Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.8%

Time: 10.7s

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: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u - 0.25}{0.75}\\ \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 + \frac{\left(u - 0.25\right) \cdot \left(u - 0.25\right)}{-0.5625}\right) \cdot \left(1 - t\_0\right)}{t\_0 + 1}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (/ (- u 0.25) 0.75)))
   (*
    (* 3.0 s)
    (*
     -0.5
     (log
      (/
       (* (+ 1.0 (/ (* (- u 0.25) (- u 0.25)) -0.5625)) (- 1.0 t_0))
       (+ t_0 1.0)))))))

C

float code(float s, float u) {
	float t_0 = (u - 0.25f) / 0.75f;
	return (3.0f * s) * (-0.5f * logf((((1.0f + (((u - 0.25f) * (u - 0.25f)) / -0.5625f)) * (1.0f - t_0)) / (t_0 + 1.0f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    t_0 = (u - 0.25e0) / 0.75e0
    code = (3.0e0 * s) * ((-0.5e0) * log((((1.0e0 + (((u - 0.25e0) * (u - 0.25e0)) / (-0.5625e0))) * (1.0e0 - t_0)) / (t_0 + 1.0e0))))
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(u - Float32(0.25)) / Float32(0.75))
	return Float32(Float32(Float32(3.0) * s) * Float32(Float32(-0.5) * log(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(Float32(u - Float32(0.25)) * Float32(u - Float32(0.25))) / Float32(-0.5625))) * Float32(Float32(1.0) - t_0)) / Float32(t_0 + Float32(1.0))))))
end

MATLAB

function tmp = code(s, u)
	t_0 = (u - single(0.25)) / single(0.75);
	tmp = (single(3.0) * s) * (single(-0.5) * log((((single(1.0) + (((u - single(0.25)) * (u - single(0.25))) / single(-0.5625))) * (single(1.0) - t_0)) / (t_0 + single(1.0)))));
end

Derivation

1. Initial program 95.7%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower-log.f32 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f32 96.4

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(-0.5 \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lift--.f32 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. unpow2 N/A

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

   7. lift-pow.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

6. Applied rewrites 96.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left(\frac{\left(1 - {\left(\frac{u - 0.25}{-0.75}\right)}^{2}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u - 0.25}{0.75} + 1}\right)}\right) \]
7. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right)}{\color{blue}{\frac{-9}{16}}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\frac{u - \frac{1}{4}}{\frac{3}{4}} + 1}\right)\right) \]

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. metadata-eval 96.9

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 - \frac{\left(u - 0.25\right) \cdot \left(-\left(u - 0.25\right)\right)}{\color{blue}{-0.5625}}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u - 0.25}{0.75} + 1}\right)\right) \]

8. Applied rewrites 96.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 - \color{blue}{\frac{\left(u - 0.25\right) \cdot \left(-\left(u - 0.25\right)\right)}{-0.5625}}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u - 0.25}{0.75} + 1}\right)\right) \]

9. Final simplification 96.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 + \frac{\left(u - 0.25\right) \cdot \left(u - 0.25\right)}{-0.5625}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u - 0.25}{0.75} + 1}\right)\right) \]
10. Add Preprocessing

Alternative 2: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 + \frac{\left(u - 0.25\right) \cdot \left(u - 0.25\right)}{-0.5625}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u}{0.75} - -0.6666666666666666}\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* 3.0 s)
  (*
   -0.5
   (log
    (/
     (*
      (+ 1.0 (/ (* (- u 0.25) (- u 0.25)) -0.5625))
      (- 1.0 (/ (- u 0.25) 0.75)))
     (- (/ u 0.75) -0.6666666666666666))))))

