\leadsto \color{blue}{s \cdot \left(3 \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

   3. log-rec 96.9%

      \[\leadsto s \cdot \left(3 \cdot \color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]

   4. neg-mul-1 96.9%

      \[\leadsto s \cdot \left(3 \cdot \color{blue}{\left(-1 \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]

   5. associate-*r* 96.9%

      \[\leadsto s \cdot \color{blue}{\left(\left(3 \cdot -1\right) \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)} \]

   6. metadata-eval 96.9%

      \[\leadsto s \cdot \left(\color{blue}{-3} \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right) \]

   7. sub-neg 96.9%

      \[\leadsto s \cdot \left(-3 \cdot \log \color{blue}{\left(1 + \left(-\frac{u - 0.25}{0.75}\right)\right)}\right) \]

   8. log1p-def 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)}\right) \]

   9. distribute-neg-frac 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right) \]

   10. neg-mul-1 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot \left(u - 0.25\right)}}{0.75}\right)\right) \]

   11. associate-/l* 98.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{\frac{0.75}{u - 0.25}}}\right)\right) \]

   12. associate-/r/ 98.1%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{0.75} \cdot \left(u - 0.25\right)}\right)\right) \]

   13. sub-neg 98.1%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{-1}{0.75} \cdot \color{blue}{\left(u + \left(-0.25\right)\right)}\right)\right) \]

   14. distribute-lft-in 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{0.75} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)}\right)\right) \]

   15. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)\right)\right) \]

   16. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + \color{blue}{-1.3333333333333333} \cdot \left(-0.25\right)\right)\right) \]

   17. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + -1.3333333333333333 \cdot \color{blue}{-0.25}\right)\right) \]

   18. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + \color{blue}{0.3333333333333333}\right)\right) \]

3. Simplified 97.0%

   \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right)} \]
4. Step-by-step derivation

   1. rewrite-binary32/binary64 99.3%

      \[\leadsto \color{blue}{s \cdot \langle \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right) \rangle_{\text{binary64}}} \]

5. Applied rewrite-once 99.3%

   \[\leadsto s \cdot \color{blue}{\langle \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right)} \rangle_{\text{binary64}}} \]

6. Final simplification 99.3%

   \[\leadsto s \cdot \langle \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right) \rangle_{\text{binary64}} \]

Alternative 3: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-3 \cdot \langle \left( \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right) \right)_{\text{binary64}} \rangle_{\text{binary32}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (*
   -3.0
   (cast
    (!
     :precision
     binary64
     (log1p (+ (* -1.3333333333333333 u) 0.3333333333333333)))))))

C

float code(float s, float u) {
	double tmp = log1p(((-1.3333333333333333 * ((double) u)) + 0.3333333333333333));
	return s * (-3.0f * ((float) tmp));
}

Julia

function code(s, u)
	tmp = log1p(Float64(Float64(-1.3333333333333333 * Float64(u)) + 0.3333333333333333))
	return Float32(s * Float32(Float32(-3.0) * Float32(tmp)))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. *-commutative 95.9%

      \[\leadsto \color{blue}{\left(s \cdot 3\right)} \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

   2. associate-*l* 95.9%

      \[\leadsto \color{blue}{s \cdot \left(3 \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

   3. log-rec 96.9%

      \[\leadsto s \cdot \left(3 \cdot \color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]

   4. neg-mul-1 96.9%

      \[\leadsto s \cdot \left(3 \cdot \color{blue}{\left(-1 \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]

   5. associate-*r* 96.9%

      \[\leadsto s \cdot \color{blue}{\left(\left(3 \cdot -1\right) \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)} \]

   6. metadata-eval 96.9%

      \[\leadsto s \cdot \left(\color{blue}{-3} \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right) \]

   7. sub-neg 96.9%

      \[\leadsto s \cdot \left(-3 \cdot \log \color{blue}{\left(1 + \left(-\frac{u - 0.25}{0.75}\right)\right)}\right) \]

