HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 15.0s

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

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (fma
    0.16666666666666666
    (/ (* u 8.0) (* v v))
    (fma
     -2.0
     (- 1.0 u)
     (fma (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) (/ 0.5 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(0.16666666666666666f, ((u * 8.0f) / (v * v)), fmaf(-2.0f, (1.0f - u), fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), (0.5f / 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(0.16666666666666666), Float32(Float32(u * Float32(8.0)) / Float32(v * v)), fma(Float32(-2.0), Float32(Float32(1.0) - u), fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(Float32(0.5) / 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

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

   5. Simplified 77.7%

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

   6. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 74.0

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

   8. Simplified 74.0%

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

   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 3: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.800000011920929:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right) - \frac{\mathsf{fma}\left(u, -2, \frac{\mathsf{fma}\left(u, 1.3333333333333333, \frac{u}{v} \cdot 0.6666666666666666\right)}{-v}\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.800000011920929)
   (-
    (fma u 2.0 -1.0)
    (/
     (fma
      u
      -2.0
      (/ (fma u 1.3333333333333333 (* (/ u v) 0.6666666666666666)) (- v)))
     v))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.800000011920929f) {
		tmp = fmaf(u, 2.0f, -1.0f) - (fmaf(u, -2.0f, (fmaf(u, 1.3333333333333333f, ((u / v) * 0.6666666666666666f)) / -v)) / v);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-0.800000011920929))
		tmp = Float32(fma(u, Float32(2.0), Float32(-1.0)) - Float32(fma(u, Float32(-2.0), Float32(fma(u, Float32(1.3333333333333333), Float32(Float32(u / v) * Float32(0.6666666666666666))) / Float32(-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)))))) < -0.800000012

   1. Initial program 93.1%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 73.3

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

   5. Simplified 73.3%

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

   6. Taylor expanded in v around -inf

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

   8. Simplified 67.6%

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

   if -0.800000012 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

Alternative 4: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.800000011920929:\\ \;\;\;\;1 + \mathsf{fma}\left(u, \frac{2}{v} + \left(2 + \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v \cdot v}\right), -2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.800000011920929)
   (+
    1.0
    (fma
     u
     (+
      (/ 2.0 v)
      (+ 2.0 (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) (* v v))))
     -2.0))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.800000011920929f) {
		tmp = 1.0f + fmaf(u, ((2.0f / v) + (2.0f + ((1.3333333333333333f + (0.6666666666666666f / v)) / (v * v)))), -2.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-0.800000011920929))
		tmp = Float32(Float32(1.0) + fma(u, Float32(Float32(Float32(2.0) / v) + Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / Float32(v * v)))), Float32(-2.0)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.800000012

   1. Initial program 93.1%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 73.3

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

   5. Simplified 73.3%

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

   6. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

   8. Simplified 66.9%

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

   9. Taylor expanded in u around -inf

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

       1. distribute-rgt-in N/A

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

       2. distribute-lft-in N/A

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

   11. Simplified 67.4%

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

   if -0.800000012 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

Alternative 5: 90.8% accurate, 0.8× speedup

Math

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

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 -1.3333333333333333 (/ u 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(-1.3333333333333333f, (u / 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(Float32(-1.3333333333333333), Float32(u / 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%

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

   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 6: 90.8% accurate, 0.8× speedup

Math

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

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) (/ 1.3333333333333333 (* 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 / v) + (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) / v) + Float32(Float32(1.3333333333333333) / Float32(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

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

   5. Simplified 77.7%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 69.2

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

   8. Simplified 69.2%

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

   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 7: 90.8% accurate, 0.9× speedup

Math

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

FPCore

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

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

      1. sub-neg N/A

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

   8. Simplified 69.1%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f32 69.2

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

   10. Applied egg-rr 69.2%

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

   11. Taylor expanded in v around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f32 69.2

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

   13. Simplified 69.2%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.4%

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

Alternative 8: 90.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

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

      1. sub-neg N/A

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

   8. Simplified 69.1%

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

   9. Taylor expanded in v around 0

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

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

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 69.1

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

   11. Simplified 69.1%

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

   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: 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. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

