HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 14.6s

Alternatives: 19

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

   3. pow-lowering-pow.f32 N/A

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

   4. exp-1-e N/A

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

   5. E-lowering-E.f32 N/A

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

   6. /-lowering-/.f32 99.5

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

4. Applied egg-rr 99.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. log-lowering-log.f32 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. --lowering--.f32 N/A

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

   8. pow-to-exp N/A

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

   9. log-E N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. exp-lowering-exp.f32 N/A

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

   13. /-lowering-/.f32 99.5

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

6. Applied egg-rr 99.5%

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

   1. +-lowering-+.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. --lowering--.f32 N/A

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

   4. exp-lowering-exp.f32 N/A

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

   5. /-lowering-/.f32 99.6

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 91.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/ (* (- u) (+ (fma u 2.0 -2.0) (/ -1.3333333333333333 v))) v)))
   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 = 1.0f + fmaf(-2.0f, (1.0f - u), ((-u * (fmaf(u, 2.0f, -2.0f) + (-1.3333333333333333f / 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(-1.0))
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(Float32(-u) * Float32(fma(u, Float32(2.0), Float32(-2.0)) + Float32(Float32(-1.3333333333333333) / v))) / 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)))))) < -1

   1. Initial program 94.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 77.8%

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

   6. Taylor expanded in u around 0

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

      1. *-lowering-*.f32 N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. +-lowering-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 75.9

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

   8. Simplified 75.9%

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

   9. Taylor expanded in v around inf

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. accelerator-lowering-fma.f32 74.0

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

   11. Simplified 74.0%

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

   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. Simplified 93.4%

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

4. Final simplification 92.3%

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

Alternative 3: 90.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (+ (fma u 2.0 (fma 1.3333333333333333 (/ u (* v v)) -1.0)) (/ (* u 2.0) v))
   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(u, 2.0f, fmaf(1.3333333333333333f, (u / (v * v)), -1.0f)) + ((u * 2.0f) / 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(-1.0))
		tmp = Float32(fma(u, Float32(2.0), fma(Float32(1.3333333333333333), Float32(u / Float32(v * v)), Float32(-1.0))) + Float32(Float32(u * Float32(2.0)) / 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)))))) < -1

   1. Initial program 94.1%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. accelerator-lowering-expm1.f32 N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{expm1}\left(2 \cdot \frac{1}{v}\right)}, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\color{blue}{\frac{2}{v}}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}\right) \]

      15. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{\color{blue}{-2}}{v}\right) \]

      16. /-lowering-/.f32 77.2

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \color{blue}{\frac{-2}{v}}\right) \]

   5. Simplified 77.2%

      \[\leadsto 1 + v \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{-2}{v}\right)} \]

   6. Taylor expanded in v around -inf

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

   8. Simplified 70.5%

      \[\leadsto 1 + \color{blue}{\left(\mathsf{fma}\left(u, 2, -2\right) - \frac{\mathsf{fma}\left(u, -2, \frac{\mathsf{fma}\left(\frac{u}{v}, 0.6666666666666666, u \cdot 1.3333333333333333\right)}{-v}\right)}{v}\right)} \]

   9. Taylor expanded in v around inf

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

       1. associate--r+ N/A

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

       2. cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. +-lowering-+.f32 N/A

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

       5. +-commutative N/A

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

       6. associate--l+ N/A

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

       7. *-commutative N/A

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

       8. accelerator-lowering-fma.f32 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(u, 2, \frac{4}{3} \cdot \frac{u}{{v}^{2}} - 1\right)} + 2 \cdot \frac{u}{v} \]

       9. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(u, 2, \color{blue}{\frac{4}{3} \cdot \frac{u}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}\right) + 2 \cdot \frac{u}{v} \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(u, 2, \frac{4}{3} \cdot \frac{u}{{v}^{2}} + \color{blue}{-1}\right) + 2 \cdot \frac{u}{v} \]

       11. accelerator-lowering-fma.f32 N/A

           \[\leadsto \mathsf{fma}\left(u, 2, \color{blue}{\mathsf{fma}\left(\frac{4}{3}, \frac{u}{{v}^{2}}, -1\right)}\right) + 2 \cdot \frac{u}{v} \]