   8. log1p-def 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)}\right) \]

   9. distribute-neg-frac 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right) \]

   10. neg-mul-1 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot \left(u - 0.25\right)}}{0.75}\right)\right) \]

   11. associate-/l* 98.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{\frac{0.75}{u - 0.25}}}\right)\right) \]

   12. associate-/r/ 98.1%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{0.75} \cdot \left(u - 0.25\right)}\right)\right) \]

   13. sub-neg 98.1%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{-1}{0.75} \cdot \color{blue}{\left(u + \left(-0.25\right)\right)}\right)\right) \]

   14. distribute-lft-in 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{0.75} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)}\right)\right) \]

   15. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)\right)\right) \]

   16. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + \color{blue}{-1.3333333333333333} \cdot \left(-0.25\right)\right)\right) \]

   17. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + -1.3333333333333333 \cdot \color{blue}{-0.25}\right)\right) \]

   18. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + \color{blue}{0.3333333333333333}\right)\right) \]

3. Simplified 97.0%

   \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right)} \]
4. Step-by-step derivation

   1. rewrite-binary32/binary64 98.9%

      \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \langle \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right) \rangle_{\text{binary64}}\right)} \]

5. Applied rewrite-once 98.9%

   \[\leadsto s \cdot \left(-3 \cdot \color{blue}{\langle \color{blue}{\mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)} \rangle_{\text{binary64}}}\right) \]

6. Final simplification 98.9%

   \[\leadsto s \cdot \left(-3 \cdot \langle \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right) \rangle_{\text{binary64}}\right) \]

Alternative 4: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -3 \cdot \left(s \cdot \log \left(1.3333333333333333 - u \cdot 1.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* -3.0 (* s (log (- 1.3333333333333333 (* u 1.3333333333333333))))))

C

float code(float s, float u) {
	return -3.0f * (s * logf((1.3333333333333333f - (u * 1.3333333333333333f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (-3.0e0) * (s * log((1.3333333333333333e0 - (u * 1.3333333333333333e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(-3.0) * Float32(s * log(Float32(Float32(1.3333333333333333) - Float32(u * Float32(1.3333333333333333))))))
end

MATLAB

function tmp = code(s, u)
	tmp = single(-3.0) * (s * log((single(1.3333333333333333) - (u * single(1.3333333333333333)))));
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. remove-double-neg 95.9%

      \[\leadsto \color{blue}{-\left(-\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

   2. distribute-rgt-neg-in 95.9%

      \[\leadsto -\color{blue}{\left(3 \cdot s\right) \cdot \left(-\log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

   3. distribute-rgt-neg-in 95.9%

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(-\left(-\log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)\right)} \]

   4. log-rec 96.8%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\left(-\color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right)\right) \]

   5. remove-double-neg 96.8%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\color{blue}{\log \left(1 - \frac{u - 0.25}{0.75}\right)}\right) \]

   6. div-sub 95.8%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\log \left(1 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right)\right) \]

   7. metadata-eval 95.8%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\log \left(1 - \left(\frac{u}{0.75} - \color{blue}{0.3333333333333333}\right)\right)\right) \]

3. Simplified 95.8%

   \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(-\log \left(1 - \left(\frac{u}{0.75} - 0.3333333333333333\right)\right)\right)} \]

4. Taylor expanded in s around 0 96.4%

   \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \left(1.3333333333333333 - 1.3333333333333333 \cdot u\right)\right)} \]

5. Final simplification 96.4%

   \[\leadsto -3 \cdot \left(s \cdot \log \left(1.3333333333333333 - u \cdot 1.3333333333333333\right)\right) \]

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -3 \cdot \left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* -3.0 (* s (log1p (* (- 0.25 u) 1.3333333333333333)))))

C

float code(float s, float u) {
	return -3.0f * (s * log1pf(((0.25f - u) * 1.3333333333333333f)));
}

