Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 95.9%

Time: 9.8s

Alternatives: 13

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: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.1%

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

3. Final simplification 96.1%

   \[\leadsto \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \cdot \left(s \cdot 3\right) \]
4. Add Preprocessing

Alternative 2: 26.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \frac{u - 0.25}{0.75} \leq 0.9100000262260437:\\ \;\;\;\;\left(\log 0.6666666666666666 \cdot s\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(0.3333333333333333 + -1.3333333333333333 \cdot u\right) \cdot -3\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (/ (- u 0.25) 0.75)) 0.9100000262260437)
   (* (* (log 0.6666666666666666) s) -3.0)
   (* (* (log1p (+ 0.3333333333333333 (* -1.3333333333333333 u))) -3.0) s)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - ((u - 0.25f) / 0.75f)) <= 0.9100000262260437f) {
		tmp = (logf(0.6666666666666666f) * s) * -3.0f;
	} else {
		tmp = (log1pf((0.3333333333333333f + (-1.3333333333333333f * u))) * -3.0f) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))) <= Float32(0.9100000262260437))
		tmp = Float32(Float32(log(Float32(0.6666666666666666)) * s) * Float32(-3.0));
	else
		tmp = Float32(Float32(log1p(Float32(Float32(0.3333333333333333) + Float32(Float32(-1.3333333333333333) * u))) * Float32(-3.0)) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (/.f32 (-.f32 u #s(literal 1/4 binary32)) #s(literal 3/4 binary32))) < 0.910000026

   1. Initial program 96.6%

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

      1. lift--.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

      2. sub-neg N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

      4. lower-+.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

      5. lift-/.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

      7. div-inv N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) + 1}\right) \]

      11. metadata-eval 96.5

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

   4. Applied rewrites 96.5%

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

   6. Applied rewrites 9.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(s \cdot 3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), -1, 0\right)} \]

   7. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \frac{2}{3}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(s \cdot \log \frac{2}{3}\right) \cdot -3} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(s \cdot \log \frac{2}{3}\right) \cdot -3} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\log \frac{2}{3} \cdot s\right)} \cdot -3 \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\log \frac{2}{3} \cdot s\right)} \cdot -3 \]

      5. lower-log.f32 29.4

         \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]

   9. Applied rewrites 29.4%

      \[\leadsto \color{blue}{\left(\log 0.6666666666666666 \cdot s\right) \cdot -3} \]

   if 0.910000026 < (-.f32 #s(literal 1 binary32) (/.f32 (-.f32 u #s(literal 1/4 binary32)) #s(literal 3/4 binary32)))

   1. Initial program 93.0%

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

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

      6. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

      2. *-commutative N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}}\right)\right) \cdot s \]

      3. lower-*.f32 45.5

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

   6. Applied rewrites 45.5%

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(u - 0.25\right) \cdot -1.3333333333333333}\right)\right) \cdot s \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}}\right)\right) \cdot s \]

      2. *-commutative N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

      3. lift--.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

      4. sub-neg N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}\right)\right) \cdot s \]

      5. distribute-lft-in N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u + \frac{-4}{3} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) \cdot s \]

      6. metadata-eval N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot u + \frac{-4}{3} \cdot \color{blue}{\frac{-1}{4}}\right)\right) \cdot s \]

      7. metadata-eval N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot u + \color{blue}{\frac{1}{3}}\right)\right) \cdot s \]

      8. lower-+.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u + \frac{1}{3}}\right)\right) \cdot s \]

