HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 14.7s

Alternatives: 24

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.5

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

5. Applied rewrites 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 2: 94.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{-1}{-1 - \frac{2}{v}}\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5289999842643738)
   (+
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
       -0.5
       (/
        (fma
         0.041666666666666664
         (/ (* u (fma u (fma u (fma u -96.0 192.0) -112.0) 16.0)) v)
         (*
          (fma
           (* (- 1.0 u) (- 1.0 u))
           (fma (- 1.0 u) 16.0 -24.0)
           (fma 8.0 (- u) 8.0))
          -0.16666666666666666))
        (- v)))
      (- v)))
    1.0)
   (+ (* v (log (+ u (* (- 1.0 u) (/ -1.0 (- -1.0 (/ 2.0 v))))))) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5289999842643738f) {
		tmp = fmaf(-2.0f, (1.0f - u), (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, (fmaf(0.041666666666666664f, ((u * fmaf(u, fmaf(u, fmaf(u, -96.0f, 192.0f), -112.0f), 16.0f)) / v), (fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), 16.0f, -24.0f), fmaf(8.0f, -u, 8.0f)) * -0.16666666666666666f)) / -v)) / -v)) + 1.0f;
	} else {
		tmp = (v * logf((u + ((1.0f - u) * (-1.0f / (-1.0f - (2.0f / v))))))) + 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5289999842643738))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(-0.5), Float32(fma(Float32(0.041666666666666664), Float32(Float32(u * fma(u, fma(u, fma(u, Float32(-96.0), Float32(192.0)), Float32(-112.0)), Float32(16.0))) / v), Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), fma(Float32(8.0), Float32(-u), Float32(8.0))) * Float32(-0.16666666666666666))) / Float32(-v))) / Float32(-v))) + Float32(1.0));
	else
		tmp = Float32(Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(Float32(2.0) / v))))))) + 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)))))) < -0.528999984

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 70.1%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \color{blue}{\left(u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right) + 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \color{blue}{\mathsf{fma}\left(u, u \cdot \left(192 + -96 \cdot u\right) - 112, 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \left(192 + -96 \cdot u\right) + \left(\mathsf{neg}\left(112\right)\right)}, 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      5. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, u \cdot \left(192 + -96 \cdot u\right) + \color{blue}{-112}, 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, 192 + -96 \cdot u, -112\right)}, 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{-96 \cdot u + 192}, -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{u \cdot -96} + 192, -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      9. lower-fma.f32 70.1

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, -96, 192\right)}, -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) \]

   8. Applied rewrites 70.1%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) \]

   if -0.528999984 < (*.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. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. lift-/.f32 N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. pow-neg N/A

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

      8. lower-/.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. exp-1-e N/A

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

      11. lower-E.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in v around inf

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

      1. log-E N/A

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

      2. lower-+.f32 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 96.4

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

   7. Applied rewrites 96.4%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{-1}{-1 - \frac{2}{v}}\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5289999842643738)
   (+
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
       -0.5
       (/
        (fma
         0.041666666666666664
         (/ (* u (fma u (fma u (fma u -96.0 192.0) -112.0) 16.0)) v)
         (*
          (fma
           (* (- 1.0 u) (- 1.0 u))
           (fma (- 1.0 u) 16.0 -24.0)
           (fma 8.0 (- u) 8.0))
          -0.16666666666666666))
        (- v)))
      (- v)))
    1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5289999842643738f) {
		tmp = fmaf(-2.0f, (1.0f - u), (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, (fmaf(0.041666666666666664f, ((u * fmaf(u, fmaf(u, fmaf(u, -96.0f, 192.0f), -112.0f), 16.0f)) / v), (fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), 16.0f, -24.0f), fmaf(8.0f, -u, 8.0f)) * -0.16666666666666666f)) / -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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5289999842643738))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(-0.5), Float32(fma(Float32(0.041666666666666664), Float32(Float32(u * fma(u, fma(u, fma(u, Float32(-96.0), Float32(192.0)), Float32(-112.0)), Float32(16.0))) / v), Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), fma(Float32(8.0), Float32(-u), Float32(8.0))) * Float32(-0.16666666666666666))) / Float32(-v))) / Float32(-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)))))) < -0.528999984