       12. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{fma}\left(u, 2, \mathsf{fma}\left(\frac{4}{3}, \color{blue}{\frac{u}{{v}^{2}}}, -1\right)\right) + 2 \cdot \frac{u}{v} \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(u, 2, \mathsf{fma}\left(\frac{4}{3}, \frac{u}{\color{blue}{v \cdot v}}, -1\right)\right) + 2 \cdot \frac{u}{v} \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{fma}\left(u, 2, \mathsf{fma}\left(\frac{4}{3}, \frac{u}{\color{blue}{v \cdot v}}, -1\right)\right) + 2 \cdot \frac{u}{v} \]

       15. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(u, 2, \mathsf{fma}\left(\frac{4}{3}, \frac{u}{v \cdot v}, -1\right)\right) + \color{blue}{\frac{2 \cdot u}{v}} \]

       16. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{fma}\left(u, 2, \mathsf{fma}\left(\frac{4}{3}, \frac{u}{v \cdot v}, -1\right)\right) + \color{blue}{\frac{2 \cdot u}{v}} \]

       17. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(u, 2, \mathsf{fma}\left(\frac{4}{3}, \frac{u}{v \cdot v}, -1\right)\right) + \frac{\color{blue}{u \cdot 2}}{v} \]

       18. *-lowering-*.f32 69.3

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

   11. Simplified 69.3%

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

   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. Simplified 93.4%

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

Alternative 4: 90.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (- (fma u 2.0 -1.0) (/ (fma u (/ -1.3333333333333333 v) (* u -2.0)) v))
   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(u, 2.0f, -1.0f) - (fmaf(u, (-1.3333333333333333f / v), (u * -2.0f)) / 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(-1.0))
		tmp = Float32(fma(u, Float32(2.0), Float32(-1.0)) - Float32(fma(u, Float32(Float32(-1.3333333333333333) / v), Float32(u * Float32(-2.0))) / 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)))))) < -1

   1. Initial program 94.1%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. accelerator-lowering-expm1.f32 N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{expm1}\left(2 \cdot \frac{1}{v}\right)}, \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\color{blue}{\frac{2}{v}}\right), \mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}\right) \]

      15. metadata-eval N/A

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{\color{blue}{-2}}{v}\right) \]

      16. /-lowering-/.f32 77.2

          \[\leadsto 1 + v \cdot \mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \color{blue}{\frac{-2}{v}}\right) \]

   5. Simplified 77.2%

      \[\leadsto 1 + v \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{-2}{v}\right)} \]

   6. Taylor expanded in v around -inf

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

   8. Simplified 69.2%

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

   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. Simplified 93.4%

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

Alternative 5: 90.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (fma u (+ 2.0 (/ (+ 2.0 (/ 1.3333333333333333 v)) v)) -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(u, (2.0f + ((2.0f + (1.3333333333333333f / v)) / v)), -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(u, Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v)), 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 94.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 77.8%

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

   6. Taylor expanded in u around 0

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

      1. *-lowering-*.f32 N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. +-lowering-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 75.9

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

   8. Simplified 75.9%

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

   9. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 - -1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right) - 1} \]
   10. Step-by-step derivation

       1. sub-neg N/A

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

       2. cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. *-lft-identity N/A

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

       5. +-commutative N/A

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

       6. remove-double-neg N/A

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

       7. mul-1-neg N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-in N/A

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

       10. metadata-eval N/A

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

       11. sub-neg N/A

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

       12. mul-1-neg N/A

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

       13. metadata-eval N/A

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

   11. Simplified 69.2%

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

   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. Simplified 93.4%

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

Alternative 6: 90.6% accurate, 0.9× speedup

Math

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

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) (fma (- u) (/ (fma u 2.0 -2.0) v) 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), fmaf(-u, (fmaf(u, 2.0f, -2.0f) / v), 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), fma(Float32(-u), Float32(fma(u, Float32(2.0), Float32(-2.0)) / v), 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 94.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 77.8%