      9. lower-*.f32 45.4

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

   8. Applied rewrites 45.4%

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333 \cdot u + 0.3333333333333333}\right)\right) \cdot s \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{u - 0.25}{0.75} \leq 0.9100000262260437:\\ \;\;\;\;\left(\log 0.6666666666666666 \cdot s\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(0.3333333333333333 + -1.3333333333333333 \cdot u\right) \cdot -3\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 27.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \frac{u - 0.25}{0.75} \leq 0.9100000262260437:\\ \;\;\;\;\left(\log 0.6666666666666666 \cdot s\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right) \cdot -3\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (/ (- u 0.25) 0.75)) 0.9100000262260437)
   (* (* (log 0.6666666666666666) s) -3.0)
   (* (* (log1p (* -1.3333333333333333 (- u 0.25))) -3.0) s)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - ((u - 0.25f) / 0.75f)) <= 0.9100000262260437f) {
		tmp = (logf(0.6666666666666666f) * s) * -3.0f;
	} else {
		tmp = (log1pf((-1.3333333333333333f * (u - 0.25f))) * -3.0f) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))) <= Float32(0.9100000262260437))
		tmp = Float32(Float32(log(Float32(0.6666666666666666)) * s) * Float32(-3.0));
	else
		tmp = Float32(Float32(log1p(Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25)))) * Float32(-3.0)) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (/.f32 (-.f32 u #s(literal 1/4 binary32)) #s(literal 3/4 binary32))) < 0.910000026

   1. Initial program 96.6%

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

      1. lift--.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

      2. sub-neg N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

      4. lower-+.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

      5. lift-/.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

      7. div-inv N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) + 1}\right) \]

      11. metadata-eval 96.5

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

   4. Applied rewrites 96.5%

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

   6. Applied rewrites 9.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(s \cdot 3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), -1, 0\right)} \]

   7. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \frac{2}{3}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(s \cdot \log \frac{2}{3}\right) \cdot -3} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(s \cdot \log \frac{2}{3}\right) \cdot -3} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\log \frac{2}{3} \cdot s\right)} \cdot -3 \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\log \frac{2}{3} \cdot s\right)} \cdot -3 \]

      5. lower-log.f32 29.4

         \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]

   9. Applied rewrites 29.4%

      \[\leadsto \color{blue}{\left(\log 0.6666666666666666 \cdot s\right) \cdot -3} \]

   if 0.910000026 < (-.f32 #s(literal 1 binary32) (/.f32 (-.f32 u #s(literal 1/4 binary32)) #s(literal 3/4 binary32)))

   1. Initial program 93.0%

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

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

      6. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{u - 0.25}{0.75} \leq 0.9100000262260437:\\ \;\;\;\;\left(\log 0.6666666666666666 \cdot s\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right) \cdot -3\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 27.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \frac{u - 0.25}{0.75} \leq 0.9100000262260437:\\ \;\;\;\;\left(\log 0.6666666666666666 \cdot s\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(\left(-0.25 + u\right) \cdot -1.3333333333333333\right) \cdot -3\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (/ (- u 0.25) 0.75)) 0.9100000262260437)
   (* (* (log 0.6666666666666666) s) -3.0)
   (* (* (log1p (* (+ -0.25 u) -1.3333333333333333)) -3.0) s)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - ((u - 0.25f) / 0.75f)) <= 0.9100000262260437f) {
		tmp = (logf(0.6666666666666666f) * s) * -3.0f;
	} else {
		tmp = (log1pf(((-0.25f + u) * -1.3333333333333333f)) * -3.0f) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))) <= Float32(0.9100000262260437))
		tmp = Float32(Float32(log(Float32(0.6666666666666666)) * s) * Float32(-3.0));
	else
		tmp = Float32(Float32(log1p(Float32(Float32(Float32(-0.25) + u) * Float32(-1.3333333333333333))) * Float32(-3.0)) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (/.f32 (-.f32 u #s(literal 1/4 binary32)) #s(literal 3/4 binary32))) < 0.910000026

   1. Initial program 96.6%

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

      1. lift--.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

      2. sub-neg N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

      4. lower-+.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

      5. lift-/.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1}\right) \]

      6. distribute-neg-frac2 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

      7. div-inv N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) + 1}\right) \]

      11. metadata-eval 96.5

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

   4. Applied rewrites 96.5%

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

   6. Applied rewrites 9.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(s \cdot 3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), -1, 0\right)} \]

   7. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \frac{2}{3}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(s \cdot \log \frac{2}{3}\right) \cdot -3} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(s \cdot \log \frac{2}{3}\right) \cdot -3} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\log \frac{2}{3} \cdot s\right)} \cdot -3 \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\log \frac{2}{3} \cdot s\right)} \cdot -3 \]