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 70.1%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \color{blue}{\left(u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right) + 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \color{blue}{\mathsf{fma}\left(u, u \cdot \left(192 + -96 \cdot u\right) - 112, 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \left(192 + -96 \cdot u\right) + \left(\mathsf{neg}\left(112\right)\right)}, 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      5. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, u \cdot \left(192 + -96 \cdot u\right) + \color{blue}{-112}, 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, 192 + -96 \cdot u, -112\right)}, 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{-96 \cdot u + 192}, -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{u \cdot -96} + 192, -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      9. lower-fma.f32 70.1

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, -96, 192\right)}, -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) \]

   8. Applied rewrites 70.1%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) \]

   if -0.528999984 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5289999842643738)
   (+
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       (fma (- 1.0 u) -4.0 4.0)
       (* (- 1.0 u) -0.5)
       (/
        (fma
         0.041666666666666664
         (/ (* u (fma u (fma u (fma u -96.0 192.0) -112.0) 16.0)) v)
         (fma
          -0.16666666666666666
          (fma (- 1.0 u) (* (- 1.0 u) (fma (- 1.0 u) 16.0 -24.0)) (* u -8.0))
          -1.3333333333333333))
        (- v)))
      (- v)))
    1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5289999842643738f) {
		tmp = fmaf(-2.0f, (1.0f - u), (fmaf(fmaf((1.0f - u), -4.0f, 4.0f), ((1.0f - u) * -0.5f), (fmaf(0.041666666666666664f, ((u * fmaf(u, fmaf(u, fmaf(u, -96.0f, 192.0f), -112.0f), 16.0f)) / v), fmaf(-0.16666666666666666f, fmaf((1.0f - u), ((1.0f - u) * fmaf((1.0f - u), 16.0f, -24.0f)), (u * -8.0f)), -1.3333333333333333f)) / -v)) / -v)) + 1.0f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5289999842643738))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)), Float32(Float32(Float32(1.0) - u) * Float32(-0.5)), Float32(fma(Float32(0.041666666666666664), Float32(Float32(u * fma(u, fma(u, fma(u, Float32(-96.0), Float32(192.0)), Float32(-112.0)), Float32(16.0))) / v), fma(Float32(-0.16666666666666666), fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0))), Float32(u * Float32(-8.0))), Float32(-1.3333333333333333))) / Float32(-v))) / Float32(-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)))))) < -0.528999984

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 70.1%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right)} \]

   7. Applied rewrites 70.1%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, 192, -112\right), 16\right)\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), -1.3333333333333333\right)\right)}{-v}\right)}}{-v}\right) \]

   8. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)}}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)}}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \color{blue}{\left(u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right) + 16\right)}}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \color{blue}{\mathsf{fma}\left(u, u \cdot \left(192 + -96 \cdot u\right) - 112, 16\right)}}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \left(192 + -96 \cdot u\right) + \left(\mathsf{neg}\left(112\right)\right)}, 16\right)}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      5. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, u \cdot \left(192 + -96 \cdot u\right) + \color{blue}{-112}, 16\right)}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, 192 + -96 \cdot u, -112\right)}, 16\right)}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{-96 \cdot u + 192}, -112\right), 16\right)}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{u \cdot -96} + 192, -112\right), 16\right)}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      9. lower-fma.f32 70.1

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, -96, 192\right)}, -112\right), 16\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), -1.3333333333333333\right)\right)}{-v}\right)}{-v}\right) \]

   10. Applied rewrites 70.1%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), -1.3333333333333333\right)\right)}{-v}\right)}{-v}\right) \]