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

   6. Taylor expanded in u around 0

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

      1. *-lowering-*.f32 N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. +-lowering-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 75.9

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

   8. Simplified 75.9%

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

   9. Taylor expanded in v around inf

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. associate-+l+ N/A

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

       4. accelerator-lowering-fma.f32 N/A

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

       5. --lowering--.f32 N/A

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

       6. associate-/l* N/A

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

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

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

       8. mul-1-neg N/A

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

       9. accelerator-lowering-fma.f32 N/A

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

       10. mul-1-neg N/A

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

       11. neg-lowering-neg.f32 N/A

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

       12. /-lowering-/.f32 N/A

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

       13. sub-neg N/A

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

       14. *-commutative N/A

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

       15. metadata-eval N/A

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

       16. accelerator-lowering-fma.f32 67.1

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

   11. Simplified 67.1%

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

   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. Simplified 93.4%

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

Alternative 7: 90.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (fma (* u 4.0) (/ 0.5 v) (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((u * 4.0f), (0.5f / v), 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(u * Float32(4.0)), Float32(Float32(0.5) / v), 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 94.1%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. *-lowering-*.f32 N/A

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

      13. --lowering--.f32 N/A

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

      14. accelerator-lowering-fma.f32 N/A

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

      15. --lowering--.f32 N/A

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

      16. /-lowering-/.f32 N/A

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

   5. Simplified 67.3%

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

   6. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 66.7

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

   8. Simplified 66.7%

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

   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. Simplified 93.4%

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

Alternative 8: 90.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (fma u (+ 2.0 (/ 2.0 v)) -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(u, (2.0f + (2.0f / v)), -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(u, Float32(Float32(2.0) + Float32(Float32(2.0) / v)), 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 94.1%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. *-lowering-*.f32 N/A

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

      13. --lowering--.f32 N/A

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

      14. accelerator-lowering-fma.f32 N/A

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

      15. --lowering--.f32 N/A

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

      16. /-lowering-/.f32 N/A

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

   5. Simplified 67.3%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-lowering-+.f32 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f32 66.7

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

   8. Simplified 66.7%

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

   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. Simplified 93.4%

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

Alternative 9: 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 99.6%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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. accelerator-lowering-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. log-lowering-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. accelerator-lowering-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. exp-lowering-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. /-lowering-/.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. --lowering--.f32 99.5

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

5. Simplified 99.5%

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

Alternative 10: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

   3. pow-lowering-pow.f32 N/A

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

   4. exp-1-e N/A

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

   5. E-lowering-E.f32 N/A

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

   6. /-lowering-/.f32 99.5

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

4. Applied egg-rr 99.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. log-lowering-log.f32 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. --lowering--.f32 N/A

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

   8. pow-to-exp N/A

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

   9. log-E N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. exp-lowering-exp.f32 N/A

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

   13. /-lowering-/.f32 99.5

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

6. Applied egg-rr 99.5%

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

   1. +-lowering-+.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. --lowering--.f32 N/A

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

   4. exp-lowering-exp.f32 N/A

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

   5. /-lowering-/.f32 99.6

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in u around 0

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

    1. metadata-eval N/A

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

    2. distribute-neg-frac N/A

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

    3. metadata-eval N/A

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

    4. associate-*r/ N/A

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

    5. exp-lowering-exp.f32 N/A

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

    6. associate-*r/ N/A

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

    7. metadata-eval N/A

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

    8. distribute-neg-frac N/A

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

    9. metadata-eval N/A

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

    10. /-lowering-/.f32 96.3

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

11. Simplified 96.3%

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

12. Final simplification 96.3%

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

Alternative 11: 91.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1 - u, -2, -1\right)\\ t_1 := \left(1 - u\right) \cdot \left(1 - u\right)\\ \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(4, t\_1, -1\right), v, \mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(-8, u, 8\right)\right)}{v \cdot 6}\right) \cdot \left(-t\_0\right)\right)}{v \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (fma (- 1.0 u) -2.0 -1.0)) (t_1 (* (- 1.0 u) (- 1.0 u))))
   (if (<= v 0.30000001192092896)
     1.0
     (/
      (fma
       (fma 4.0 t_1 -1.0)
       v
       (*
        (fma
         (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
         -0.5
         (/ (fma t_1 (fma (- 1.0 u) 16.0 -24.0) (fma -8.0 u 8.0)) (* v 6.0)))
        (- t_0)))
      (* v t_0)))))