      5. lower-log.f32 29.4

         \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]

   9. Applied rewrites 29.4%

      \[\leadsto \color{blue}{\left(\log 0.6666666666666666 \cdot s\right) \cdot -3} \]

   if 0.910000026 < (-.f32 #s(literal 1 binary32) (/.f32 (-.f32 u #s(literal 1/4 binary32)) #s(literal 3/4 binary32)))

   1. Initial program 93.0%

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

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

      6. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   4. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
   5. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right) \cdot 1}\right)\right) \cdot s \]

      2. lift-*.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)} \cdot 1\right)\right) \cdot s \]

      3. associate-*l* N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(\left(u - \frac{1}{4}\right) \cdot 1\right)}\right)\right) \cdot s \]

      4. lower-*.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(\left(u - \frac{1}{4}\right) \cdot 1\right)}\right)\right) \cdot s \]

      5. lower-*.f32 46.0

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

   6. Applied rewrites 46.0%

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333 \cdot \left(\left(u - 0.25\right) \cdot 1\right)}\right)\right) \cdot s \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(\left(u - \frac{1}{4}\right) \cdot 1\right)}\right)\right) \cdot s \]

      2. *-rgt-identity 46.5

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

      3. lift--.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

      4. sub-neg N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}\right)\right) \cdot s \]

      5. +-commutative N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)}\right)\right) \cdot s \]

      6. lower-+.f32 N/A

         \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)}\right)\right) \cdot s \]

      7. metadata-eval 45.5

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

   8. Applied rewrites 45.5%

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \color{blue}{\left(-0.25 + u\right)}\right)\right) \cdot s \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{u - 0.25}{0.75} \leq 0.9100000262260437:\\ \;\;\;\;\left(\log 0.6666666666666666 \cdot s\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(\left(-0.25 + u\right) \cdot -1.3333333333333333\right) \cdot -3\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{-1.3333333333333333 \cdot \left(u - 0.25\right) + 1}\right) \cdot \left(s \cdot 3\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (log (/ 1.0 (+ (* -1.3333333333333333 (- u 0.25)) 1.0))) (* s 3.0)))

C

float code(float s, float u) {
	return logf((1.0f / ((-1.3333333333333333f * (u - 0.25f)) + 1.0f))) * (s * 3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = log((1.0e0 / (((-1.3333333333333333e0) * (u - 0.25e0)) + 1.0e0))) * (s * 3.0e0)
end function

Julia

function code(s, u)
	return Float32(log(Float32(Float32(1.0) / Float32(Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25))) + Float32(1.0)))) * Float32(s * Float32(3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = log((single(1.0) / ((single(-1.3333333333333333) * (u - single(0.25))) + single(1.0)))) * (s * single(3.0));
end

Derivation

1. Initial program 96.1%

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

   1. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   2. sub-neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   7. div-inv N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) + 1}\right) \]

   11. metadata-eval 96.0

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

4. Applied rewrites 96.0%

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

5. Final simplification 96.0%

   \[\leadsto \log \left(\frac{1}{-1.3333333333333333 \cdot \left(u - 0.25\right) + 1}\right) \cdot \left(s \cdot 3\right) \]
6. Add Preprocessing

Alternative 6: 95.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{1.3333333333333333 - 1.3333333333333333 \cdot u}\right) \cdot \left(s \cdot 3\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (log (/ 1.0 (- 1.3333333333333333 (* 1.3333333333333333 u)))) (* s 3.0)))

C

float code(float s, float u) {
	return logf((1.0f / (1.3333333333333333f - (1.3333333333333333f * u)))) * (s * 3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = log((1.0e0 / (1.3333333333333333e0 - (1.3333333333333333e0 * u)))) * (s * 3.0e0)
end function

Julia

function code(s, u)
	return Float32(log(Float32(Float32(1.0) / Float32(Float32(1.3333333333333333) - Float32(Float32(1.3333333333333333) * u)))) * Float32(s * Float32(3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = log((single(1.0) / (single(1.3333333333333333) - (single(1.3333333333333333) * u)))) * (s * single(3.0));
end

