Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.8% → 98.9%

Time: 49.9s

Alternatives: 14

Speedup: 2.8×

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

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

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)
use fmin_fmax_functions
    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: 60.8% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9599999785423279:\\ \;\;\;\;s \cdot \left(-0.5 \cdot \log \left(\mathsf{fma}\left(u, 4, -1\right) \cdot \mathsf{fma}\left(u, 4, -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot s, u, 4 \cdot s\right) \cdot u\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  (if (<= (- 1.0 (* 4.0 u)) 0.9599999785423279)
  (* s (* -0.5 (log (* (fma u 4.0 -1.0) (fma u 4.0 -1.0)))))
  (*
   (fma (* (fma (fma 64.0 u 21.333333333333332) u 8.0) s) u (* 4.0 s))
   u)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9599999785423279f) {
		tmp = s * (-0.5f * logf((fmaf(u, 4.0f, -1.0f) * fmaf(u, 4.0f, -1.0f))));
	} else {
		tmp = fmaf((fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f) * s), u, (4.0f * s)) * u;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9599999785423279))
		tmp = Float32(s * Float32(Float32(-0.5) * log(Float32(fma(u, Float32(4.0), Float32(-1.0)) * fma(u, Float32(4.0), Float32(-1.0))))));
	else
		tmp = Float32(fma(Float32(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)) * s), u, Float32(Float32(4.0) * s)) * u);
	end
	return 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 60.8%

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

   3. Applied rewrites 63.2%

      \[\leadsto s \cdot \color{blue}{\left(-0.5 \cdot \log \left(\mathsf{fma}\left(u, 4, -1\right) \cdot \mathsf{fma}\left(u, 4, -1\right)\right)\right)} \]

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

   1. Initial program 60.8%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.5%

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

   4. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. add-flip N/A

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

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

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. sub-flip N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f32 N/A

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

   6. Applied rewrites 94.0%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

   8. Applied rewrites 93.5%

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.968999981880188:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot s, u, 4 \cdot s\right) \cdot u\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  (if (<= (- 1.0 (* 4.0 u)) 0.968999981880188)
  (* (- (log (fma -4.0 u 1.0))) s)
  (*
   (fma (* (fma (fma 64.0 u 21.333333333333332) u 8.0) s) u (* 4.0 s))
   u)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.968999981880188f) {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	} else {
		tmp = fmaf((fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f) * s), u, (4.0f * s)) * u;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.968999981880188))
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	else
		tmp = Float32(fma(Float32(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)) * s), u, Float32(Float32(4.0) * s)) * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.968999982

   1. Initial program 60.8%

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

   3. Applied rewrites 63.2%

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

   if 0.968999982 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 60.8%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.5%

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

   4. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. add-flip N/A

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

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

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. sub-flip N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f32 N/A

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

   6. Applied rewrites 94.0%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

   8. Applied rewrites 93.5%

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

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[s \cdot \left(-0.5 \cdot \mathsf{log1p}\left(u \cdot \left(16 \cdot u - 8\right)\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (* s (* -0.5 (log1p (* u (- (* 16.0 u) 8.0))))))

C

float code(float s, float u) {
	return s * (-0.5f * log1pf((u * ((16.0f * u) - 8.0f))));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-0.5) * log1p(Float32(u * Float32(Float32(Float32(16.0) * u) - Float32(8.0))))))
end

Derivation

1. Initial program 60.8%

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

3. Applied rewrites 63.2%

   \[\leadsto s \cdot \color{blue}{\left(-0.5 \cdot \log \left(\mathsf{fma}\left(u, 4, -1\right) \cdot \mathsf{fma}\left(u, 4, -1\right)\right)\right)} \]
4. Step-by-step derivation

   1. lower-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. fabs-sqr N/A

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

   4. lift-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lift-log.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lift-*.f32 N/A