   if -0.528999984 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -96, 192\right), -112\right), 16\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, -112, 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5289999842643738)
   (+
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
       -0.5
       (/
        (fma
         0.041666666666666664
         (/ (* u (fma u -112.0 16.0)) v)
         (*
          (fma
           (* (- 1.0 u) (- 1.0 u))
           (fma (- 1.0 u) 16.0 -24.0)
           (fma 8.0 (- u) 8.0))
          -0.16666666666666666))
        (- v)))
      (- v)))
    1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5289999842643738f) {
		tmp = fmaf(-2.0f, (1.0f - u), (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, (fmaf(0.041666666666666664f, ((u * fmaf(u, -112.0f, 16.0f)) / v), (fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), 16.0f, -24.0f), fmaf(8.0f, -u, 8.0f)) * -0.16666666666666666f)) / -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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5289999842643738))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(-0.5), Float32(fma(Float32(0.041666666666666664), Float32(Float32(u * fma(u, Float32(-112.0), Float32(16.0))) / v), Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), fma(Float32(8.0), Float32(-u), Float32(8.0))) * Float32(-0.16666666666666666))) / Float32(-v))) / Float32(-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)))))) < -0.528999984

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 70.1%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + -112 \cdot u\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + -112 \cdot u\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \color{blue}{\left(-112 \cdot u + 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \left(\color{blue}{u \cdot -112} + 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, \mathsf{neg}\left(u\right), 8\right)\right) \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      4. lower-fma.f32 68.7

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \color{blue}{\mathsf{fma}\left(u, -112, 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) \]

   8. Applied rewrites 68.7%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, -112, 16\right)}}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) \]

   if -0.528999984 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, -112, 16\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, -112, 16\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5289999842643738)
   (+
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       (fma (- 1.0 u) -4.0 4.0)
       (* (- 1.0 u) -0.5)
       (/
        (fma
         0.041666666666666664
         (/ (* u (fma u -112.0 16.0)) v)
         (fma
          -0.16666666666666666
          (fma (- 1.0 u) (* (- 1.0 u) (fma (- 1.0 u) 16.0 -24.0)) (* u -8.0))
          -1.3333333333333333))
        (- v)))
      (- v)))
    1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5289999842643738f) {
		tmp = fmaf(-2.0f, (1.0f - u), (fmaf(fmaf((1.0f - u), -4.0f, 4.0f), ((1.0f - u) * -0.5f), (fmaf(0.041666666666666664f, ((u * fmaf(u, -112.0f, 16.0f)) / v), fmaf(-0.16666666666666666f, fmaf((1.0f - u), ((1.0f - u) * fmaf((1.0f - u), 16.0f, -24.0f)), (u * -8.0f)), -1.3333333333333333f)) / -v)) / -v)) + 1.0f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5289999842643738))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)), Float32(Float32(Float32(1.0) - u) * Float32(-0.5)), Float32(fma(Float32(0.041666666666666664), Float32(Float32(u * fma(u, Float32(-112.0), Float32(16.0))) / v), fma(Float32(-0.16666666666666666), fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0))), Float32(u * Float32(-8.0))), Float32(-1.3333333333333333))) / Float32(-v))) / Float32(-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)))))) < -0.528999984

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 70.1%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right)} \]

   7. Applied rewrites 70.1%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, 192, -112\right), 16\right)\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), -1.3333333333333333\right)\right)}{-v}\right)}}{-v}\right) \]

   8. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + -112 \cdot u\right)}}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{\color{blue}{u \cdot \left(16 + -112 \cdot u\right)}}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \color{blue}{\left(-112 \cdot u + 16\right)}}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot \frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{24}, \frac{u \cdot \left(\color{blue}{u \cdot -112} + 16\right)}{v}, \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), \frac{-4}{3}\right)\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      4. lower-fma.f32 68.7

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \color{blue}{\mathsf{fma}\left(u, -112, 16\right)}}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), -1.3333333333333333\right)\right)}{-v}\right)}{-v}\right) \]

   10. Applied rewrites 68.7%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, -112, 16\right)}}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), -1.3333333333333333\right)\right)}{-v}\right)}{-v}\right) \]

