Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.7%

Time: 8.2s

Alternatives: 11

Speedup: 1.1×

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: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return -((3.0f * s) * logf((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 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

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

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. sqr-pow N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)} \cdot {\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   6. log-pow N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{3 \cdot s}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{s \cdot 3}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(s \cdot \color{blue}{\frac{3}{2}}\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   13. lower-log.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   14. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   15. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   16. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   17. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left(s \cdot 1.5\right) \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{-2}\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-2}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-2}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{s \cdot \left(\frac{3}{2} \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-2}\right)\right)} \]

   4. lift-log.f32 N/A

      \[\leadsto s \cdot \left(\frac{3}{2} \cdot \color{blue}{\log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-2}\right)}\right) \]

   5. log-pow-rev N/A

      \[\leadsto s \cdot \color{blue}{\log \left({\left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-2}\right)}^{\frac{3}{2}}\right)} \]

   6. lift-pow.f32 N/A

      \[\leadsto s \cdot \log \left({\color{blue}{\left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-2}\right)}}^{\frac{3}{2}}\right) \]

   7. pow-pow N/A

      \[\leadsto s \cdot \log \color{blue}{\left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{\left(-2 \cdot \frac{3}{2}\right)}\right)} \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. pow-pow N/A

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

   11. inv-pow N/A

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

   12. lift--.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. lift--.f32 N/A

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

   15. log-pow-rev N/A

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

6. Applied rewrites 96.7%

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

Alternative 2: 25.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log 0.75 + u\right) \cdot s\right) \cdot 3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (+ (log 0.75) u) s) 3.0))

C

float code(float s, float u) {
	return ((logf(0.75f) + u) * s) * 3.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((log(0.75e0) + u) * s) * 3.0e0
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(log(Float32(0.75)) + u) * s) * Float32(3.0))
end

MATLAB

function tmp = code(s, u)
	tmp = ((log(single(0.75)) + u) * s) * single(3.0);
end

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. lower-log.f32 25.8

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

5. Applied rewrites 25.8%

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

Alternative 3: 7.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log 0.75 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* 3.0 s) (log 0.75)))

C

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

Fortran

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

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(0.75)))
end

MATLAB

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

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 7.3%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{0.75} \]
6. Add Preprocessing

Alternative 4: 7.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 0.75 \cdot s\right) \cdot 3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (log 0.75) s) 3.0))

C

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

Fortran

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

Julia

function code(s, u)
	return Float32(Float32(log(Float32(0.75)) * s) * Float32(3.0))
end

MATLAB

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

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-log.f32 7.3

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

5. Applied rewrites 7.3%

   \[\leadsto \color{blue}{\left(\log 0.75 \cdot s\right) \cdot 3} \]
6. Add Preprocessing

Alternative 5: 8.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 1.5\right) \cdot \mathsf{fma}\left(\frac{0}{0}, 2, \frac{0}{0}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s 1.5) (fma (/ 0.0 0.0) 2.0 (/ 0.0 0.0))))

C

float code(float s, float u) {
	return (s * 1.5f) * fmaf((0.0f / 0.0f), 2.0f, (0.0f / 0.0f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(1.5)) * fma(Float32(Float32(0.0) / Float32(0.0)), Float32(2.0), Float32(Float32(0.0) / Float32(0.0))))
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. sqr-pow N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)} \cdot {\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   6. log-pow N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{3 \cdot s}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{s \cdot 3}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(s \cdot \color{blue}{\frac{3}{2}}\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   13. lower-log.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   14. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   15. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   16. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   17. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left(s \cdot 1.5\right) \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{-2}\right)} \]

6. Applied rewrites -0.0%

   \[\leadsto \left(s \cdot 1.5\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0}{0}, 2, \frac{0}{0}\right)} \]
7. Add Preprocessing

Alternative 6: 10.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 1.5\right) \cdot \mathsf{fma}\left(3, \frac{0}{0}, \frac{0}{0}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s 1.5) (fma 3.0 (/ 0.0 0.0) (/ 0.0 0.0))))

C

float code(float s, float u) {
	return (s * 1.5f) * fmaf(3.0f, (0.0f / 0.0f), (0.0f / 0.0f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(1.5)) * fma(Float32(3.0), Float32(Float32(0.0) / Float32(0.0)), Float32(Float32(0.0) / Float32(0.0))))
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. sqr-pow N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)} \cdot {\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   6. log-pow N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{3 \cdot s}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{s \cdot 3}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(s \cdot \color{blue}{\frac{3}{2}}\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   13. lower-log.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   14. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   15. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   16. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   17. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left(s \cdot 1.5\right) \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{-2}\right)} \]

6. Applied rewrites -0.0%

   \[\leadsto \left(s \cdot 1.5\right) \cdot \color{blue}{\mathsf{fma}\left(3, \frac{0}{0}, \frac{0}{0}\right)} \]
7. Add Preprocessing

Alternative 7: 9.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 1.5\right) \cdot \mathsf{fma}\left(1.5, \frac{0}{0}, \frac{0}{0}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s 1.5) (fma 1.5 (/ 0.0 0.0) (/ 0.0 0.0))))

C

float code(float s, float u) {
	return (s * 1.5f) * fmaf(1.5f, (0.0f / 0.0f), (0.0f / 0.0f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(1.5)) * fma(Float32(1.5), Float32(Float32(0.0) / Float32(0.0)), Float32(Float32(0.0) / Float32(0.0))))
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. sqr-pow N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)} \cdot {\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   6. log-pow N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{3 \cdot s}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{s \cdot 3}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(s \cdot \color{blue}{\frac{3}{2}}\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   13. lower-log.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   14. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   15. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   16. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   17. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left(s \cdot 1.5\right) \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{-2}\right)} \]