C

float code(float u, float v) {
	float t_0 = fmaf((1.0f - u), -2.0f, -1.0f);
	float t_1 = (1.0f - u) * (1.0f - u);
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(fmaf(4.0f, t_1, -1.0f), v, (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, (fmaf(t_1, fmaf((1.0f - u), 16.0f, -24.0f), fmaf(-8.0f, u, 8.0f)) / (v * 6.0f))) * -t_0)) / (v * t_0);
	}
	return tmp;
}

Julia

function code(u, v)
	t_0 = fma(Float32(Float32(1.0) - u), Float32(-2.0), Float32(-1.0))
	t_1 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(fma(Float32(4.0), t_1, Float32(-1.0)), v, Float32(fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(-0.5), Float32(fma(t_1, fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), fma(Float32(-8.0), u, Float32(8.0))) / Float32(v * Float32(6.0)))) * Float32(-t_0))) / Float32(v * t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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. Simplified 93.7%

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

   if 0.300000012 < v

   1. Initial program 94.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 74.4%

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

   7. Applied egg-rr 74.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(4, \left(1 - u\right) \cdot \left(1 - u\right), -1\right), v, \mathsf{fma}\left(1 - u, -2, -1\right) \cdot \left(-\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(-8, u, 8\right)\right)}{v \cdot 6}\right)\right)\right)}{\mathsf{fma}\left(1 - u, -2, -1\right) \cdot v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(4, \left(1 - u\right) \cdot \left(1 - u\right), -1\right), v, \mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(-8, u, 8\right)\right)}{v \cdot 6}\right) \cdot \left(-\mathsf{fma}\left(1 - u, -2, -1\right)\right)\right)}{v \cdot \mathsf{fma}\left(1 - u, -2, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, -4, 4\right) \cdot 0.5, v \cdot \mathsf{fma}\left(u, 2, -1\right)\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(u, -8, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right), -1.3333333333333333\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (/
    (fma
     v
     (fma (- 1.0 u) (* (fma (- 1.0 u) -4.0 4.0) 0.5) (* v (fma u 2.0 -1.0)))
     (fma
      -0.16666666666666666
      (fma u -8.0 (* (* (- 1.0 u) (- 1.0 u)) (fma (- 1.0 u) 16.0 -24.0)))
      -1.3333333333333333))
    (* v v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(v, fmaf((1.0f - u), (fmaf((1.0f - u), -4.0f, 4.0f) * 0.5f), (v * fmaf(u, 2.0f, -1.0f))), fmaf(-0.16666666666666666f, fmaf(u, -8.0f, (((1.0f - u) * (1.0f - u)) * fmaf((1.0f - u), 16.0f, -24.0f))), -1.3333333333333333f)) / (v * v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(v, fma(Float32(Float32(1.0) - u), Float32(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)) * Float32(0.5)), Float32(v * fma(u, Float32(2.0), Float32(-1.0)))), fma(Float32(-0.16666666666666666), fma(u, Float32(-8.0), Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)) * fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)))), Float32(-1.3333333333333333))) / Float32(v * v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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. Simplified 93.7%

