HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.2s

Alternatives: 12

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.5:\\ \;\;\;\;1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{u \cdot \left(2 + \mathsf{fma}\left(u, -2 + \frac{-4}{v}, \frac{1.3333333333333333}{v}\right)\right)}{-v}\right)}{-v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.5)
   (+
    1.0
    (*
     v
     (/
      (fma
       (- 1.0 u)
       2.0
       (/
        (* u (+ 2.0 (fma u (+ -2.0 (/ -4.0 v)) (/ 1.3333333333333333 v))))
        (- v)))
      (- v))))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.5f) {
		tmp = 1.0f + (v * (fmaf((1.0f - u), 2.0f, ((u * (2.0f + fmaf(u, (-2.0f + (-4.0f / v)), (1.3333333333333333f / v)))) / -v)) / -v));
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-0.5))
		tmp = Float32(Float32(1.0) + Float32(v * Float32(fma(Float32(Float32(1.0) - u), Float32(2.0), Float32(Float32(u * Float32(Float32(2.0) + fma(u, Float32(Float32(-2.0) + Float32(Float32(-4.0) / v)), Float32(Float32(1.3333333333333333) / v)))) / Float32(-v))) / Float32(-v))));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.5

   1. Initial program 92.8%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + \frac{1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Applied rewrites 73.8%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{u \cdot \left(2 + \left(-1 \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right)\right) + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.1%

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{u \cdot \left(2 + \mathsf{fma}\left(u, -2 + \frac{-4}{v}, \frac{1.3333333333333333}{v}\right)\right)}{-v}\right)}{-v} \]

   if -0.5 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.5199999809265137:\\ \;\;\;\;1 + \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-2}{v}, u + -1, 2\right), -2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.5199999809265137)
   (+ 1.0 (fma u (fma (/ -2.0 v) (+ u -1.0) 2.0) -2.0))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.5199999809265137f) {
		tmp = 1.0f + fmaf(u, fmaf((-2.0f / v), (u + -1.0f), 2.0f), -2.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-0.5199999809265137))
		tmp = Float32(Float32(1.0) + fma(u, fma(Float32(Float32(-2.0) / v), Float32(u + Float32(-1.0)), Float32(2.0)), Float32(-2.0)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.519999981

   1. Initial program 92.4%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower--.f32 N/A

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

      13. lower-fma.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lower-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Applied rewrites 66.0%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{0.5}{v}, \mathsf{fma}\left(-2, -u, -2\right)\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \left(u \cdot \left(2 + \left(-2 \cdot \frac{u}{v} + 2 \cdot \frac{1}{v}\right)\right) - \color{blue}{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 66.0%

      \[\leadsto 1 + \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(\frac{-2}{v}, -1 + u, 2\right)}, -2\right) \]

   if -0.519999981 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.5199999809265137:\\ \;\;\;\;1 + \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-2}{v}, u + -1, 2\right), -2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.5199999809265137:\\ \;\;\;\;1 + \mathsf{fma}\left(u, 2 + \frac{2}{v}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.5199999809265137)
   (+ 1.0 (fma u (+ 2.0 (/ 2.0 v)) -2.0))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.5199999809265137f) {
		tmp = 1.0f + fmaf(u, (2.0f + (2.0f / v)), -2.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-0.5199999809265137))
		tmp = Float32(Float32(1.0) + fma(u, Float32(Float32(2.0) + Float32(Float32(2.0) / v)), Float32(-2.0)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.519999981

   1. Initial program 92.4%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower--.f32 N/A

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

      13. lower-fma.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lower-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Applied rewrites 66.0%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{0.5}{v}, \mathsf{fma}\left(-2, -u, -2\right)\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) - \color{blue}{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.2%

      \[\leadsto 1 + \mathsf{fma}\left(u, \color{blue}{2 + \frac{2}{v}}, -2\right) \]

   if -0.519999981 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (fma -2.0 (- 1.0 u) 1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 92.7%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{1 + -2 \cdot \left(1 - u\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-2 \cdot \left(1 - u\right) + 1} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right)} \]

      3. lower--.f32 56.0

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{1 - u}, 1\right) \]

   5. Applied rewrites 56.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right)} \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (fma v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))) 1.0))

C

float code(float u, float v) {
	return fmaf(v, logf((u + ((1.0f - u) * expf((-2.0f / v))))), 1.0f);
}

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v))))), Float32(1.0))
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{\color{blue}{\mathsf{neg}\left(2\right)}}{v}}, 1 - u, u\right)\right), 1\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\color{blue}{\mathsf{neg}\left(\frac{2}{v}\right)}}, 1 - u, u\right)\right), 1\right) \]

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)}, 1 - u, u\right)\right), 1\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}}, 1 - u, u\right)\right), 1\right) \]

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.4

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right), 1\right) \]

8. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right) \]
9. Add Preprocessing

Alternative 7: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (fma v (log (fma (exp (/ -2.0 v)) (- 1.0 u) u)) 1.0))