   if -0.528999984 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{u \cdot \mathsf{fma}\left(u, -112, 16\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{-v}\right)}{-v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(u, \left(2 - u \cdot \left(\frac{4}{v \cdot v} + \left(\frac{2}{v} + \frac{4.666666666666667}{v \cdot \left(v \cdot v\right)}\right)\right)\right) + \frac{\frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} - -2}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -1.0)
   (fma
    u
    (+
     (-
      2.0
      (*
       u
       (+ (/ 4.0 (* v v)) (+ (/ 2.0 v) (/ 4.666666666666667 (* v (* v v)))))))
     (/ (- (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v) -2.0) v))
    -1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -1.0f) {
		tmp = fmaf(u, ((2.0f - (u * ((4.0f / (v * v)) + ((2.0f / v) + (4.666666666666667f / (v * (v * v))))))) + ((((1.3333333333333333f + (0.6666666666666666f / v)) / v) - -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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(u, Float32(Float32(Float32(2.0) - Float32(u * Float32(Float32(Float32(4.0) / Float32(v * v)) + Float32(Float32(Float32(2.0) / v) + Float32(Float32(4.666666666666667) / Float32(v * Float32(v * v))))))) + Float32(Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v) - 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 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 72.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right)} \]

   7. Applied rewrites 72.8%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, 192, -112\right), 16\right)\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), -1.3333333333333333\right)\right)}{-v}\right)}}{-v}\right) \]

   8. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(-1 \cdot \left(u \cdot \left(4 \cdot \frac{1}{{v}^{2}} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{3}} + 2 \cdot \frac{1}{v}\right)\right)\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} - 2}{v}\right)\right) - 1} \]

   10. Applied rewrites 73.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(u, \left(2 - u \cdot \left(\frac{4}{v \cdot v} + \left(\frac{2}{v} + \frac{4.666666666666667}{v \cdot \left(v \cdot v\right)}\right)\right)\right) - \frac{\frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{-v} + -2}{v}, -1\right)} \]

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(u, \left(2 - u \cdot \left(\frac{4}{v \cdot v} + \left(\frac{2}{v} + \frac{4.666666666666667}{v \cdot \left(v \cdot v\right)}\right)\right)\right) + \frac{\frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} - -2}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \left(\frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} - \mathsf{fma}\left(u, \frac{4}{v} + \left(2 + \frac{4.666666666666667}{v \cdot v}\right), -2\right)\right)}{v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -1.0)
   (+
    (fma
     -2.0
     (- 1.0 u)
     (/
      (*
       u
       (-
        (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v)
        (fma u (+ (/ 4.0 v) (+ 2.0 (/ 4.666666666666667 (* v v)))) -2.0)))
      v))
    1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -1.0f) {
		tmp = fmaf(-2.0f, (1.0f - u), ((u * (((1.3333333333333333f + (0.6666666666666666f / v)) / v) - fmaf(u, ((4.0f / v) + (2.0f + (4.666666666666667f / (v * v)))), -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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(u * Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v) - fma(u, Float32(Float32(Float32(4.0) / v) + Float32(Float32(2.0) + Float32(Float32(4.666666666666667) / Float32(v * v)))), 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 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 72.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right) - 2\right)}}{\mathsf{neg}\left(v\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right) - 2\right)}}{\mathsf{neg}\left(v\right)}\right) \]

      2. associate--l+ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \color{blue}{\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + \left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right) - 2\right)\right)}}{\mathsf{neg}\left(v\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right) - 2\right) + -1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v}\right)}}{\mathsf{neg}\left(v\right)}\right) \]

      4. mul-1-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \left(\left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right) - 2\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v}\right)\right)}\right)}{\mathsf{neg}\left(v\right)}\right) \]

      5. unsub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right) - 2\right) - \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v}\right)}}{\mathsf{neg}\left(v\right)}\right) \]

      6. lower--.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \frac{14}{3} \cdot \frac{1}{{v}^{2}}\right)\right) - 2\right) - \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v}\right)}}{\mathsf{neg}\left(v\right)}\right) \]

   8. Applied rewrites 72.1%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{u \cdot \left(\mathsf{fma}\left(u, \frac{4}{v} + \left(\frac{4.666666666666667}{v \cdot v} + 2\right), -2\right) - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}}{-v}\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \left(\frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} - \mathsf{fma}\left(u, \frac{4}{v} + \left(2 + \frac{4.666666666666667}{v \cdot v}\right), -2\right)\right)}{v}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.8%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 69.2%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 91.2%