Derivation

1. Initial program 96.1%

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

   1. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   2. sub-neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   7. div-inv N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) + 1}\right) \]

   11. metadata-eval 96.0

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

4. Applied rewrites 96.0%

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

6. Applied rewrites 10.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\mathsf{fma}\left(1.3333333333333333, u - 0.25, 1\right)}}\right) \]

8. Applied rewrites 95.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1.3333333333333333 - 1.3333333333333333 \cdot u}}\right) \]

9. Final simplification 95.9%

   \[\leadsto \log \left(\frac{1}{1.3333333333333333 - 1.3333333333333333 \cdot u}\right) \cdot \left(s \cdot 3\right) \]
10. Add Preprocessing

Alternative 7: 9.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

4. Applied rewrites 34.7%

   \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

   2. *-commutative N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}}\right)\right) \cdot s \]

   3. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}\right)\right) \cdot s \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \frac{4}{3}\right)}\right)\right) \cdot s \]

   5. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\frac{1}{\frac{3}{4}}}\right)\right)\right) \cdot s \]

   6. div-inv N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)\right) \cdot s \]

   7. distribute-neg-frac N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}}\right)\right) \cdot s \]

   8. lower-/.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}}\right)\right) \cdot s \]

   9. lift--.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(u - \frac{1}{4}\right)}\right)}{\frac{3}{4}}\right)\right) \cdot s \]

   10. sub-neg N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}\right)}{\frac{3}{4}}\right)\right) \cdot s \]

   11. distribute-neg-in N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(\mathsf{neg}\left(u\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)}}{\frac{3}{4}}\right)\right) \cdot s \]

   12. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{\left(\mathsf{neg}\left(u\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{4}}\right)\right)}{\frac{3}{4}}\right)\right) \cdot s \]

   13. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{\left(\mathsf{neg}\left(u\right)\right) + \color{blue}{\frac{1}{4}}}{\frac{3}{4}}\right)\right) \cdot s \]

   14. +-commutative N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)}}{\frac{3}{4}}\right)\right) \cdot s \]

   15. sub-neg N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{4} - u}}{\frac{3}{4}}\right)\right) \cdot s \]

   16. lift--.f32 34.9

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

6. Applied rewrites 34.8%

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

7. Final simplification 34.8%

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

Alternative 8: 9.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{log1p}\left(\frac{u}{-0.75} - -0.3333333333333333\right) \cdot -3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (log1p (- (/ u -0.75) -0.3333333333333333)) -3.0) s))

C

float code(float s, float u) {
	return (log1pf(((u / -0.75f) - -0.3333333333333333f)) * -3.0f) * s;
}

Julia

function code(s, u)
	return Float32(Float32(log1p(Float32(Float32(u / Float32(-0.75)) - Float32(-0.3333333333333333))) * Float32(-3.0)) * s)
end

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

4. Applied rewrites 34.7%

   \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

   2. *-commutative N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}}\right)\right) \cdot s \]

   3. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}\right)\right) \cdot s \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \frac{4}{3}\right)}\right)\right) \cdot s \]

   5. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\frac{1}{\frac{3}{4}}}\right)\right)\right) \cdot s \]

   6. div-inv N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)\right) \cdot s \]

   7. distribute-neg-frac2 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}}\right)\right) \cdot s \]

   8. lift--.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{u - \frac{1}{4}}}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right) \cdot s \]

   9. div-sub N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)} - \frac{\frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}}\right)\right) \cdot s \]

   10. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)} - \frac{\frac{1}{4}}{\color{blue}{\frac{-3}{4}}}\right)\right) \cdot s \]

   11. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)} - \color{blue}{\frac{-1}{3}}\right)\right) \cdot s \]

   12. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right) \cdot s \]

   13. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)} - \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4}}{\frac{3}{4}}}\right)\right)\right)\right) \cdot s \]

   14. lower--.f32 N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)} - \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)}\right)\right) \cdot s \]

   15. lower-/.f32 N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}} - \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right)\right) \cdot s \]

   16. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{u}{\color{blue}{\frac{-3}{4}}} - \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right)\right) \cdot s \]