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

   10. fabs-sqr N/A

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

   11. pow2 N/A

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

   12. lift-fma.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. sum-square-pow N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. lower-log1p.f32 N/A

       \[\leadsto s \cdot \left(\frac{-1}{2} \cdot \color{blue}{\mathsf{log1p}\left({\left(u \cdot 4\right)}^{2} + 2 \cdot \left(\left(u \cdot 4\right) \cdot -1\right)\right)}\right) \]

5. Applied rewrites 98.9%

   \[\leadsto s \cdot \left(-0.5 \cdot \color{blue}{\mathsf{log1p}\left(\left(-4 \cdot u\right) \cdot \mathsf{fma}\left(-4, u, 2\right)\right)}\right) \]

6. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(-0.5 \cdot \mathsf{log1p}\left(\color{blue}{u \cdot \left(16 \cdot u - 8\right)}\right)\right) \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto s \cdot \left(\frac{-1}{2} \cdot \mathsf{log1p}\left(u \cdot \color{blue}{\left(16 \cdot u - 8\right)}\right)\right) \]

   2. lower--.f32 N/A

      \[\leadsto s \cdot \left(\frac{-1}{2} \cdot \mathsf{log1p}\left(u \cdot \left(16 \cdot u - \color{blue}{8}\right)\right)\right) \]

   3. lower-*.f32 98.9%

      \[\leadsto s \cdot \left(-0.5 \cdot \mathsf{log1p}\left(u \cdot \left(16 \cdot u - 8\right)\right)\right) \]

8. Applied rewrites 98.9%

   \[\leadsto s \cdot \left(-0.5 \cdot \mathsf{log1p}\left(\color{blue}{u \cdot \left(16 \cdot u - 8\right)}\right)\right) \]
9. Add Preprocessing

Alternative 4: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.968999981880188:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right)\right) \cdot u\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  (if (<= (- 1.0 (* 4.0 u)) 0.968999981880188)
  (* (- (log (fma -4.0 u 1.0))) s)
  (* (* s (fma (fma (fma 64.0 u 21.333333333333332) u 8.0) u 4.0)) u)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.968999981880188f) {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	} else {
		tmp = (s * fmaf(fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f), u, 4.0f)) * u;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.968999981880188))
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	else
		tmp = Float32(Float32(s * fma(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)), u, Float32(4.0))) * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.968999982

   1. Initial program 60.8%

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

   3. Applied rewrites 63.2%

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

   if 0.968999982 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 60.8%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.5%

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

   4. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. add-flip N/A

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

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

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. sub-flip N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f32 N/A

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

   6. Applied rewrites 94.0%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

   8. Applied rewrites 93.5%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f32 N/A

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

      9. lower-fma.f32 93.3%

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

   10. Applied rewrites 93.3%

       \[\leadsto \left(s \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right)\right) \cdot u \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9850000143051147:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right) \cdot s, u, 4 \cdot s\right) \cdot u\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  (if (<= (- 1.0 (* 4.0 u)) 0.9850000143051147)
  (* (- (log (fma -4.0 u 1.0))) s)
  (* (fma (* (fma 21.333333333333332 u 8.0) s) u (* 4.0 s)) u)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9850000143051147f) {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	} else {
		tmp = fmaf((fmaf(21.333333333333332f, u, 8.0f) * s), u, (4.0f * s)) * u;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9850000143051147))
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	else
		tmp = Float32(fma(Float32(fma(Float32(21.333333333333332), u, Float32(8.0)) * s), u, Float32(Float32(4.0) * s)) * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.985000014

   1. Initial program 60.8%

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

   3. Applied rewrites 63.2%

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

   if 0.985000014 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 60.8%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.5%

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

   4. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. add-flip N/A

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

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

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. sub-flip N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f32 N/A