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

Alternative 10: 91.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(-0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1.3333333333333333 \cdot \left(u \cdot \left(1 - u\right)\right)}{v}\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5289999842643738)
   (-
    (fma (- 1.0 u) -2.0 1.0)
    (/
     (fma
      -0.5
      (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
      (/ (* -1.3333333333333333 (* u (- 1.0 u))) v))
     v))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5289999842643738f) {
		tmp = fmaf((1.0f - u), -2.0f, 1.0f) - (fmaf(-0.5f, ((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), ((-1.3333333333333333f * (u * (1.0f - u))) / v)) / v);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5289999842643738))
		tmp = Float32(fma(Float32(Float32(1.0) - u), Float32(-2.0), Float32(1.0)) - Float32(fma(Float32(-0.5), Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(Float32(Float32(-1.3333333333333333) * Float32(u * Float32(Float32(1.0) - u))) / 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)))))) < -0.528999984

   1. Initial program 93.8%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 65.9%

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

   9. Taylor expanded in u around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 64.0

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

   11. Applied rewrites 64.0%

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

   12. Taylor expanded in v around -inf

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

       1. associate-+r+ N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f32 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f32 N/A

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

       8. lower--.f32 N/A

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

       9. lower-/.f32 N/A

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

   14. Applied rewrites 64.1%

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

   if -0.528999984 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5289999842643738:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(-0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{-1.3333333333333333 \cdot \left(u \cdot \left(1 - u\right)\right)}{v}\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		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 * Float32(0.6666666666666666)) / v)) / 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)))))) < -1

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in v around -inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 N/A

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

   8. Applied rewrites 66.8%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 91.1%

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

Alternative 12: 91.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -1.0f) {
		tmp = fmaf((u / (v * v)), (1.3333333333333333f + (0.6666666666666666f / v)), 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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(Float32(u / Float32(v * v)), Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)), 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 93.8%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-fma.f32 N/A

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

   8. Applied rewrites 66.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 91.1%

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

Alternative 13: 91.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -1.0f) {
		tmp = fmaf(u, (2.0f + ((((1.3333333333333333f + (0.6666666666666666f / v)) / v) - -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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(u, Float32(Float32(2.0) + Float32(Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v) - 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 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 72.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{-v}\right)} \]

   7. Applied rewrites 72.8%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, -4, 4\right), \left(1 - u\right) \cdot -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, 192, -112\right), 16\right)\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), -8 \cdot u\right), -1.3333333333333333\right)\right)}{-v}\right)}}{-v}\right) \]

   8. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f32 N/A

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

   10. Applied rewrites 66.8%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 91.1%

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

Alternative 14: 91.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = Float32(fma(v, fma(v, fma(Float32(2.0), u, Float32(-1.0)), Float32(u * Float32(2.0))), Float32(u * Float32(1.3333333333333333))) / 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)))))) < -1

   1. Initial program 93.8%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f32 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

       6. lower-+.f32 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f32 N/A

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

       9. distribute-lft-in N/A

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

       10. metadata-eval N/A

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

       11. neg-mul-1 N/A

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

       12. lower-+.f32 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. distribute-neg-frac N/A

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

       16. metadata-eval N/A

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

       17. lower-/.f32 63.2

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

   11. Applied rewrites 63.2%

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

   12. Taylor expanded in v around 0

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

   14. Applied rewrites 63.8%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 90.8%

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

Alternative 15: 91.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.8%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

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

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

       4. mul-1-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f32 N/A

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

   11. Applied rewrites 63.6%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 90.8%

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

Alternative 16: 90.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -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 + (expf((-2.0f / v)) * (1.0f - u))))) <= -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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= 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 93.8%

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

   3. Taylor expanded in v around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 60.3%

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

   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. lower-fma.f32 N/A

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

      4. lower-+.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. lower-/.f32 61.8

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

   8. Applied rewrites 61.8%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 90.7%

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

Alternative 17: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(u, Float32(2.0), 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 93.8%

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

   3. Taylor expanded in v around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot u - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto u \cdot 2 + \color{blue}{-1} \]

      4. lower-fma.f32 51.6

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

   8. Applied rewrites 51.6%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 89.9%

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

Alternative 18: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. pow-neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u around 0