   5. Simplified 67.1%

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

   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 10: 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 11: 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 12: 91.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * (fmaf(((1.0f - u) * (1.0f - u)), (0.16666666666666666f * fmaf((1.0f - u), -16.0f, 24.0f)), fmaf(v, fmaf(v, fmaf(u, 2.0f, -2.0f), ((1.0f - u) * fmaf(0.5f, ((1.0f - u) * -4.0f), 2.0f))), fmaf(u, 1.3333333333333333f, -1.3333333333333333f))) / (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(Float32(1.0) + Float32(v * Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), Float32(Float32(0.16666666666666666) * fma(Float32(Float32(1.0) - u), Float32(-16.0), Float32(24.0))), fma(v, fma(v, fma(u, Float32(2.0), Float32(-2.0)), Float32(Float32(Float32(1.0) - u) * fma(Float32(0.5), Float32(Float32(Float32(1.0) - u) * Float32(-4.0)), Float32(2.0)))), fma(u, Float32(1.3333333333333333), Float32(-1.3333333333333333)))) / Float32(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. 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. Taylor expanded in v around inf

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

   7. Simplified 74.2%

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

   8. Taylor expanded in v around 0

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

   10. Simplified 74.8%

       \[\leadsto 1 + v \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(1 - u, -4, 4\right) \cdot \left(\left(1 - u\right) \cdot 0.5\right)\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, -16, 24\right) \cdot 0.16666666666666666, \mathsf{fma}\left(-1.3333333333333333, -u, -1.3333333333333333\right)\right)\right)}{v \cdot v}}}{v} \]

   11. Taylor expanded in v around 0

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

   13. Simplified 74.9%

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

Alternative 13: 91.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(1 - u, -4, 4\right) \cdot \left(\left(1 - u\right) \cdot 0.5\right)\right), u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 2.6666666666666665, -4\right), 1.3333333333333333\right)\right)}{v \cdot v}}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+
    1.0
    (*
     v
     (/
      (/
       (fma
        v
        (fma v (fma u 2.0 -2.0) (* (fma (- 1.0 u) -4.0 4.0) (* (- 1.0 u) 0.5)))
        (* u (fma u (fma u 2.6666666666666665 -4.0) 1.3333333333333333)))
       (* v v))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * ((fmaf(v, fmaf(v, fmaf(u, 2.0f, -2.0f), (fmaf((1.0f - u), -4.0f, 4.0f) * ((1.0f - u) * 0.5f))), (u * fmaf(u, fmaf(u, 2.6666666666666665f, -4.0f), 1.3333333333333333f))) / (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(Float32(1.0) + Float32(v * Float32(Float32(fma(v, fma(v, fma(u, Float32(2.0), Float32(-2.0)), Float32(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)) * Float32(Float32(Float32(1.0) - u) * Float32(0.5)))), Float32(u * fma(u, fma(u, Float32(2.6666666666666665), Float32(-4.0)), Float32(1.3333333333333333)))) / Float32(v * 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. Taylor expanded in v around inf

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

   7. Simplified 74.2%

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

   8. Taylor expanded in v around 0

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

   10. Simplified 74.8%

       \[\leadsto 1 + v \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(1 - u, -4, 4\right) \cdot \left(\left(1 - u\right) \cdot 0.5\right)\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, -16, 24\right) \cdot 0.16666666666666666, \mathsf{fma}\left(-1.3333333333333333, -u, -1.3333333333333333\right)\right)\right)}{v \cdot v}}}{v} \]

   11. Taylor expanded in u around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. accelerator-lowering-fma.f32 74.8

          \[\leadsto 1 + v \cdot \frac{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(1 - u, -4, 4\right) \cdot \left(\left(1 - u\right) \cdot 0.5\right)\right), u \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, 2.6666666666666665, -4\right)}, 1.3333333333333333\right)\right)}{v \cdot v}}{v} \]

   13. Simplified 74.8%

       \[\leadsto 1 + v \cdot \frac{\frac{\mathsf{fma}\left(v, \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(1 - u, -4, 4\right) \cdot \left(\left(1 - u\right) \cdot 0.5\right)\right), \color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 2.6666666666666665, -4\right), 1.3333333333333333\right)}\right)}{v \cdot v}}{v} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 91.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   5. Simplified 74.3%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

   7. Applied egg-rr 74.4%

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

4. Final simplification 92.5%

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

Alternative 15: 91.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   5. Simplified 74.3%

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

      1. *-commutative N/A

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

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

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

   7. Applied egg-rr 74.3%

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

Alternative 16: 91.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   5. Simplified 74.3%

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

   6. Taylor expanded in u around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 74.3

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

   8. Simplified 74.3%

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

Alternative 17: 91.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   5. Simplified 74.3%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 72.9%

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