   17. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3}}\right)\right)\right)\right) \cdot s \]

   18. metadata-eval 34.7

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

6. Applied rewrites 34.9%

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

7. Final simplification 34.7%

   \[\leadsto \left(\mathsf{log1p}\left(\frac{u}{-0.75} - -0.3333333333333333\right) \cdot -3\right) \cdot s \]
8. Add Preprocessing

Alternative 9: 9.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

4. Applied rewrites 34.7%

   \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

   2. *-commutative N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}}\right)\right) \cdot s \]

   3. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}\right)\right) \cdot s \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \frac{4}{3}\right)}\right)\right) \cdot s \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right) \cdot \frac{4}{3}}\right)\right) \cdot s \]

   6. lift--.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\left(\mathsf{neg}\left(\color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \cdot \frac{4}{3}\right)\right) \cdot s \]

   7. sub-neg N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\left(\mathsf{neg}\left(\color{blue}{\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}\right)\right) \cdot \frac{4}{3}\right)\right) \cdot s \]

   8. distribute-neg-in N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\left(\mathsf{neg}\left(u\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right)} \cdot \frac{4}{3}\right)\right) \cdot s \]

   9. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\left(\left(\mathsf{neg}\left(u\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{4}}\right)\right)\right) \cdot \frac{4}{3}\right)\right) \cdot s \]

   10. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\left(\left(\mathsf{neg}\left(u\right)\right) + \color{blue}{\frac{1}{4}}\right) \cdot \frac{4}{3}\right)\right) \cdot s \]

   11. +-commutative N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right)} \cdot \frac{4}{3}\right)\right) \cdot s \]

   12. sub-neg N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{4} - u\right)} \cdot \frac{4}{3}\right)\right) \cdot s \]

   13. lift--.f32 N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{4} - u\right)} \cdot \frac{4}{3}\right)\right) \cdot s \]

   14. *-rgt-identity N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\left(\frac{1}{4} - u\right) \cdot \frac{4}{3}\right) \cdot 1}\right)\right) \cdot s \]

   15. *-commutative N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{4}{3} \cdot \left(\frac{1}{4} - u\right)\right)} \cdot 1\right)\right) \cdot s \]

   16. associate-*l* N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{4}{3} \cdot \left(\left(\frac{1}{4} - u\right) \cdot 1\right)}\right)\right) \cdot s \]

   17. lift--.f32 N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \left(\color{blue}{\left(\frac{1}{4} - u\right)} \cdot 1\right)\right)\right) \cdot s \]

   18. sub-neg N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \left(\color{blue}{\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right)} \cdot 1\right)\right)\right) \cdot s \]

   19. +-commutative N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(u\right)\right) + \frac{1}{4}\right)} \cdot 1\right)\right)\right) \cdot s \]

   20. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \left(\left(\left(\mathsf{neg}\left(u\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)}\right) \cdot 1\right)\right)\right) \cdot s \]

   21. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \left(\left(\left(\mathsf{neg}\left(u\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \cdot 1\right)\right)\right) \cdot s \]

   22. distribute-neg-in N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right)} \cdot 1\right)\right)\right) \cdot s \]

   23. sub-neg N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \cdot 1\right)\right)\right) \cdot s \]

   24. lift--.f32 N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \cdot 1\right)\right)\right) \cdot s \]

   25. lower-*.f32 N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{4}{3} \cdot \left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right) \cdot 1\right)}\right)\right) \cdot s \]

   26. lower-*.f32 N/A

       \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\frac{4}{3} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right) \cdot 1\right)}\right)\right) \cdot s \]

6. Applied rewrites 34.8%

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

7. Final simplification 34.8%

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

Alternative 10: 14.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

4. Applied rewrites 34.7%

   \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
5. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right) \cdot 1}\right)\right) \cdot s \]

   2. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)} \cdot 1\right)\right) \cdot s \]

   3. associate-*l* N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(\left(u - \frac{1}{4}\right) \cdot 1\right)}\right)\right) \cdot s \]