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

   if 0.300000012 < v

   1. Initial program 94.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 74.4%

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

   6. Taylor expanded in v around 0

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

   8. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(1 - u, 0.5 \cdot \mathsf{fma}\left(1 - u, -4, 4\right), v \cdot \mathsf{fma}\left(u, 2, -1\right)\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(u, -8, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right), -1.3333333333333333\right)\right)}{v \cdot v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, -4, 4\right) \cdot 0.5, v \cdot \mathsf{fma}\left(u, 2, -1\right)\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(u, -8, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right), -1.3333333333333333\right)\right)}{v \cdot v}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(0.5, \mathsf{fma}\left(u, 4, -4\right), 2\right), \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), v\right)\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (/
    (fma
     v
     (fma (- 1.0 u) (fma 0.5 (fma u 4.0 -4.0) 2.0) (fma v (fma u 2.0 -2.0) v))
     (fma
      -0.16666666666666666
      (fma (* (- 1.0 u) (- 1.0 u)) (fma (- 1.0 u) 16.0 -24.0) (* u -8.0))
      -1.3333333333333333))
    (* v v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(v, fmaf((1.0f - u), fmaf(0.5f, fmaf(u, 4.0f, -4.0f), 2.0f), fmaf(v, fmaf(u, 2.0f, -2.0f), v)), fmaf(-0.16666666666666666f, fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), 16.0f, -24.0f), (u * -8.0f)), -1.3333333333333333f)) / (v * v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(v, fma(Float32(Float32(1.0) - u), fma(Float32(0.5), fma(u, Float32(4.0), Float32(-4.0)), Float32(2.0)), fma(v, fma(u, Float32(2.0), Float32(-2.0)), v)), fma(Float32(-0.16666666666666666), fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), Float32(u * Float32(-8.0))), Float32(-1.3333333333333333))) / Float32(v * v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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. Simplified 93.7%

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

   if 0.300000012 < v

   1. Initial program 94.2%

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

      1. *-lft-identity N/A

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

      2. exp-prod N/A

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

      3. pow-lowering-pow.f32 N/A

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

      4. exp-1-e N/A

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

      5. E-lowering-E.f32 N/A

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

      6. /-lowering-/.f32 93.3

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

   4. Applied egg-rr 93.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. log-lowering-log.f32 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. --lowering--.f32 N/A

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

      8. pow-to-exp N/A

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

      9. log-E N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. exp-lowering-exp.f32 N/A

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

      13. /-lowering-/.f32 93.4

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

   6. Applied egg-rr 93.4%

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

   7. Taylor expanded in v around -inf

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

   9. Simplified 74.5%

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

   10. Taylor expanded in v around 0

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

   12. Simplified 74.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(0.5, \mathsf{fma}\left(u, 4, -4\right), 2\right), \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), v\right)\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{v \cdot v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 91.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -16, 24\right), -8\right)\right) \cdot \frac{0.16666666666666666}{v}\right)}{-v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
       -0.5
       (* (* u (fma u (fma u -16.0 24.0) -8.0)) (/ 0.16666666666666666 v)))
      (- v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, ((u * fmaf(u, fmaf(u, -16.0f, 24.0f), -8.0f)) * (0.16666666666666666f / v))) / -v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(-0.5), Float32(Float32(u * fma(u, fma(u, Float32(-16.0), Float32(24.0)), Float32(-8.0))) * Float32(Float32(0.16666666666666666) / v))) / Float32(-v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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. Simplified 93.7%

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

   if 0.300000012 < v

   1. Initial program 94.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 74.4%

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

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \color{blue}{\left(u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)\right)} \cdot \frac{\frac{1}{6}}{v}\right)}{v}\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f32 74.4

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

   8. Simplified 74.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -16, 24\right), -8\right)\right) \cdot \frac{0.16666666666666666}{v}\right)}{-v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u, -\mathsf{fma}\left(u, 4, -1.3333333333333333\right), v \cdot \mathsf{fma}\left(u, -\mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(v, \mathsf{fma}\left(-2, -u, -2\right), v\right)\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (/
    (fma
     u
     (- (fma u 4.0 -1.3333333333333333))
     (* v (fma u (- (fma u 2.0 -2.0)) (fma v (fma -2.0 (- u) -2.0) v))))
    (* v v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, -fmaf(u, 4.0f, -1.3333333333333333f), (v * fmaf(u, -fmaf(u, 2.0f, -2.0f), fmaf(v, fmaf(-2.0f, -u, -2.0f), v)))) / (v * v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(u, Float32(-fma(u, Float32(4.0), Float32(-1.3333333333333333))), Float32(v * fma(u, Float32(-fma(u, Float32(2.0), Float32(-2.0))), fma(v, fma(Float32(-2.0), Float32(-u), Float32(-2.0)), v)))) / Float32(v * v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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. Simplified 93.7%