C

float code(float u, float v) {
	return fmaf(v, logf(fmaf(expf((-2.0f / v)), (1.0f - u), u)), 1.0f);
}

Julia

function code(u, v)
	return fma(v, log(fma(exp(Float32(Float32(-2.0) / v)), Float32(Float32(1.0) - u), u)), Float32(1.0))
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{\color{blue}{\mathsf{neg}\left(2\right)}}{v}}, 1 - u, u\right)\right), 1\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\color{blue}{\mathsf{neg}\left(\frac{2}{v}\right)}}, 1 - u, u\right)\right), 1\right) \]

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)}, 1 - u, u\right)\right), 1\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}}, 1 - u, u\right)\right), 1\right) \]

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.4

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), 1\right)} \]
6. Add Preprocessing

Alternative 8: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, \log \left(u + e^{\frac{-2}{v}}\right), 1\right) \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (fma v (log (+ u (exp (/ -2.0 v)))) 1.0))

C

float code(float u, float v) {
	return fmaf(v, logf((u + expf((-2.0f / v)))), 1.0f);
}

Julia

function code(u, v)
	return fma(v, log(Float32(u + exp(Float32(Float32(-2.0) / v)))), Float32(1.0))
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{\color{blue}{\mathsf{neg}\left(2\right)}}{v}}, 1 - u, u\right)\right), 1\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\color{blue}{\mathsf{neg}\left(\frac{2}{v}\right)}}, 1 - u, u\right)\right), 1\right) \]

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)}, 1 - u, u\right)\right), 1\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}}, 1 - u, u\right)\right), 1\right) \]

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.4

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right), 1\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v}} + u\right), 1\right) \]
9. Step-by-step derivation

10. Applied rewrites 95.5%

    \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v}} + u\right), 1\right) \]

11. Final simplification 95.5%

    \[\leadsto \mathsf{fma}\left(v, \log \left(u + e^{\frac{-2}{v}}\right), 1\right) \]
12. Add Preprocessing

Alternative 9: 91.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1 - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \mathsf{fma}\left(u, -16, -8\right)\right), \frac{0.16666666666666666}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (fma
    -2.0
    (- 1.0 u)
    (-
     1.0
     (/
      (fma
       (* (- 1.0 u) (+ 8.0 (* (- 1.0 u) (fma u -16.0 -8.0))))
       (/ 0.16666666666666666 v)
       (* -0.5 (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), (1.0f - (fmaf(((1.0f - u) * (8.0f + ((1.0f - u) * fmaf(u, -16.0f, -8.0f)))), (0.16666666666666666f / v), (-0.5f * ((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)))) / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(1.0) - Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(8.0) + Float32(Float32(Float32(1.0) - u) * fma(u, Float32(-16.0), Float32(-8.0))))), Float32(Float32(0.16666666666666666) / v), Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))))) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{1} \]

   if 0.100000001 < v

   1. Initial program 93.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{\color{blue}{\mathsf{neg}\left(2\right)}}{v}}, 1 - u, u\right)\right), 1\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\color{blue}{\mathsf{neg}\left(\frac{2}{v}\right)}}, 1 - u, u\right)\right), 1\right) \]

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)}, 1 - u, u\right)\right), 1\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}}, 1 - u, u\right)\right), 1\right) \]

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 91.9

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

   5. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.2%

      \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right), 1\right) \]

   8. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   10. Applied rewrites 72.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1 - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \mathsf{fma}\left(u, -16, -8\right)\right), \frac{0.16666666666666666}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right)\right)}{v}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \mathsf{fma}\left(u, -16, -8\right)\right), \frac{0.16666666666666666}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (fma
      (* (- 1.0 u) (+ 8.0 (* (- 1.0 u) (fma u -16.0 -8.0))))
      (/ 0.16666666666666666 v)
      (* -0.5 (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - (fmaf(((1.0f - u) * (8.0f + ((1.0f - u) * fmaf(u, -16.0f, -8.0f)))), (0.16666666666666666f / v), (-0.5f * ((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(8.0) + Float32(Float32(Float32(1.0) - u) * fma(u, Float32(-16.0), Float32(-8.0))))), Float32(Float32(0.16666666666666666) / v), Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{1} \]

   if 0.100000001 < v

   1. Initial program 93.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

      \[\leadsto 1 + v \cdot \log \color{blue}{\left(e^{\frac{-2}{v}} + u \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-+r+ N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

         \[\leadsto 1 + v \cdot \log \left(e^{\frac{-2}{v}} \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(u\right)\right)}\right) + u\right) \]

      10. sub-neg N/A

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

      11. lower-fma.f32 N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   8. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \mathsf{fma}\left(u, -16, -8\right)\right), \frac{0.16666666666666666}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right)\right)}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 86.6% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 1.0)

C

float code(float u, float v) {
	return 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

function code(u, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 87.0%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Alternative 12: 6.0% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

float code(float u, float v) {
	return -1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = -1.0e0
end function

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 5.8%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))