Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.0% → 98.5%

Time: 7.6s

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: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9599999785423279:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9599999785423279)
     (* s (log (/ 1.0 t_0)))
     (*
      (* (- (* (- (* (- (* -64.0 u) 21.333333333333332) u) 8.0) u) 4.0) u)
      (- s)))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9599999785423279f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = (((((((-64.0f * u) - 21.333333333333332f) * u) - 8.0f) * 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) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 - (4.0e0 * u)
    if (t_0 <= 0.9599999785423279e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = ((((((((-64.0e0) * u) - 21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u) * -s
    end if
    code = tmp
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9599999785423279))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-64.0) * u) - Float32(21.333333333333332)) * u) - Float32(8.0)) * u) - Float32(4.0)) * u) * Float32(-s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	t_0 = single(1.0) - (single(4.0) * u);
	tmp = single(0.0);
	if (t_0 <= single(0.9599999785423279))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = (((((((single(-64.0) * u) - single(21.333333333333332)) * u) - single(8.0)) * u) - single(4.0)) * u) * -s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.959999979

   1. Initial program 95.6%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 54.9%

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

   3. Taylor expanded in u around inf

      \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{u} - 4\right)}}\right) \]
   4. 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 54.2

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

   5. Applied rewrites 54.2%

      \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{u} - 4\right) \cdot u}}\right) \]
   6. 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 54.2

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

      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 55.7

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

   7. Applied rewrites 55.7%

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

   8. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 99.0%

       \[\leadsto \left(-\color{blue}{\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u}\right) \cdot s \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9599999785423279:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((((((((-64.0e0) * u) - 21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u) * -s
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (((((((single(-64.0) * u) - single(21.333333333333332)) * u) - single(8.0)) * u) - single(4.0)) * u) * -s;
end

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in u around inf

   \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{u} - 4\right)}}\right) \]
4. 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 60.4

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

5. Applied rewrites 60.4%

   \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{u} - 4\right) \cdot u}}\right) \]
6. 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 60.4

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

   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 61.8

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

7. Applied rewrites 61.8%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 93.3%

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

11. Final simplification 93.3%

    \[\leadsto \left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right) \]
12. Add Preprocessing

Alternative 3: 91.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (- (* (- (* -21.333333333333332 u) 8.0) u) 4.0) u) (- s)))

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((((((-21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u) * -s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(Float32(Float32(-21.333333333333332) * u) - Float32(8.0)) * u) - Float32(4.0)) * u) * Float32(-s))
end

MATLAB

function tmp = code(s, u)
	tmp = (((((single(-21.333333333333332) * u) - single(8.0)) * u) - single(4.0)) * u) * -s;
end

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in u around inf

   \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{u} - 4\right)}}\right) \]
4. 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 60.4

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

5. Applied rewrites 60.4%

   \[\leadsto s \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{u} - 4\right) \cdot u}}\right) \]
6. 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 60.4

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

   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 61.8

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

7. Applied rewrites 61.8%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 91.2

      \[\leadsto \left(-\left(\left(\color{blue}{-21.333333333333332 \cdot u} - 8\right) \cdot u - 4\right) \cdot u\right) \cdot s \]

10. Applied rewrites 91.2%

    \[\leadsto \left(-\color{blue}{\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u}\right) \cdot s \]

11. Final simplification 91.2%

    \[\leadsto \left(\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right) \]
12. Add Preprocessing

Alternative 4: 91.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(-s\right) \cdot \left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right)\right) \cdot u \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (- s) (- (* (- (* -21.333333333333332 u) 8.0) u) 4.0)) u))

C

float code(float s, float u) {
	return (-s * ((((-21.333333333333332f * 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 * (((((-21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0)) * u
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-s) * Float32(Float32(Float32(Float32(Float32(-21.333333333333332) * u) - Float32(8.0)) * u) - Float32(4.0))) * u)
end

MATLAB

function tmp = code(s, u)
	tmp = (-s * ((((single(-21.333333333333332) * u) - single(8.0)) * u) - single(4.0))) * u;
end

Derivation

1. Initial program 61.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 42.0%

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

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 73.2

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

7. Applied rewrites 73.2%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 73.1

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

10. Applied rewrites 72.9%

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

11. Taylor expanded in s around -inf

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

13. Applied rewrites 91.2%

    \[\leadsto \left(\left(-s\right) \cdot \left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right)\right) \cdot u \]
14. Add Preprocessing

Alternative 5: 91.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(-s\right) \cdot u\right) \cdot \left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (- s) u) (- (* (- (* -21.333333333333332 u) 8.0) u) 4.0)))

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (-s * u) * (((((-21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-s) * u) * Float32(Float32(Float32(Float32(Float32(-21.333333333333332) * u) - Float32(8.0)) * u) - Float32(4.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = (-s * u) * ((((single(-21.333333333333332) * u) - single(8.0)) * u) - single(4.0));
end

Derivation

1. Initial program 61.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 43.2%

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

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 73.2

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

7. Applied rewrites 73.2%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 73.1

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

10. Applied rewrites 73.1%

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

11. Taylor expanded in s around -inf

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

13. Applied rewrites 90.9%

    \[\leadsto \left(\left(-s\right) \cdot u\right) \cdot \color{blue}{\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right)} \]
14. Add Preprocessing

Alternative 6: 87.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(8 \cdot u\right) \cdot u + u \cdot 4\right) \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (((8.0e0 * u) * u) + (u * 4.0e0))
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * (((single(8.0) * u) * u) + (u * single(4.0)));
end

Derivation

1. Initial program 61.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 42.5%

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

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 73.2

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

7. Applied rewrites 73.4%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 73.1

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

10. Applied rewrites 72.9%

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

12. Applied rewrites 86.7%

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

Alternative 7: 87.1% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 40.6%

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

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 73.4

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

7. Applied rewrites 73.2%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 73.1

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

10. Applied rewrites 72.9%

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

12. Applied rewrites 86.6%

    \[\leadsto s \cdot \left(\left(4 - -8 \cdot u\right) \cdot u\right) \]
13. Add Preprocessing

Alternative 8: 74.2% 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 61.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.1

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

5. Applied rewrites 73.1%

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

Reproduce

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