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

   if 0.300000012 < v

   1. Initial program 94.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 74.4%

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

   6. Taylor expanded in u around 0

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

      1. *-lowering-*.f32 N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. +-lowering-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 72.7

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

   8. Simplified 72.7%

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

   9. Taylor expanded in v around 0

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

       1. /-lowering-/.f32 N/A

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

   11. Simplified 73.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u, -\mathsf{fma}\left(u, 4, -1.3333333333333333\right), v \cdot \mathsf{fma}\left(u, -\mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(v, \mathsf{fma}\left(-2, -u, -2\right), v\right)\right)\right)}{v \cdot v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 90.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) + \frac{u \cdot \left(\frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+
    (fma -2.0 (- 1.0 u) 1.0)
    (/ (* u (- (/ (fma u -4.0 1.3333333333333333) v) (fma u 2.0 -2.0))) v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) + ((u * ((fmaf(u, -4.0f, 1.3333333333333333f) / v) - fmaf(u, 2.0f, -2.0f))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) + Float32(Float32(u * Float32(Float32(fma(u, Float32(-4.0), Float32(1.3333333333333333)) / v) - fma(u, Float32(2.0), Float32(-2.0)))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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. Simplified 93.7%

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

   if 0.300000012 < v

   1. Initial program 94.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 74.4%

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

   6. Taylor expanded in u around 0

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

      1. *-lowering-*.f32 N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. +-lowering-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 72.7

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

   8. Simplified 72.7%

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

   9. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - -1 \cdot \left(u \cdot \left(2 \cdot u - 2\right)\right)}{v}\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

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

       2. +-lowering-+.f32 N/A

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

       3. +-commutative N/A

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

       4. accelerator-lowering-fma.f32 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right)} + -1 \cdot \frac{-1 \cdot \frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - -1 \cdot \left(u \cdot \left(2 \cdot u - 2\right)\right)}{v} \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2, \color{blue}{1 - u}, 1\right) + -1 \cdot \frac{-1 \cdot \frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - -1 \cdot \left(u \cdot \left(2 \cdot u - 2\right)\right)}{v} \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - -1 \cdot \left(u \cdot \left(2 \cdot u - 2\right)\right)\right)}{v}} \]

       7. mul-1-neg N/A

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

       8. distribute-lft-out-- N/A

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

       9. mul-1-neg N/A

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

       10. remove-double-neg N/A

           \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) + \frac{\color{blue}{\frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - u \cdot \left(2 \cdot u - 2\right)}}{v} \]

       11. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) + \color{blue}{\frac{\frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - u \cdot \left(2 \cdot u - 2\right)}{v}} \]

   11. Simplified 72.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) + \frac{u \cdot \left(\frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 90.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(u, 4, -1.3333333333333333\right)\right)}{v \cdot \left(-v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/
      (* u (fma v (fma u 2.0 -2.0) (fma u 4.0 -1.3333333333333333)))
      (* v (- v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), ((u * fmaf(v, fmaf(u, 2.0f, -2.0f), fmaf(u, 4.0f, -1.3333333333333333f))) / (v * -v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(u * fma(v, fma(u, Float32(2.0), Float32(-2.0)), fma(u, Float32(4.0), Float32(-1.3333333333333333)))) / Float32(v * Float32(-v)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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. Simplified 93.7%

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

   if 0.300000012 < v

   1. Initial program 94.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 74.4%

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

   6. Taylor expanded in u around 0

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

      1. *-lowering-*.f32 N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. +-lowering-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 72.7

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

   8. Simplified 72.7%

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

   9. Taylor expanded in v around 0

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

       1. /-lowering-/.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \color{blue}{\frac{-1 \cdot \left(u \cdot \left(v \cdot \left(2 \cdot u - 2\right)\right)\right) - u \cdot \left(4 \cdot u - \frac{4}{3}\right)}{{v}^{2}}}\right) \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{-1 \cdot \left(u \cdot \left(v \cdot \left(2 \cdot u - 2\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right) \cdot \left(4 \cdot u - \frac{4}{3}\right)}}{{v}^{2}}\right) \]