C

float code(float s, float u) {
	return (3.0f * s) * (-0.5f * logf((((1.0f + (((u - 0.25f) * (u - 0.25f)) / -0.5625f)) * (1.0f - ((u - 0.25f) / 0.75f))) / ((u / 0.75f) - -0.6666666666666666f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * ((-0.5e0) * log((((1.0e0 + (((u - 0.25e0) * (u - 0.25e0)) / (-0.5625e0))) * (1.0e0 - ((u - 0.25e0) / 0.75e0))) / ((u / 0.75e0) - (-0.6666666666666666e0)))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(Float32(-0.5) * log(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(Float32(u - Float32(0.25)) * Float32(u - Float32(0.25))) / Float32(-0.5625))) * Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75)))) / Float32(Float32(u / Float32(0.75)) - Float32(-0.6666666666666666))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * (single(-0.5) * log((((single(1.0) + (((u - single(0.25)) * (u - single(0.25))) / single(-0.5625))) * (single(1.0) - ((u - single(0.25)) / single(0.75)))) / ((u / single(0.75)) - single(-0.6666666666666666)))));
end

Derivation

1. Initial program 95.7%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower-log.f32 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f32 96.4

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(-0.5 \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lift--.f32 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. unpow2 N/A

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

   7. lift-pow.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

6. Applied rewrites 96.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left(\frac{\left(1 - {\left(\frac{u - 0.25}{-0.75}\right)}^{2}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u - 0.25}{0.75} + 1}\right)}\right) \]
7. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right)}{\color{blue}{\frac{-9}{16}}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\frac{u - \frac{1}{4}}{\frac{3}{4}} + 1}\right)\right) \]

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. metadata-eval 96.9

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 - \frac{\left(u - 0.25\right) \cdot \left(-\left(u - 0.25\right)\right)}{\color{blue}{-0.5625}}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u - 0.25}{0.75} + 1}\right)\right) \]

8. Applied rewrites 96.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 - \color{blue}{\frac{\left(u - 0.25\right) \cdot \left(-\left(u - 0.25\right)\right)}{-0.5625}}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u - 0.25}{0.75} + 1}\right)\right) \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(-\left(u - \frac{1}{4}\right)\right)}{\frac{-9}{16}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}} + 1}}\right)\right) \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(-\left(u - \frac{1}{4}\right)\right)}{\frac{-9}{16}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}} + 1}\right)\right) \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(-\left(u - \frac{1}{4}\right)\right)}{\frac{-9}{16}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\frac{\color{blue}{u - \frac{1}{4}}}{\frac{3}{4}} + 1}\right)\right) \]

   4. div-sub N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(-\left(u - \frac{1}{4}\right)\right)}{\frac{-9}{16}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\color{blue}{\left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)} + 1}\right)\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(-\left(u - \frac{1}{4}\right)\right)}{\frac{-9}{16}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\left(\color{blue}{\frac{u}{\frac{3}{4}}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right) + 1}\right)\right) \]

   6. associate-+l- N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(-\left(u - \frac{1}{4}\right)\right)}{\frac{-9}{16}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\color{blue}{\frac{u}{\frac{3}{4}} - \left(\frac{\frac{1}{4}}{\frac{3}{4}} - 1\right)}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(-\left(u - \frac{1}{4}\right)\right)}{\frac{-9}{16}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\frac{u}{\frac{3}{4}} - \left(\color{blue}{\frac{1}{3}} - 1\right)}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{-1}{2} \cdot \log \left(\frac{\left(1 - \frac{\left(u - \frac{1}{4}\right) \cdot \left(-\left(u - \frac{1}{4}\right)\right)}{\frac{-9}{16}}\right) \cdot \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}{\frac{u}{\frac{3}{4}} - \color{blue}{\frac{-2}{3}}}\right)\right) \]

   9. lift--.f32 96.7

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 - \frac{\left(u - 0.25\right) \cdot \left(-\left(u - 0.25\right)\right)}{-0.5625}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\color{blue}{\frac{u}{0.75} - -0.6666666666666666}}\right)\right) \]

10. Applied rewrites 96.7%

    \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 - \frac{\left(u - 0.25\right) \cdot \left(-\left(u - 0.25\right)\right)}{-0.5625}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\color{blue}{\frac{u}{0.75} - -0.6666666666666666}}\right)\right) \]

11. Final simplification 96.7%

    \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\frac{\left(1 + \frac{\left(u - 0.25\right) \cdot \left(u - 0.25\right)}{-0.5625}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)}{\frac{u}{0.75} - -0.6666666666666666}\right)\right) \]
12. Add Preprocessing

