Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.0% → 98.0%

Time: 9.0s

Alternatives: 8

Speedup: 11.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.05000000074505806:\\ \;\;\;\;s \cdot \left(\left(\frac{\frac{\frac{4}{u} + 8}{u} + 21.333333333333332}{u} + 64\right) \cdot {u}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* 4.0 u) 0.05000000074505806)
   (*
    s
    (*
     (+ (/ (+ (/ (+ (/ 4.0 u) 8.0) u) 21.333333333333332) u) 64.0)
     (pow u 4.0)))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if ((4.0f * u) <= 0.05000000074505806f) {
		tmp = s * (((((((4.0f / u) + 8.0f) / u) + 21.333333333333332f) / u) + 64.0f) * powf(u, 4.0f));
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((4.0e0 * u) <= 0.05000000074505806e0) then
        tmp = s * (((((((4.0e0 / u) + 8.0e0) / u) + 21.333333333333332e0) / u) + 64.0e0) * (u ** 4.0e0))
    else
        tmp = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(4.0) * u) <= Float32(0.05000000074505806))
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(4.0) / u) + Float32(8.0)) / u) + Float32(21.333333333333332)) / u) + Float32(64.0)) * (u ^ Float32(4.0))));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(4.0) * u) <= single(0.05000000074505806))
		tmp = s * (((((((single(4.0) / u) + single(8.0)) / u) + single(21.333333333333332)) / u) + single(64.0)) * (u ^ single(4.0)));
	else
		tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.0500000007

   1. Initial program 56.1%

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

      1. lift-log.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. flip-- N/A

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

      6. +-lft-identity N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in u around 0

      \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(-64 \cdot u - 16\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(-64 \cdot u - 16\right) \cdot {u}^{2}}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      2. unpow2 N/A

         \[\leadsto s \cdot \frac{\left(-64 \cdot u - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      3. associate-*r* N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right)} \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      6. lower--.f32 N/A

         \[\leadsto s \cdot \frac{\left(\color{blue}{\left(-64 \cdot u - 16\right)} \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      7. lower-*.f32 79.7

         \[\leadsto s \cdot \frac{\left(\left(\color{blue}{-64 \cdot u} - 16\right) \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   7. Applied rewrites 79.7%

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   8. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      3. +-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

      6. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

      7. *-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

      9. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

      10. lower-fma.f32 80.2

          \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

   10. Applied rewrites 79.8%

       \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]

   11. Taylor expanded in u around inf

       \[\leadsto s \cdot \left({u}^{4} \cdot \color{blue}{\left(64 + \left(\frac{64}{3} \cdot \frac{1}{u} + \left(4 \cdot \frac{1}{{u}^{3}} + \frac{8}{{u}^{2}}\right)\right)\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 98.3%

       \[\leadsto s \cdot \left(\left(\frac{\frac{\frac{4}{u} + 8}{u} + 21.333333333333332}{u} + 64\right) \cdot \color{blue}{{u}^{4}}\right) \]

   if 0.0500000007 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 96.5%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.004600000102072954:\\ \;\;\;\;s \cdot \frac{\left(\left(\left(-234.66666666666666 \cdot u - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{1}{u} - 4\right) \cdot u\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* 4.0 u) 0.004600000102072954)
   (*
    s
    (/
     (* (* (- (* (- (* -234.66666666666666 u) 64.0) u) 16.0) u) u)
     (* (- (* -8.0 u) 4.0) u)))
   (* (log (* (- (/ 1.0 u) 4.0) u)) (- s))))