Julia

function code(s, u)
	return Float32(Float32(-3.0) * Float32(s * log1p(Float32(Float32(Float32(0.25) - u) * Float32(1.3333333333333333)))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. *-commutative 95.9%

      \[\leadsto \color{blue}{\left(s \cdot 3\right)} \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

   2. associate-*l* 95.9%

      \[\leadsto \color{blue}{s \cdot \left(3 \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

   3. log-rec 96.9%

      \[\leadsto s \cdot \left(3 \cdot \color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]

   4. neg-mul-1 96.9%

      \[\leadsto s \cdot \left(3 \cdot \color{blue}{\left(-1 \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]

   5. associate-*r* 96.9%

      \[\leadsto s \cdot \color{blue}{\left(\left(3 \cdot -1\right) \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)} \]

   6. metadata-eval 96.9%

      \[\leadsto s \cdot \left(\color{blue}{-3} \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right) \]

   7. sub-neg 96.9%

      \[\leadsto s \cdot \left(-3 \cdot \log \color{blue}{\left(1 + \left(-\frac{u - 0.25}{0.75}\right)\right)}\right) \]

   8. log1p-def 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)}\right) \]

   9. distribute-neg-frac 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right) \]

   10. neg-mul-1 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot \left(u - 0.25\right)}}{0.75}\right)\right) \]

   11. associate-/l* 98.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{\frac{0.75}{u - 0.25}}}\right)\right) \]

   12. associate-/r/ 98.1%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{0.75} \cdot \left(u - 0.25\right)}\right)\right) \]

   13. sub-neg 98.1%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{-1}{0.75} \cdot \color{blue}{\left(u + \left(-0.25\right)\right)}\right)\right) \]

   14. distribute-lft-in 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{0.75} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)}\right)\right) \]

   15. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)\right)\right) \]

   16. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + \color{blue}{-1.3333333333333333} \cdot \left(-0.25\right)\right)\right) \]

   17. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + -1.3333333333333333 \cdot \color{blue}{-0.25}\right)\right) \]

   18. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + \color{blue}{0.3333333333333333}\right)\right) \]

3. Simplified 97.0%

   \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right)} \]
4. Step-by-step derivation

   1. metadata-eval 97.0%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(-1.3333333333333333\right)} \cdot u + 0.3333333333333333\right)\right) \]

   2. metadata-eval 97.0%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\color{blue}{\frac{1}{0.75}}\right) \cdot u + 0.3333333333333333\right)\right) \]

   3. distribute-lft-neg-in 97.0%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(-\frac{1}{0.75} \cdot u\right)} + 0.3333333333333333\right)\right) \]

   4. *-commutative 97.0%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\color{blue}{u \cdot \frac{1}{0.75}}\right) + 0.3333333333333333\right)\right) \]

   5. div-inv 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\color{blue}{\frac{u}{0.75}}\right) + 0.3333333333333333\right)\right) \]

   6. metadata-eval 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\frac{u}{0.75}\right) + \color{blue}{\left(--0.3333333333333333\right)}\right)\right) \]

   7. metadata-eval 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\frac{u}{0.75}\right) + \left(-\color{blue}{\left(-0.3333333333333333\right)}\right)\right)\right) \]

   8. distribute-neg-in 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{-\left(\frac{u}{0.75} + \left(-0.3333333333333333\right)\right)}\right)\right) \]

   9. sub-neg 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-\color{blue}{\left(\frac{u}{0.75} - 0.3333333333333333\right)}\right)\right) \]

   10. metadata-eval 96.7%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-\left(\frac{u}{0.75} - \color{blue}{\frac{0.25}{0.75}}\right)\right)\right) \]

   11. div-sub 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-\color{blue}{\frac{u - 0.25}{0.75}}\right)\right) \]

   12. distribute-frac-neg 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right) \]

   13. sub-neg 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{-\color{blue}{\left(u + \left(-0.25\right)\right)}}{0.75}\right)\right) \]