   4. lower-*.f32 N/A

      \[\leadsto \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(\left(u - \frac{1}{4}\right) \cdot 1\right)}\right)\right) \cdot s \]

   5. lower-*.f32 34.9

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

6. Applied rewrites 34.8%

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

7. Final simplification 34.8%

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

Alternative 11: 28.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \log 1.5 \cdot \left(s \cdot 3\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log 1.5) (* s 3.0)))

C

float code(float s, float u) {
	return logf(1.5f) * (s * 3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = log(1.5e0) * (s * 3.0e0)
end function

Julia

function code(s, u)
	return Float32(log(Float32(1.5)) * Float32(s * Float32(3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = log(single(1.5)) * (s * single(3.0));
end

Derivation

1. Initial program 96.1%

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

   1. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   2. sub-neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   7. div-inv N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) + 1}\right) \]

   11. metadata-eval 96.0

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

4. Applied rewrites 96.0%

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

6. Applied rewrites 10.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\mathsf{fma}\left(1.3333333333333333, u - 0.25, 1\right)}}\right) \]

7. Taylor expanded in u around 0

   \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\frac{3}{2}} \]
8. Step-by-step derivation

9. Applied rewrites 28.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{1.5} \]

10. Final simplification 28.6%

    \[\leadsto \log 1.5 \cdot \left(s \cdot 3\right) \]
11. Add Preprocessing

Alternative 12: 28.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 0.6666666666666666 \cdot s\right) \cdot -3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (log 0.6666666666666666) s) -3.0))

C

float code(float s, float u) {
	return (logf(0.6666666666666666f) * s) * -3.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (log(0.6666666666666666e0) * s) * (-3.0e0)
end function

Julia

function code(s, u)
	return Float32(Float32(log(Float32(0.6666666666666666)) * s) * Float32(-3.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (log(single(0.6666666666666666)) * s) * single(-3.0);
end

Derivation

1. Initial program 96.1%

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

   1. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   2. sub-neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   7. div-inv N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) + 1}\right) \]

   11. metadata-eval 96.0

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

4. Applied rewrites 96.0%

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

6. Applied rewrites 10.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(s \cdot 3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), -1, 0\right)} \]

7. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \frac{2}{3}\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot \log \frac{2}{3}\right) \cdot -3} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(s \cdot \log \frac{2}{3}\right) \cdot -3} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\log \frac{2}{3} \cdot s\right)} \cdot -3 \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\log \frac{2}{3} \cdot s\right)} \cdot -3 \]

   5. lower-log.f32 28.6

      \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]

9. Applied rewrites 28.6%

   \[\leadsto \color{blue}{\left(\log 0.6666666666666666 \cdot s\right) \cdot -3} \]
10. Add Preprocessing

Alternative 13: 28.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 1.5 \cdot s\right) \cdot 3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (log 1.5) s) 3.0))

C

float code(float s, float u) {
	return (logf(1.5f) * s) * 3.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (log(1.5e0) * s) * 3.0e0
end function

Julia

function code(s, u)
	return Float32(Float32(log(Float32(1.5)) * s) * Float32(3.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (log(single(1.5)) * s) * single(3.0);
end

Derivation

1. Initial program 96.1%

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

   1. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   2. sub-neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1}\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   7. div-inv N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} + 1}\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) + 1}\right) \]

   11. metadata-eval 96.0

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

4. Applied rewrites 96.0%

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

6. Applied rewrites 10.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\mathsf{fma}\left(1.3333333333333333, u - 0.25, 1\right)}}\right) \]

7. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{2}\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot \log \frac{3}{2}\right) \cdot 3} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(s \cdot \log \frac{3}{2}\right) \cdot 3} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\log \frac{3}{2} \cdot s\right)} \cdot 3 \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\log \frac{3}{2} \cdot s\right)} \cdot 3 \]

   5. lower-log.f32 28.6

      \[\leadsto \left(\color{blue}{\log 1.5} \cdot s\right) \cdot 3 \]

9. Applied rewrites 28.6%

   \[\leadsto \color{blue}{\left(\log 1.5 \cdot s\right) \cdot 3} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024264 
(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))))))