C

float code(float s, float u) {
	float tmp;
	if ((4.0f * u) <= 0.004600000102072954f) {
		tmp = s * (((((((-234.66666666666666f * u) - 64.0f) * u) - 16.0f) * u) * u) / (((-8.0f * u) - 4.0f) * u));
	} else {
		tmp = logf((((1.0f / u) - 4.0f) * u)) * -s;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((4.0e0 * u) <= 0.004600000102072954e0) then
        tmp = s * ((((((((-234.66666666666666e0) * u) - 64.0e0) * u) - 16.0e0) * u) * u) / ((((-8.0e0) * u) - 4.0e0) * u))
    else
        tmp = log((((1.0e0 / u) - 4.0e0) * u)) * -s
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(4.0) * u) <= Float32(0.004600000102072954))
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-234.66666666666666) * u) - Float32(64.0)) * u) - Float32(16.0)) * u) * u) / Float32(Float32(Float32(Float32(-8.0) * u) - Float32(4.0)) * u)));
	else
		tmp = Float32(log(Float32(Float32(Float32(Float32(1.0) / u) - Float32(4.0)) * u)) * Float32(-s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(4.0) * u) <= single(0.004600000102072954))
		tmp = s * (((((((single(-234.66666666666666) * u) - single(64.0)) * u) - single(16.0)) * u) * u) / (((single(-8.0) * u) - single(4.0)) * u));
	else
		tmp = log((((single(1.0) / u) - single(4.0)) * u)) * -s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.0046000001

   1. Initial program 52.3%

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

      1. lift-log.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. flip-- N/A

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

      6. +-lft-identity N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in u around 0

      \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(-64 \cdot u - 16\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(-64 \cdot u - 16\right) \cdot {u}^{2}}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      2. unpow2 N/A

         \[\leadsto s \cdot \frac{\left(-64 \cdot u - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      3. associate-*r* N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right)} \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      6. lower--.f32 N/A

         \[\leadsto s \cdot \frac{\left(\color{blue}{\left(-64 \cdot u - 16\right)} \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      7. lower-*.f32 83.5

         \[\leadsto s \cdot \frac{\left(\left(\color{blue}{-64 \cdot u} - 16\right) \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   7. Applied rewrites 83.5%

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   8. Taylor expanded in u around 0

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{u \cdot \left(-8 \cdot u - 4\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

      3. lower--.f32 N/A

         \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right)} \cdot u} \]

      4. lower-*.f32 96.7

         \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\left(\color{blue}{-8 \cdot u} - 4\right) \cdot u} \]

   10. Applied rewrites 96.7%

       \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

   11. Taylor expanded in u around 0

       \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right)}}{\left(-8 \cdot u - 4\right) \cdot u} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto s \cdot \frac{\color{blue}{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot {u}^{2}}}{\left(-8 \cdot u - 4\right) \cdot u} \]

       2. unpow2 N/A

          \[\leadsto s \cdot \frac{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\left(-8 \cdot u - 4\right) \cdot u} \]

       3. associate-*r* N/A

          \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]

       4. lower-*.f32 N/A

          \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]

       5. lower-*.f32 N/A

          \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right)} \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       6. lower--.f32 N/A

          \[\leadsto s \cdot \frac{\left(\color{blue}{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right)} \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       7. *-commutative N/A

          \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right) \cdot u} - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       8. lower-*.f32 N/A

          \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right) \cdot u} - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       9. lower--.f32 N/A

          \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right)} \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       10. lower-*.f32 97.1

           \[\leadsto s \cdot \frac{\left(\left(\left(\color{blue}{-234.66666666666666 \cdot u} - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

   13. Applied rewrites 97.1%

       \[\leadsto s \cdot \frac{\color{blue}{\left(\left(\left(-234.66666666666666 \cdot u - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]

   if 0.0046000001 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 92.9%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