Alternative 3: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower-log.f32 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f32 96.4

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(-0.5 \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 96.5%

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

7. Final simplification 96.5%

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

Alternative 4: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return (-s * 3.0f) * 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 = (-s * 3.0e0) * log((1.0e0 - ((u - 0.25e0) / 0.75e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower-log.f32 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f32 96.4

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(-0.5 \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 96.4

      \[\leadsto \color{blue}{\left(s \cdot 3\right)} \cdot \left(-0.5 \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)\right) \]

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. log-pow-rev N/A

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

   7. lift-pow.f32 N/A

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

   8. pow-pow N/A

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

   9. metadata-eval N/A

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

   10. inv-pow N/A

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

   11. log-rec N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-log.f32 96.4

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

6. Applied rewrites 96.4%

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

7. Final simplification 96.4%

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

Alternative 5: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(-s\right) \cdot \log \left(\frac{u}{-0.75} - -1.3333333333333333\right)\right) \cdot 3 \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (- s) (log (- (/ u -0.75) -1.3333333333333333))) 3.0))

C

float code(float s, float u) {
	return (-s * logf(((u / -0.75f) - -1.3333333333333333f))) * 3.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (-s * log(((u / (-0.75e0)) - (-1.3333333333333333e0)))) * 3.0e0
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-s) * log(Float32(Float32(u / Float32(-0.75)) - Float32(-1.3333333333333333)))) * Float32(3.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (-s * log(((u / single(-0.75)) - single(-1.3333333333333333)))) * single(3.0);
end

Derivation

1. Initial program 95.7%

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

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 32.7%

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

   1. lift-log1p.f32 N/A

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

   2. lower-log.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. lift--.f32 N/A

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

   6. div-sub N/A

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

   7. associate-+l- N/A

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

   8. lower--.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 94.7

       \[\leadsto \left(\left(-s\right) \cdot \log \left(\frac{u}{-0.75} - \color{blue}{-1.3333333333333333}\right)\right) \cdot 3 \]

6. Applied rewrites 94.7%

   \[\leadsto \left(\left(-s\right) \cdot \color{blue}{\log \left(\frac{u}{-0.75} - -1.3333333333333333\right)}\right) \cdot 3 \]
7. Add Preprocessing

Alternative 6: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \left(-s\right)\right) \cdot \log \left(\frac{u}{-0.75} - -1.3333333333333333\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 (- s)) (log (- (/ u -0.75) -1.3333333333333333))))

C

float code(float s, float u) {
	return (3.0f * -s) * logf(((u / -0.75f) - -1.3333333333333333f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * -s) * log(((u / (-0.75e0)) - (-1.3333333333333333e0)))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * Float32(-s)) * log(Float32(Float32(u / Float32(-0.75)) - Float32(-1.3333333333333333))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * -s) * log(((u / single(-0.75)) - single(-1.3333333333333333)));
end

Derivation

1. Initial program 95.7%

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

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 31.8%

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

   1. lift-log1p.f32 N/A

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

   2. lower-log.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. lift--.f32 N/A

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

   6. div-sub N/A

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

   7. associate-+l- N/A

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

   8. lower--.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 94.7

       \[\leadsto \left(\left(-s\right) \cdot \log \left(\frac{u}{-0.75} - \color{blue}{-1.3333333333333333}\right)\right) \cdot 3 \]

6. Applied rewrites 94.7%

   \[\leadsto \left(\left(-s\right) \cdot \color{blue}{\log \left(\frac{u}{-0.75} - -1.3333333333333333\right)}\right) \cdot 3 \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 94.7

      \[\leadsto \color{blue}{\left(3 \cdot \left(-s\right)\right)} \cdot \log \left(\frac{u}{-0.75} - -1.3333333333333333\right) \]

8. Applied rewrites 94.7%

   \[\leadsto \color{blue}{\left(3 \cdot \left(-s\right)\right) \cdot \log \left(\frac{u}{-0.75} - -1.3333333333333333\right)} \]
9. Add Preprocessing