   14. +-commutative 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{-\color{blue}{\left(\left(-0.25\right) + u\right)}}{0.75}\right)\right) \]

   15. distribute-neg-in 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(-\left(-0.25\right)\right) + \left(-u\right)}}{0.75}\right)\right) \]

   16. metadata-eval 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\left(-\color{blue}{-0.25}\right) + \left(-u\right)}{0.75}\right)\right) \]

   17. metadata-eval 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{0.25} + \left(-u\right)}{0.75}\right)\right) \]

5. Applied egg-rr 98.5%

   \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{0.25 + \left(-u\right)}{0.75}}\right)\right) \]
6. Step-by-step derivation

   1. sub-neg 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{0.25 - u}}{0.75}\right)\right) \]

7. Simplified 98.5%

   \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right)\right) \]

8. Taylor expanded in s around 0 96.5%

   \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \left(1 + 1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)} \]
9. Step-by-step derivation

   1. log1p-def 97.9%

      \[\leadsto -3 \cdot \left(s \cdot \color{blue}{\mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right)}\right) \]

10. Simplified 97.9%

    \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)} \]

11. Final simplification 97.9%

    \[\leadsto -3 \cdot \left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right) \]

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -3 \cdot \left(s \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* -3.0 (* s (log1p (/ (- 0.25 u) 0.75)))))

C

float code(float s, float u) {
	return -3.0f * (s * log1pf(((0.25f - u) / 0.75f)));
}

Julia

function code(s, u)
	return Float32(Float32(-3.0) * Float32(s * log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75)))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. remove-double-neg 95.9%

      \[\leadsto \color{blue}{-\left(-\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

   2. distribute-rgt-neg-in 95.9%

      \[\leadsto -\color{blue}{\left(3 \cdot s\right) \cdot \left(-\log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

   3. distribute-rgt-neg-in 95.9%

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(-\left(-\log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)\right)} \]

   4. log-rec 96.8%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\left(-\color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right)\right) \]

   5. remove-double-neg 96.8%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\color{blue}{\log \left(1 - \frac{u - 0.25}{0.75}\right)}\right) \]

   6. sub-neg 96.8%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\log \color{blue}{\left(1 + \left(-\frac{u - 0.25}{0.75}\right)\right)}\right) \]

   7. log1p-def 98.5%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)}\right) \]

   8. distribute-neg-frac 98.5%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right) \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)\right)} \]
4. Step-by-step derivation

   1. add-exp-log_binary32 90.7%

      \[\leadsto \color{blue}{e^{\log \left(\left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)\right)\right)}} \]

5. Applied rewrite-once 90.7%

   \[\leadsto \color{blue}{e^{\log \left(\left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)\right)\right)}} \]
6. Step-by-step derivation

   1. rem-exp-log 98.5%

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)\right)} \]

   2. distribute-rgt-neg-out 98.5%

      \[\leadsto \color{blue}{-\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)} \]

   3. distribute-lft-neg-in 98.5%

      \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)} \]

   4. log1p-expm1 98.4%

      \[\leadsto \left(-3 \cdot s\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)\right)\right)} \]

   5. log1p-def 96.8%

      \[\leadsto \left(-3 \cdot s\right) \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)\right)\right)} \]

   6. expm1-log1p 96.8%

      \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(1 + \color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right) \]

   7. distribute-frac-neg 96.8%

      \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(1 + \color{blue}{\left(-\frac{u - 0.25}{0.75}\right)}\right) \]

   8. div-sub 95.8%

      \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(1 + \left(-\color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right)\right) \]

   9. metadata-eval 95.8%

      \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(1 + \left(-\left(\frac{u}{0.75} - \color{blue}{0.3333333333333333}\right)\right)\right) \]

   10. sub-neg 95.8%

       \[\leadsto \left(-3 \cdot s\right) \cdot \log \color{blue}{\left(1 - \left(\frac{u}{0.75} - 0.3333333333333333\right)\right)} \]