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

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

      7. lift-*.f32 N/A

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

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

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

      9. lift-*.f32 N/A

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

      10. sqr-neg-rev N/A

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

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

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

      12. lift-*.f32 N/A

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

      13. lower-/.f32 N/A

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 92.3

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

   7. Applied rewrites 92.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 92.3

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-log.f32 93.2

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

   9. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\left(-\log \left(\left(\frac{1}{u} - 4\right) \cdot u\right)\right) \cdot s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.004600000102072954:\\ \;\;\;\;s \cdot \frac{\left(\left(\left(-234.66666666666666 \cdot u - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{1}{u} - 4\right) \cdot u\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.004999999888241291:\\ \;\;\;\;s \cdot \frac{\left(\left(\left(-234.66666666666666 \cdot u - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* 4.0 u) 0.004999999888241291)
   (*
    s
    (/
     (* (* (- (* (- (* -234.66666666666666 u) 64.0) u) 16.0) u) u)
     (* (- (* -8.0 u) 4.0) u)))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if ((4.0f * u) <= 0.004999999888241291f) {
		tmp = s * (((((((-234.66666666666666f * u) - 64.0f) * u) - 16.0f) * u) * u) / (((-8.0f * u) - 4.0f) * u));
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((4.0e0 * u) <= 0.004999999888241291e0) then
        tmp = s * ((((((((-234.66666666666666e0) * u) - 64.0e0) * u) - 16.0e0) * u) * u) / ((((-8.0e0) * u) - 4.0e0) * u))
    else
        tmp = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(4.0) * u) <= Float32(0.004999999888241291))
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-234.66666666666666) * u) - Float32(64.0)) * u) - Float32(16.0)) * u) * u) / Float32(Float32(Float32(Float32(-8.0) * u) - Float32(4.0)) * u)));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(4.0) * u) <= single(0.004999999888241291))
		tmp = s * (((((((single(-234.66666666666666) * u) - single(64.0)) * u) - single(16.0)) * u) * u) / (((single(-8.0) * u) - single(4.0)) * u));
	else
		tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.00499999989

   1. Initial program 52.4%

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

      1. lift-log.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. flip-- N/A

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

      6. +-lft-identity N/A

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

      7. lower-/.f32 N/A

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in u around 0

      \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(-64 \cdot u - 16\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(-64 \cdot u - 16\right) \cdot {u}^{2}}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      2. unpow2 N/A

         \[\leadsto s \cdot \frac{\left(-64 \cdot u - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      3. associate-*r* N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right)} \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      6. lower--.f32 N/A

         \[\leadsto s \cdot \frac{\left(\color{blue}{\left(-64 \cdot u - 16\right)} \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

      7. lower-*.f32 83.4

         \[\leadsto s \cdot \frac{\left(\left(\color{blue}{-64 \cdot u} - 16\right) \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   7. Applied rewrites 83.4%

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   8. Taylor expanded in u around 0

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{u \cdot \left(-8 \cdot u - 4\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

      3. lower--.f32 N/A

         \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right)} \cdot u} \]

      4. lower-*.f32 96.6

         \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\left(\color{blue}{-8 \cdot u} - 4\right) \cdot u} \]

   10. Applied rewrites 96.6%

       \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

   11. Taylor expanded in u around 0

       \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right)}}{\left(-8 \cdot u - 4\right) \cdot u} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto s \cdot \frac{\color{blue}{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot {u}^{2}}}{\left(-8 \cdot u - 4\right) \cdot u} \]

       2. unpow2 N/A

          \[\leadsto s \cdot \frac{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\left(-8 \cdot u - 4\right) \cdot u} \]

       3. associate-*r* N/A

          \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]

       4. lower-*.f32 N/A

          \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]

       5. lower-*.f32 N/A

          \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right)} \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       6. lower--.f32 N/A

          \[\leadsto s \cdot \frac{\left(\color{blue}{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right)} \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       7. *-commutative N/A

          \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right) \cdot u} - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       8. lower-*.f32 N/A

          \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right) \cdot u} - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       9. lower--.f32 N/A

          \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right)} \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

       10. lower-*.f32 97.0

           \[\leadsto s \cdot \frac{\left(\left(\left(\color{blue}{-234.66666666666666 \cdot u} - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

   13. Applied rewrites 97.0%

       \[\leadsto s \cdot \frac{\color{blue}{\left(\left(\left(-234.66666666666666 \cdot u - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]

   if 0.00499999989 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 93.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{\left(\left(\left(-234.66666666666666 \cdot u - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (/
   (* (* (- (* (- (* -234.66666666666666 u) 64.0) u) 16.0) u) u)
   (* (- (* -8.0 u) 4.0) u))))

C

float code(float s, float u) {
	return s * (((((((-234.66666666666666f * u) - 64.0f) * u) - 16.0f) * u) * u) / (((-8.0f * u) - 4.0f) * u));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((((((((-234.66666666666666e0) * u) - 64.0e0) * u) - 16.0e0) * u) * u) / ((((-8.0e0) * u) - 4.0e0) * u))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-234.66666666666666) * u) - Float32(64.0)) * u) - Float32(16.0)) * u) * u) / Float32(Float32(Float32(Float32(-8.0) * u) - Float32(4.0)) * u)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (((((((single(-234.66666666666666) * u) - single(64.0)) * u) - single(16.0)) * u) * u) / (((single(-8.0) * u) - single(4.0)) * u));
end