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

   1. rec-exp N/A

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

   2. log-E N/A

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

   3. metadata-eval N/A

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

   4. log-E N/A

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

   5. associate-*r/ N/A

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

   6. log-E N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. log-E N/A

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

   13. lower-exp.f32 N/A

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

   14. log-E N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 95.4

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

7. Applied rewrites 95.4%

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

8. Final simplification 95.4%

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

Alternative 19: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{-1}{\frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v} + -1}\right) + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  (*
   v
   (log
    (+
     u
     (*
      (- 1.0 u)
      (/
       -1.0
       (+ (/ (+ -2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v)) v) -1.0))))))
  1.0))

C

float code(float u, float v) {
	return (v * logf((u + ((1.0f - u) * (-1.0f / (((-2.0f + ((-2.0f + (-1.3333333333333333f / v)) / v)) / v) + -1.0f)))))) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (v * log((u + ((1.0e0 - u) * ((-1.0e0) / ((((-2.0e0) + (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)) / v) + (-1.0e0))))))) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)) / v) + Float32(-1.0))))))) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (v * log((u + ((single(1.0) - u) * (single(-1.0) / (((single(-2.0) + ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)) / v) + single(-1.0))))))) + single(1.0);
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. pow-neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around -inf

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

7. Applied rewrites 94.2%

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

8. Final simplification 94.2%

   \[\leadsto v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{-1}{\frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v} + -1}\right) + 1 \]
9. Add Preprocessing

Alternative 20: 93.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. pow-neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 92.6%

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

8. Final simplification 92.6%

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

Alternative 21: 91.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, u \cdot \left(\mathsf{fma}\left(2.6666666666666665, \frac{u}{v \cdot v}, \frac{-2}{v}\right) + \frac{-4}{v \cdot v}\right) - \left(-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (fma
    u
    (-
     (*
      u
      (+ (fma 2.6666666666666665 (/ u (* v v)) (/ -2.0 v)) (/ -4.0 (* v v))))
     (+ -2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v)))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, ((u * (fmaf(2.6666666666666665f, (u / (v * v)), (-2.0f / v)) + (-4.0f / (v * v)))) - (-2.0f + ((-2.0f + (-1.3333333333333333f / v)) / v))), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(u, Float32(Float32(u * Float32(fma(Float32(2.6666666666666665), Float32(u / Float32(v * v)), Float32(Float32(-2.0) / v)) + Float32(Float32(-4.0) / Float32(v * v)))) - Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v))), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

   if 0.200000003 < v

   1. Initial program 94.6%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 67.5%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 68.0%

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

Alternative 22: 91.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \left(1 - u\right) \cdot \frac{\mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, 16, -24\right), 8\right)}{v}, \left(1 - u\right) \cdot \mathsf{fma}\left(-0.5, \left(1 - u\right) \cdot -4, -2\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (fma
      0.16666666666666666
      (* (- 1.0 u) (/ (fma (- 1.0 u) (fma (- 1.0 u) 16.0 -24.0) 8.0) v))
      (* (- 1.0 u) (fma -0.5 (* (- 1.0 u) -4.0) -2.0)))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - (fmaf(0.16666666666666666f, ((1.0f - u) * (fmaf((1.0f - u), fmaf((1.0f - u), 16.0f, -24.0f), 8.0f) / v)), ((1.0f - u) * fmaf(-0.5f, ((1.0f - u) * -4.0f), -2.0f))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(fma(Float32(0.16666666666666666), Float32(Float32(Float32(1.0) - u) * Float32(fma(Float32(Float32(1.0) - u), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), Float32(8.0)) / v)), Float32(Float32(Float32(1.0) - u) * fma(Float32(-0.5), Float32(Float32(Float32(1.0) - u) * Float32(-4.0)), Float32(-2.0)))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

   if 0.200000003 < v

   1. Initial program 94.6%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 67.5%

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

   9. Taylor expanded in v around -inf

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

       1. associate-+r+ N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 N/A

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

       7. lower--.f32 N/A

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

       8. lower-/.f32 N/A

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

   11. Applied rewrites 67.7%

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

Alternative 23: 87.0% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 86.3%

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

Alternative 24: 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 99.5%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.2%

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

Reproduce

herbie shell --seed 2024216 
(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))))))))