Alternative 7: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log \left(\frac{u}{-0.75} - -1.3333333333333333\right) \cdot \left(-3\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* (log (- (/ u -0.75) -1.3333333333333333)) (- 3.0))))

C

float code(float s, float u) {
	return s * (logf(((u / -0.75f) - -1.3333333333333333f)) * -3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (log(((u / (-0.75e0)) - (-1.3333333333333333e0))) * -3.0e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(log(Float32(Float32(u / Float32(-0.75)) - Float32(-1.3333333333333333))) * Float32(-Float32(3.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (log(((u / single(-0.75)) - single(-1.3333333333333333))) * -single(3.0));
end

Derivation

1. Initial program 95.7%

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

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 32.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-log1p.f32 N/A

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

   7. log-pow-rev N/A

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

   8. lower-log.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-/.f32 N/A

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

   12. lift--.f32 N/A

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

   13. div-sub N/A

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

   14. associate-+l- N/A

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

   15. lower--.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval 94.4

       \[\leadsto \left(-s\right) \cdot \log \left({\left(\frac{u}{-0.75} - \color{blue}{-1.3333333333333333}\right)}^{3}\right) \]

6. Applied rewrites 94.4%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left({\left(\frac{u}{-0.75} - -1.3333333333333333\right)}^{3}\right)} \]
7. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. pow-to-exp N/A

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

   4. rem-log-exp N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log.f32 94.6

      \[\leadsto \left(-s\right) \cdot \left(\color{blue}{\log \left(\frac{u}{-0.75} - -1.3333333333333333\right)} \cdot 3\right) \]

8. Applied rewrites 94.6%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\log \left(\frac{u}{-0.75} - -1.3333333333333333\right) \cdot 3\right)} \]

9. Final simplification 94.6%

   \[\leadsto s \cdot \left(\log \left(\frac{u}{-0.75} - -1.3333333333333333\right) \cdot \left(-3\right)\right) \]
10. Add Preprocessing

Alternative 8: 25.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-log.f32 25.2

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

5. Applied rewrites 25.2%

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

Alternative 9: 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 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 25.2

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

5. Applied rewrites 25.2%

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

Alternative 10: 3.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \mathsf{fma}\left(3, u, \log 0.421875\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (fma 3.0 u (log 0.421875))))

C

float code(float s, float u) {
	return s * fmaf(3.0f, u, logf(0.421875f));
}

Julia

function code(s, u)
	return Float32(s * fma(Float32(3.0), u, log(Float32(0.421875))))
end

Derivation

1. Initial program 95.7%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower-log.f32 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f32 96.4

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)}\right) \]

4. Applied rewrites 96.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(-0.5 \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. log-pow-rev N/A

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

   9. log-pow-rev N/A

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

   10. lower-log.f32 N/A

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

   11. lift-pow.f32 N/A

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

   12. pow-pow N/A

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

   13. metadata-eval N/A

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

   14. pow-pow N/A

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

   15. lower-pow.f32 N/A

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

   16. metadata-eval 96.3

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

6. Applied rewrites 96.3%

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

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(\log \frac{27}{64} + 3 \cdot u\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(3 \cdot u + \log \frac{27}{64}\right)} \]

   2. lower-fma.f32 N/A

      \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(3, u, \log \frac{27}{64}\right)} \]

   3. lower-log.f32 3.4

      \[\leadsto s \cdot \mathsf{fma}\left(3, u, \color{blue}{\log 0.421875}\right) \]

9. Applied rewrites 3.4%

   \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(3, u, \log 0.421875\right)} \]
10. Add Preprocessing

Alternative 11: 7.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \log 0.421875 \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log 0.421875) s))

C

float code(float s, float u) {
	return logf(0.421875f) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = log(0.421875e0) * s
end function

Julia

function code(s, u)
	return Float32(log(Float32(0.421875)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = log(single(0.421875)) * s;
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 7.4

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

5. Applied rewrites 7.4%

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

7. Applied rewrites 7.4%

   \[\leadsto \color{blue}{\log 0.421875 \cdot s} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(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))))))