Derivation

1. Initial program 62.1%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. log-div N/A

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

   4. metadata-eval N/A

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

   5. flip-- N/A

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

   6. +-lft-identity N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 46.1%

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

5. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(-64 \cdot u - 16\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(-64 \cdot u - 16\right) \cdot {u}^{2}}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   2. unpow2 N/A

      \[\leadsto s \cdot \frac{\left(-64 \cdot u - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   3. associate-*r* N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right)} \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   6. lower--.f32 N/A

      \[\leadsto s \cdot \frac{\left(\color{blue}{\left(-64 \cdot u - 16\right)} \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   7. lower-*.f32 73.6

      \[\leadsto s \cdot \frac{\left(\left(\color{blue}{-64 \cdot u} - 16\right) \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

7. Applied rewrites 73.6%

   \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

8. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{u \cdot \left(-8 \cdot u - 4\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

   3. lower--.f32 N/A

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right)} \cdot u} \]

   4. lower-*.f32 85.8

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\left(\color{blue}{-8 \cdot u} - 4\right) \cdot u} \]

10. Applied rewrites 85.8%

    \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

11. Taylor expanded in u around 0

    \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right)}}{\left(-8 \cdot u - 4\right) \cdot u} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto s \cdot \frac{\color{blue}{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot {u}^{2}}}{\left(-8 \cdot u - 4\right) \cdot u} \]

    2. unpow2 N/A

       \[\leadsto s \cdot \frac{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\left(-8 \cdot u - 4\right) \cdot u} \]

    3. associate-*r* N/A

       \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]

    4. lower-*.f32 N/A

       \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]

    5. lower-*.f32 N/A

       \[\leadsto s \cdot \frac{\color{blue}{\left(\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right) \cdot u\right)} \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

    6. lower--.f32 N/A

       \[\leadsto s \cdot \frac{\left(\color{blue}{\left(u \cdot \left(\frac{-704}{3} \cdot u - 64\right) - 16\right)} \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

    7. *-commutative N/A

       \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right) \cdot u} - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

    8. lower-*.f32 N/A

       \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right) \cdot u} - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

    9. lower--.f32 N/A

       \[\leadsto s \cdot \frac{\left(\left(\color{blue}{\left(\frac{-704}{3} \cdot u - 64\right)} \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

    10. lower-*.f32 87.1

        \[\leadsto s \cdot \frac{\left(\left(\left(\color{blue}{-234.66666666666666 \cdot u} - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \]

13. Applied rewrites 87.1%

    \[\leadsto s \cdot \frac{\color{blue}{\left(\left(\left(-234.66666666666666 \cdot u - 64\right) \cdot u - 16\right) \cdot u\right) \cdot u}}{\left(-8 \cdot u - 4\right) \cdot u} \]
14. Add Preprocessing

Alternative 5: 85.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\left(-8 \cdot u - 4\right) \cdot u} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (/ (* (* (- (* -64.0 u) 16.0) u) u) (* (- (* -8.0 u) 4.0) u))))

C

float code(float s, float u) {
	return s * (((((-64.0f * u) - 16.0f) * u) * u) / (((-8.0f * u) - 4.0f) * u));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((((((-64.0e0) * u) - 16.0e0) * u) * u) / ((((-8.0e0) * u) - 4.0e0) * u))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(-64.0) * u) - Float32(16.0)) * u) * u) / Float32(Float32(Float32(Float32(-8.0) * u) - Float32(4.0)) * u)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (((((single(-64.0) * u) - single(16.0)) * u) * u) / (((single(-8.0) * u) - single(4.0)) * u));
end