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

   6. Applied rewrites 94.0%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

   8. Applied rewrites 93.5%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 91.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right) \cdot s, u, 4 \cdot s\right) \cdot u \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.004000000189989805:\\ \;\;\;\;\left(u - \mathsf{fma}\left(\mathsf{fma}\left(-21.333333333333332, u, -8\right), u, -3\right) \cdot u\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  (if (<= u 0.004000000189989805)
  (* (- u (* (fma (fma -21.333333333333332 u -8.0) u -3.0) u)) s)
  (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.004000000189989805f) {
		tmp = (u - (fmaf(fmaf(-21.333333333333332f, u, -8.0f), u, -3.0f) * u)) * s;
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.004000000189989805))
		tmp = Float32(Float32(u - Float32(fma(fma(Float32(-21.333333333333332), u, Float32(-8.0)), u, Float32(-3.0)) * u)) * s);
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00400000019

   1. Initial program 60.8%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 91.2%

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

   4. Applied rewrites 91.2%

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

   6. Applied rewrites 91.2%

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

      1. lift-fma.f32 N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right) \cdot u + \left(3 + 1\right)\right) \cdot u\right) \cdot s \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right) \cdot u + 3\right) + 1\right) \cdot u\right) \cdot s \]

      4. lower-+.f32 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right) \cdot u + 3\right) + 1\right) \cdot u\right) \cdot s \]

      5. lower-fma.f32 91.1%

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 3\right) + 1\right) \cdot u\right) \cdot s \]

   8. Applied rewrites 91.1%

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 3\right) + 1\right) \cdot u\right) \cdot s \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right), u, 3\right) + 1\right) \cdot \color{blue}{u}\right) \cdot s \]

      2. *-commutative N/A

         \[\leadsto \left(u \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right), u, 3\right) + 1\right)}\right) \cdot s \]

      3. lift-+.f32 N/A

         \[\leadsto \left(u \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right), u, 3\right) + \color{blue}{1}\right)\right) \cdot s \]

      4. +-commutative N/A

         \[\leadsto \left(u \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right), u, 3\right)}\right)\right) \cdot s \]

      5. distribute-rgt-in N/A

         \[\leadsto \left(1 \cdot u + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right), u, 3\right) \cdot u}\right) \cdot s \]

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

         \[\leadsto \left(1 \cdot u - \color{blue}{\left(\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right), u, 3\right)\right)\right) \cdot u}\right) \cdot s \]

      7. *-lft-identity N/A

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

      8. lower--.f32 N/A

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

      9. lower-*.f32 N/A

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

   10. Applied rewrites 91.3%

       \[\leadsto \left(u - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-21.333333333333332, u, -8\right), u, -3\right) \cdot u}\right) \cdot s \]

   if 0.00400000019 < u

   1. Initial program 60.8%

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

   3. Applied rewrites 63.2%

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

Alternative 7: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.004000000189989805:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 3\right), u, u\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  (if (<= u 0.004000000189989805)
  (* (fma (fma (fma 21.333333333333332 u 8.0) u 3.0) u u) s)
  (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.004000000189989805f) {
		tmp = fmaf(fmaf(fmaf(21.333333333333332f, u, 8.0f), u, 3.0f), u, u) * s;
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.004000000189989805))
		tmp = Float32(fma(fma(fma(Float32(21.333333333333332), u, Float32(8.0)), u, Float32(3.0)), u, u) * s);
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00400000019

   1. Initial program 60.8%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 91.2%

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

   4. Applied rewrites 91.2%

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

   6. Applied rewrites 91.2%

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

      1. lift-fma.f32 N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right) \cdot u + \left(3 + 1\right)\right) \cdot u\right) \cdot s \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right) \cdot u + 3\right) + 1\right) \cdot u\right) \cdot s \]

      4. lower-+.f32 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right) \cdot u + 3\right) + 1\right) \cdot u\right) \cdot s \]

      5. lower-fma.f32 91.1%

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 3\right) + 1\right) \cdot u\right) \cdot s \]

   8. Applied rewrites 91.1%

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 3\right) + 1\right) \cdot u\right) \cdot s \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right), u, 3\right) + 1\right) \cdot \color{blue}{u}\right) \cdot s \]