6. Applied rewrites -0.0%

   \[\leadsto \left(s \cdot 1.5\right) \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{0}{0}, \frac{0}{0}\right)} \]
7. Add Preprocessing

Alternative 8: 2.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 1.5\right) \cdot \mathsf{fma}\left(0.5, \frac{0}{0}, \frac{0}{0}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s 1.5) (fma 0.5 (/ 0.0 0.0) (/ 0.0 0.0))))

C

float code(float s, float u) {
	return (s * 1.5f) * fmaf(0.5f, (0.0f / 0.0f), (0.0f / 0.0f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(1.5)) * fma(Float32(0.5), Float32(Float32(0.0) / Float32(0.0)), Float32(Float32(0.0) / Float32(0.0))))
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. sqr-pow N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)} \cdot {\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   6. log-pow N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{3 \cdot s}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{s \cdot 3}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(s \cdot \color{blue}{\frac{3}{2}}\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   13. lower-log.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   14. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   15. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   16. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   17. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left(s \cdot 1.5\right) \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{-2}\right)} \]

6. Applied rewrites -0.0%

   \[\leadsto \left(s \cdot 1.5\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{0}{0}, \frac{0}{0}\right)} \]
7. Add Preprocessing

Alternative 9: 10.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(s \cdot 2, \frac{0}{0}, \mathsf{NAN}\left(\right) \cdot s\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (fma (* s 2.0) (/ 0.0 0.0) (* (NAN) s)))

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. sqr-pow N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)} \cdot {\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   6. log-pow N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{3 \cdot s}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{s \cdot 3}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(s \cdot \color{blue}{\frac{3}{2}}\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   13. lower-log.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   14. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   15. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   16. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   17. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left(s \cdot 1.5\right) \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{-2}\right)} \]

6. Applied rewrites -0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot 2, \frac{0}{0}, \frac{0}{0} \cdot s\right)} \]

7. Taylor expanded in s around 0

   \[\leadsto \mathsf{fma}\left(s \cdot 2, \frac{0}{0}, \color{blue}{s \cdot \mathsf{NAN}\left(\right)}\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(s \cdot 2, \frac{0}{0}, \color{blue}{\mathsf{NAN}\left(\right) \cdot s}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(s \cdot 2, \frac{0}{0}, \color{blue}{\mathsf{NAN}\left(\right) \cdot s}\right) \]

   3. lower-NAN.f32 -0.0

      \[\leadsto \mathsf{fma}\left(s \cdot 2, \frac{0}{0}, \color{blue}{\mathsf{NAN}\left(\right)} \cdot s\right) \]

9. Applied rewrites -0.0%

   \[\leadsto \mathsf{fma}\left(s \cdot 2, \frac{0}{0}, \color{blue}{\mathsf{NAN}\left(\right) \cdot s}\right) \]
10. Add Preprocessing

Alternative 10: 6.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{NAN}\left(\right) \cdot s, 1.5, \frac{0}{0} \cdot s\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (fma (* (NAN) s) 1.5 (* (/ 0.0 0.0) s)))

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. sqr-pow N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)} \cdot {\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   6. log-pow N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{3 \cdot s}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{s \cdot 3}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(s \cdot \color{blue}{\frac{3}{2}}\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   13. lower-log.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   14. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   15. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   16. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   17. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left(s \cdot 1.5\right) \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{-2}\right)} \]

6. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{0} \cdot s, 1.5, \frac{0}{0} \cdot s\right)} \]

7. Taylor expanded in s around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{s \cdot \mathsf{NAN}\left(\right)}, \frac{3}{2}, \frac{0}{0} \cdot s\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{NAN}\left(\right) \cdot s}, \frac{3}{2}, \frac{0}{0} \cdot s\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{NAN}\left(\right) \cdot s}, \frac{3}{2}, \frac{0}{0} \cdot s\right) \]

   3. lower-NAN.f32 -0.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{NAN}\left(\right)} \cdot s, 1.5, \frac{0}{0} \cdot s\right) \]

9. Applied rewrites -0.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{NAN}\left(\right) \cdot s}, 1.5, \frac{0}{0} \cdot s\right) \]
10. Add Preprocessing

Alternative 11: -0.0% accurate, 139.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{NAN}\left(\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (NAN))

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. sqr-pow N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)} \cdot {\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   5. pow-prod-down N/A

      \[\leadsto \log \color{blue}{\left({\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}^{\left(\frac{3 \cdot s}{2}\right)}\right)} \]

   6. log-pow N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{3 \cdot s}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{3 \cdot s}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{s \cdot 3}}{2} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(s \cdot \frac{3}{2}\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(s \cdot \color{blue}{\frac{3}{2}}\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   13. lower-log.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   14. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   15. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left(\color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}} \cdot \frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   16. lift-/.f32 N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   17. inv-pow N/A

       \[\leadsto \left(s \cdot \frac{3}{2}\right) \cdot \log \left({\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1} \cdot \color{blue}{{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}^{-1}}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left(s \cdot 1.5\right) \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{-2}\right)} \]

6. Applied rewrites -0.0%

   \[\leadsto \color{blue}{\frac{s \cdot 0}{0}} \]

7. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\mathsf{NAN}\left(\right)} \]
8. Step-by-step derivation

   1. lower-NAN.f32 -0.0

      \[\leadsto \color{blue}{\mathsf{NAN}\left(\right)} \]

9. Applied rewrites -0.0%

   \[\leadsto \color{blue}{\mathsf{NAN}\left(\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))))))