Derivation

1. Initial program 62.1%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. log-div N/A

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

   4. metadata-eval N/A

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

   5. flip-- N/A

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

   6. +-lft-identity N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 45.3%

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

5. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(-64 \cdot u - 16\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(-64 \cdot u - 16\right) \cdot {u}^{2}}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   2. unpow2 N/A

      \[\leadsto s \cdot \frac{\left(-64 \cdot u - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   3. associate-*r* N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right)} \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   6. lower--.f32 N/A

      \[\leadsto s \cdot \frac{\left(\color{blue}{\left(-64 \cdot u - 16\right)} \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   7. lower-*.f32 73.6

      \[\leadsto s \cdot \frac{\left(\left(\color{blue}{-64 \cdot u} - 16\right) \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

7. Applied rewrites 73.6%

   \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

8. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{u \cdot \left(-8 \cdot u - 4\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

   3. lower--.f32 N/A

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right)} \cdot u} \]

   4. lower-*.f32 85.8

      \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\left(\color{blue}{-8 \cdot u} - 4\right) \cdot u} \]

10. Applied rewrites 85.8%

    \[\leadsto s \cdot \frac{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]
11. Add Preprocessing

Alternative 6: 53.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u\right) \cdot u + 4 \cdot u\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (+ (* (* (fma (fma 64.0 u 21.333333333333332) u 8.0) u) u) (* 4.0 u))))

C

float code(float s, float u) {
	return s * (((fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f) * u) * u) + (4.0f * u));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)) * u) * u) + Float32(Float32(4.0) * u)))
end

Derivation

1. Initial program 62.1%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. log-div N/A

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

   4. metadata-eval N/A

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

   5. flip-- N/A

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

   6. +-lft-identity N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 46.0%

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

5. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(-64 \cdot u - 16\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(-64 \cdot u - 16\right) \cdot {u}^{2}}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   2. unpow2 N/A

      \[\leadsto s \cdot \frac{\left(-64 \cdot u - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   3. associate-*r* N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right)} \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   6. lower--.f32 N/A

      \[\leadsto s \cdot \frac{\left(\color{blue}{\left(-64 \cdot u - 16\right)} \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   7. lower-*.f32 73.6

      \[\leadsto s \cdot \frac{\left(\left(\color{blue}{-64 \cdot u} - 16\right) \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

7. Applied rewrites 73.6%

   \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

8. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

   3. +-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

   4. *-commutative N/A

      \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

   6. +-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

   7. *-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

   9. +-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

   10. lower-fma.f32 73.6

       \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

10. Applied rewrites 73.2%

    \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
11. Step-by-step derivation

12. Applied rewrites 87.0%

    \[\leadsto s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u\right) \cdot u + \color{blue}{4 \cdot u}\right) \]
13. Add Preprocessing

Alternative 7: 53.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u + 4\right) \cdot u\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* (+ (* (fma (fma 64.0 u 21.333333333333332) u 8.0) u) 4.0) u)))

C

float code(float s, float u) {
	return s * (((fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f) * u) + 4.0f) * u);
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)) * u) + Float32(4.0)) * u))
end

Derivation

1. Initial program 62.1%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. log-div N/A

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

   4. metadata-eval N/A

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

   5. flip-- N/A

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

   6. +-lft-identity N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 46.0%

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

5. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{\color{blue}{{u}^{2} \cdot \left(-64 \cdot u - 16\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(-64 \cdot u - 16\right) \cdot {u}^{2}}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   2. unpow2 N/A

      \[\leadsto s \cdot \frac{\left(-64 \cdot u - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   3. associate-*r* N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right)} \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   6. lower--.f32 N/A

      \[\leadsto s \cdot \frac{\left(\color{blue}{\left(-64 \cdot u - 16\right)} \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   7. lower-*.f32 73.6

      \[\leadsto s \cdot \frac{\left(\left(\color{blue}{-64 \cdot u} - 16\right) \cdot u\right) \cdot u}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

7. Applied rewrites 73.6%

   \[\leadsto s \cdot \frac{\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u}}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

8. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

   3. +-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

   4. *-commutative N/A

      \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

   6. +-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

   7. *-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

   9. +-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

   10. lower-fma.f32 73.6

       \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

10. Applied rewrites 73.2%

    \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
11. Step-by-step derivation

12. Applied rewrites 86.9%

    \[\leadsto s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u + 4\right) \cdot u\right) \]
13. Add Preprocessing

Alternative 8: 74.1% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 73.6

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

5. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2024329 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))