      2. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{64}{3}, u, 8\right), u, 3\right) + 1\right) \cdot u\right) \cdot s \]

      3. distribute-lft1-in N/A

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

      4. lower-fma.f32 91.4%

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 3\right), \color{blue}{u}, u\right) \cdot s \]

   10. Applied rewrites 91.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 3\right), \color{blue}{u}, u\right) \cdot s \]

   if 0.00400000019 < u

   1. Initial program 60.8%

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

   3. Applied rewrites 63.2%

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

Alternative 8: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.004000000189989805:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 4\right) \cdot u\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  (if (<= u 0.004000000189989805)
  (* (* (fma (fma 21.333333333333332 u 8.0) u 4.0) u) s)
  (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.004000000189989805f) {
		tmp = (fmaf(fmaf(21.333333333333332f, u, 8.0f), u, 4.0f) * u) * s;
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.004000000189989805))
		tmp = Float32(Float32(fma(fma(Float32(21.333333333333332), u, Float32(8.0)), u, Float32(4.0)) * u) * s);
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00400000019

   1. Initial program 60.8%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 91.2%

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

   4. Applied rewrites 91.2%

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

   6. Applied rewrites 91.2%

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

   if 0.00400000019 < u

   1. Initial program 60.8%

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

   3. Applied rewrites 63.2%

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

Alternative 9: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.0007999999797903001:\\ \;\;\;\;u \cdot \mathsf{fma}\left(4, s, 8 \cdot \left(s \cdot u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
  :precision binary32
  (if (<= u 0.0007999999797903001)
  (* u (fma 4.0 s (* 8.0 (* s u))))
  (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.0007999999797903001f) {
		tmp = u * fmaf(4.0f, s, (8.0f * (s * u)));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.0007999999797903001))
		tmp = Float32(u * fma(Float32(4.0), s, Float32(Float32(8.0) * Float32(s * u))));
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 7.9999998e-4

   1. Initial program 60.8%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 87.3%

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

   4. Applied rewrites 87.3%

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

   if 7.9999998e-4 < u

   1. Initial program 60.8%

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

   3. Applied rewrites 63.2%

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

Alternative 10: 87.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.8%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 87.3%

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

4. Applied rewrites 87.3%

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

Alternative 11: 87.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.8%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 91.2%

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

4. Applied rewrites 91.2%

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

6. Applied rewrites 91.2%

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

7. Taylor expanded in u around 0

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

9. Applied rewrites 87.1%

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

Alternative 12: 74.4% accurate, 2.8× speedup

Math

\[s \cdot \left(u \cdot 4\right) \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.8%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 91.2%

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

4. Applied rewrites 91.2%

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

5. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(u \cdot 4\right) \]
6. Step-by-step derivation

7. Applied rewrites 74.4%

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

Alternative 13: 74.2% accurate, 2.8× speedup

Math

\[4 \cdot \left(s \cdot u\right) \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.8%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 74.2%

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

4. Applied rewrites 74.2%

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

Alternative 14: 16.6% accurate, 19.4× speedup

Math

\[0 \]

FPCore

(FPCore (s u)
  :precision binary32
  0.0)

C

float code(float s, float u) {
	return 0.0f;
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 0.0e0
end function

Julia

function code(s, u)
	return Float32(0.0)
end

MATLAB

function tmp = code(s, u)
	tmp = single(0.0);
end

Derivation

1. Initial program 60.8%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. 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. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. distribute-rgt-neg-out N/A

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

   10. remove-double-neg N/A

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

   11. metadata-eval N/A

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

   12. distribute-lft-in N/A

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

   13. *-rgt-identity N/A

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

   14. associate--r+ N/A

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

   15. lower--.f32 N/A

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

   16. lower--.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 60.1%

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

3. Applied rewrites 60.1%

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

5. Applied rewrites 16.6%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(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))))))