   11. distribute-lft-neg-in 95.8%

       \[\leadsto \color{blue}{\left(\left(-3\right) \cdot s\right)} \cdot \log \left(1 - \left(\frac{u}{0.75} - 0.3333333333333333\right)\right) \]

   12. metadata-eval 95.8%

       \[\leadsto \left(\color{blue}{-3} \cdot s\right) \cdot \log \left(1 - \left(\frac{u}{0.75} - 0.3333333333333333\right)\right) \]

   13. sub-neg 95.8%

       \[\leadsto \left(-3 \cdot s\right) \cdot \log \color{blue}{\left(1 + \left(-\left(\frac{u}{0.75} - 0.3333333333333333\right)\right)\right)} \]

   14. metadata-eval 95.8%

       \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(1 + \left(-\left(\frac{u}{0.75} - \color{blue}{\frac{0.25}{0.75}}\right)\right)\right) \]

   15. div-sub 96.8%

       \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(1 + \left(-\color{blue}{\frac{u - 0.25}{0.75}}\right)\right) \]

   16. distribute-frac-neg 96.8%

       \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(1 + \color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right) \]

   17. expm1-log1p 96.8%

       \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\left(u - 0.25\right)}{0.75}\right)\right)}\right) \]

7. Simplified 98.5%

   \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)} \]
8. Step-by-step derivation

   1. add-exp-log_binary32 90.7%

      \[\leadsto \color{blue}{e^{\log \left(\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right)}} \]

9. Applied rewrite-once 90.7%

   \[\leadsto \color{blue}{e^{\log \left(\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right)}} \]
10. Step-by-step derivation

    1. rem-exp-log 98.5%

       \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)} \]

    2. associate-*l* 98.3%

       \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right)} \]

11. Simplified 98.3%

    \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right)} \]

12. Final simplification 98.3%

    \[\leadsto -3 \cdot \left(s \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right) \]

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* -3.0 (log1p (/ (- 0.25 u) 0.75)))))

C

float code(float s, float u) {
	return s * (-3.0f * log1pf(((0.25f - u) / 0.75f)));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75)))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. *-commutative 95.9%

      \[\leadsto \color{blue}{\left(s \cdot 3\right)} \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

   2. associate-*l* 95.9%

      \[\leadsto \color{blue}{s \cdot \left(3 \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

   3. log-rec 96.9%

      \[\leadsto s \cdot \left(3 \cdot \color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]

   4. neg-mul-1 96.9%

      \[\leadsto s \cdot \left(3 \cdot \color{blue}{\left(-1 \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]

   5. associate-*r* 96.9%

      \[\leadsto s \cdot \color{blue}{\left(\left(3 \cdot -1\right) \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)} \]

   6. metadata-eval 96.9%

      \[\leadsto s \cdot \left(\color{blue}{-3} \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right) \]

   7. sub-neg 96.9%

      \[\leadsto s \cdot \left(-3 \cdot \log \color{blue}{\left(1 + \left(-\frac{u - 0.25}{0.75}\right)\right)}\right) \]

   8. log1p-def 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)}\right) \]

   9. distribute-neg-frac 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right) \]

   10. neg-mul-1 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot \left(u - 0.25\right)}}{0.75}\right)\right) \]

   11. associate-/l* 98.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{\frac{0.75}{u - 0.25}}}\right)\right) \]

   12. associate-/r/ 98.1%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{0.75} \cdot \left(u - 0.25\right)}\right)\right) \]

   13. sub-neg 98.1%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{-1}{0.75} \cdot \color{blue}{\left(u + \left(-0.25\right)\right)}\right)\right) \]

   14. distribute-lft-in 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-1}{0.75} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)}\right)\right) \]

   15. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)\right)\right) \]

   16. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + \color{blue}{-1.3333333333333333} \cdot \left(-0.25\right)\right)\right) \]

   17. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + -1.3333333333333333 \cdot \color{blue}{-0.25}\right)\right) \]

   18. metadata-eval 97.0%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + \color{blue}{0.3333333333333333}\right)\right) \]