       3. associate-*r* N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\left(-1 \cdot u\right) \cdot \left(v \cdot \left(2 \cdot u - 2\right)\right)} + \left(\mathsf{neg}\left(u\right)\right) \cdot \left(4 \cdot u - \frac{4}{3}\right)}{{v}^{2}}\right) \]

       4. mul-1-neg N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(-1 \cdot u\right) \cdot \left(v \cdot \left(2 \cdot u - 2\right)\right) + \color{blue}{\left(-1 \cdot u\right)} \cdot \left(4 \cdot u - \frac{4}{3}\right)}{{v}^{2}}\right) \]

       5. distribute-lft-out N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\left(-1 \cdot u\right) \cdot \left(v \cdot \left(2 \cdot u - 2\right) + \left(4 \cdot u - \frac{4}{3}\right)\right)}}{{v}^{2}}\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\left(-1 \cdot u\right) \cdot \left(v \cdot \left(2 \cdot u - 2\right) + \left(4 \cdot u - \frac{4}{3}\right)\right)}}{{v}^{2}}\right) \]

       7. mul-1-neg N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\left(\mathsf{neg}\left(u\right)\right)} \cdot \left(v \cdot \left(2 \cdot u - 2\right) + \left(4 \cdot u - \frac{4}{3}\right)\right)}{{v}^{2}}\right) \]

       8. neg-lowering-neg.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\left(\mathsf{neg}\left(u\right)\right)} \cdot \left(v \cdot \left(2 \cdot u - 2\right) + \left(4 \cdot u - \frac{4}{3}\right)\right)}{{v}^{2}}\right) \]

       9. accelerator-lowering-fma.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \color{blue}{\mathsf{fma}\left(v, 2 \cdot u - 2, 4 \cdot u - \frac{4}{3}\right)}}{{v}^{2}}\right) \]

       10. sub-neg N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \mathsf{fma}\left(v, \color{blue}{2 \cdot u + \left(\mathsf{neg}\left(2\right)\right)}, 4 \cdot u - \frac{4}{3}\right)}{{v}^{2}}\right) \]

       11. *-commutative N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \mathsf{fma}\left(v, \color{blue}{u \cdot 2} + \left(\mathsf{neg}\left(2\right)\right), 4 \cdot u - \frac{4}{3}\right)}{{v}^{2}}\right) \]

       12. metadata-eval N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \mathsf{fma}\left(v, u \cdot 2 + \color{blue}{-2}, 4 \cdot u - \frac{4}{3}\right)}{{v}^{2}}\right) \]

       13. accelerator-lowering-fma.f32 N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \mathsf{fma}\left(v, \color{blue}{\mathsf{fma}\left(u, 2, -2\right)}, 4 \cdot u - \frac{4}{3}\right)}{{v}^{2}}\right) \]

       14. sub-neg N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \color{blue}{4 \cdot u + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}\right)}{{v}^{2}}\right) \]

       15. *-commutative N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \color{blue}{u \cdot 4} + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)\right)}{{v}^{2}}\right) \]

       16. metadata-eval N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), u \cdot 4 + \color{blue}{\frac{-4}{3}}\right)}{{v}^{2}}\right) \]

       17. accelerator-lowering-fma.f32 N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\mathsf{neg}\left(u\right)\right) \cdot \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \color{blue}{\mathsf{fma}\left(u, 4, \frac{-4}{3}\right)}\right)}{{v}^{2}}\right) \]

       18. unpow2 N/A

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

       19. *-lowering-*.f32 72.7

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

   11. Simplified 72.7%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \color{blue}{\frac{\left(-u\right) \cdot \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(u, 4, -1.3333333333333333\right)\right)}{v \cdot v}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(u, 4, -1.3333333333333333\right)\right)}{v \cdot \left(-v\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 86.7% 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 99.6%

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

3. Taylor expanded in v around 0

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

5. Simplified 88.4%

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

Alternative 19: 5.9% 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 99.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.6%

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

Reproduce

herbie shell --seed 2024198 
(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))))))))