3. Simplified 97.0%

   \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot u + 0.3333333333333333\right)\right)} \]
4. Step-by-step derivation

   1. metadata-eval 97.0%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(-1.3333333333333333\right)} \cdot u + 0.3333333333333333\right)\right) \]

   2. metadata-eval 97.0%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\color{blue}{\frac{1}{0.75}}\right) \cdot u + 0.3333333333333333\right)\right) \]

   3. distribute-lft-neg-in 97.0%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(-\frac{1}{0.75} \cdot u\right)} + 0.3333333333333333\right)\right) \]

   4. *-commutative 97.0%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\color{blue}{u \cdot \frac{1}{0.75}}\right) + 0.3333333333333333\right)\right) \]

   5. div-inv 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\color{blue}{\frac{u}{0.75}}\right) + 0.3333333333333333\right)\right) \]

   6. metadata-eval 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\frac{u}{0.75}\right) + \color{blue}{\left(--0.3333333333333333\right)}\right)\right) \]

   7. metadata-eval 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(-\frac{u}{0.75}\right) + \left(-\color{blue}{\left(-0.3333333333333333\right)}\right)\right)\right) \]

   8. distribute-neg-in 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{-\left(\frac{u}{0.75} + \left(-0.3333333333333333\right)\right)}\right)\right) \]

   9. sub-neg 96.7%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-\color{blue}{\left(\frac{u}{0.75} - 0.3333333333333333\right)}\right)\right) \]

   10. metadata-eval 96.7%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-\left(\frac{u}{0.75} - \color{blue}{\frac{0.25}{0.75}}\right)\right)\right) \]

   11. div-sub 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(-\color{blue}{\frac{u - 0.25}{0.75}}\right)\right) \]

   12. distribute-frac-neg 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right) \]

   13. sub-neg 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{-\color{blue}{\left(u + \left(-0.25\right)\right)}}{0.75}\right)\right) \]

   14. +-commutative 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{-\color{blue}{\left(\left(-0.25\right) + u\right)}}{0.75}\right)\right) \]

   15. distribute-neg-in 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(-\left(-0.25\right)\right) + \left(-u\right)}}{0.75}\right)\right) \]

   16. metadata-eval 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\left(-\color{blue}{-0.25}\right) + \left(-u\right)}{0.75}\right)\right) \]

   17. metadata-eval 98.5%

       \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{0.25} + \left(-u\right)}{0.75}\right)\right) \]

5. Applied egg-rr 98.5%

   \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{0.25 + \left(-u\right)}{0.75}}\right)\right) \]
6. Step-by-step derivation

   1. sub-neg 98.5%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{0.25 - u}}{0.75}\right)\right) \]

7. Simplified 98.5%

   \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right)\right) \]

8. Final simplification 98.5%

   \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right) \]

Alternative 8: 30.1% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(s \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* 3.0 (* s u)))

C

float code(float s, float u) {
	return 3.0f * (s * u);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 3.0e0 * (s * u)
end function

Julia

function code(s, u)
	return Float32(Float32(3.0) * Float32(s * u))
end

MATLAB

function tmp = code(s, u)
	tmp = single(3.0) * (s * u);
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

2. Taylor expanded in u around 0 26.2%

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot u\right) + 3 \cdot \left(s \cdot \log 0.75\right)} \]
3. Step-by-step derivation

   1. distribute-lft-out 26.2%

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log 0.75\right)} \]

   2. distribute-lft-out 26.2%

      \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \left(u + \log 0.75\right)\right)} \]

4. Simplified 26.2%

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \left(u + \log 0.75\right)\right)} \]

5. Taylor expanded in u around inf 30.2%

   \[\leadsto 3 \cdot \left(s \cdot \color{blue}{u}\right) \]

6. Final simplification 30.2%

   \[\leadsto 3 \cdot \left(s \cdot u\right) \]

Reproduce

herbie shell --seed 2023297 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))