HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 14.6s

Alternatives: 21

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f - (v * logf((1.0f / (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((1.0e0 / (u + ((1.0e0 - u) * exp(((-2.0e0) / v)))))))
end function

Julia

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}} + u\right)\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\frac{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - u \cdot u}{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{/.f32}\left(\left(\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - u \cdot u\right), \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\frac{1}{\frac{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}{\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v} + \frac{-2}{v}}\right) - u \cdot u}}\right)\right)\right) \]

   2. log-rec N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\mathsf{neg}\left(\log \left(\frac{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}{\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v} + \frac{-2}{v}}\right) - u \cdot u}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{neg.f32}\left(\log \left(\frac{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}{\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v} + \frac{-2}{v}}\right) - u \cdot u}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}{\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v} + \frac{-2}{v}}\right) - u \cdot u}\right)\right)\right)\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v} + \frac{-2}{v}}\right) - u \cdot u}{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{v}{\frac{1}{\frac{\left(1 - u\right) \cdot \left(\left(-2 + \frac{4 + \left(1 - u\right) \cdot -4}{\frac{v}{0.5}}\right) + \frac{-8 + \left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right)}{\frac{v \cdot v}{0.16666666666666666}}\right) + \frac{\left(1 - u\right) \cdot \left(-4 \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)\right) + \left(1 - u\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + 16\right) \cdot 0.041666666666666664\right)}{v \cdot \left(v \cdot v\right)}}{v}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (+ 1.0 (* v (log (* (expm1 (/ -2.0 v)) (- u)))))
   (+
    1.0
    (/
     v
     (/
      1.0
      (/
       (+
        (*
         (- 1.0 u)
         (+
          (+ -2.0 (/ (+ 4.0 (* (- 1.0 u) -4.0)) (/ v 0.5)))
          (/
           (+ -8.0 (* (- 1.0 u) (+ 24.0 (* (- 1.0 u) -16.0))))
           (/ (* v v) 0.16666666666666666))))
        (/
         (+
          (* (- 1.0 u) (* -4.0 (* (- 1.0 u) (* (- 1.0 u) (- 1.0 u)))))
          (*
           (- 1.0 u)
           (*
            (+ (* (- 1.0 u) (+ -112.0 (* (- 1.0 u) 192.0))) 16.0)
            0.041666666666666664)))
         (* v (* v v))))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
	} else {
		tmp = 1.0f + (v / (1.0f / ((((1.0f - u) * ((-2.0f + ((4.0f + ((1.0f - u) * -4.0f)) / (v / 0.5f))) + ((-8.0f + ((1.0f - u) * (24.0f + ((1.0f - u) * -16.0f)))) / ((v * v) / 0.16666666666666666f)))) + ((((1.0f - u) * (-4.0f * ((1.0f - u) * ((1.0f - u) * (1.0f - u))))) + ((1.0f - u) * ((((1.0f - u) * (-112.0f + ((1.0f - u) * 192.0f))) + 16.0f) * 0.041666666666666664f))) / (v * (v * v)))) / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u)))));
	else
		tmp = Float32(Float32(1.0) + Float32(v / Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(4.0) + Float32(Float32(Float32(1.0) - u) * Float32(-4.0))) / Float32(v / Float32(0.5)))) + Float32(Float32(Float32(-8.0) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(24.0) + Float32(Float32(Float32(1.0) - u) * Float32(-16.0))))) / Float32(Float32(v * v) / Float32(0.16666666666666666))))) + Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(-4.0) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))))) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(-112.0) + Float32(Float32(Float32(1.0) - u) * Float32(192.0)))) + Float32(16.0)) * Float32(0.041666666666666664)))) / Float32(v * Float32(v * v)))) / v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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 u around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\color{blue}{\left(u \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)\right)}\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(-1 \cdot e^{\frac{-2}{v}} + 1\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\left(\mathsf{neg}\left(e^{\frac{-2}{v}}\right)\right) + 1\right)\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\left(\mathsf{neg}\left(e^{\frac{-2}{v}}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)\right)\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} + -1\right)\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} - 1\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\mathsf{neg}\left(u \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(-1 \cdot \left(u \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(-1 \cdot u\right) \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(e^{\frac{-2}{v}} - 1\right) \cdot \left(-1 \cdot u\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(e^{\frac{-2}{v}} - 1\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      12. expm1-define N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{-2}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{\mathsf{neg}\left(2\right)}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{2}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      17. expm1-lowering-expm1.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(\frac{2}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      20. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\frac{\mathsf{neg}\left(2\right)}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\frac{-2}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      22. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

      23. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(\mathsf{neg}\left(u\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.7%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   7. Applied egg-rr 65.1%

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

   9. Applied egg-rr 65.8%

      \[\leadsto \frac{v}{\color{blue}{\frac{1}{\frac{\left(1 - u\right) \cdot \left(\left(-2 + \frac{4 + \left(1 - u\right) \cdot -4}{\frac{v}{0.5}}\right) + \frac{-8 + \left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right)}{\frac{v \cdot v}{0.16666666666666666}}\right) + \frac{\left(1 - u\right) \cdot \left(-4 \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)\right) + \left(1 - u\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + 16\right) \cdot 0.041666666666666664\right)}{v \cdot \left(v \cdot v\right)}}{v}}}} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{v}{\frac{1}{\frac{\left(1 - u\right) \cdot \left(\left(-2 + \frac{4 + \left(1 - u\right) \cdot -4}{\frac{v}{0.5}}\right) + \frac{-8 + \left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right)}{\frac{v \cdot v}{0.16666666666666666}}\right) + \frac{\left(1 - u\right) \cdot \left(-4 \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)\right) + \left(1 - u\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right) + 16\right) \cdot 0.041666666666666664\right)}{v \cdot \left(v \cdot v\right)}}{v}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 4: 91.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{v}{\frac{v}{\left(\left(1 - u\right) \cdot \left(-2 + \frac{4 + \left(1 - u\right) \cdot -4}{\frac{v}{0.5}}\right) + \left(\left(1 - u\right) \cdot \left(-8 + \left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right)\right)\right) \cdot \frac{0.16666666666666666}{v \cdot v}\right) + \frac{u \cdot \left(0.6666666666666666 + u \cdot \left(u \cdot \left(8 + u \cdot -4\right) + -4.666666666666667\right)\right)}{v \cdot \left(v \cdot v\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+
    1.0
    (/
     v
     (/
      v
      (+
       (+
        (* (- 1.0 u) (+ -2.0 (/ (+ 4.0 (* (- 1.0 u) -4.0)) (/ v 0.5))))
        (*
         (* (- 1.0 u) (+ -8.0 (* (- 1.0 u) (+ 24.0 (* (- 1.0 u) -16.0)))))
         (/ 0.16666666666666666 (* v v))))
       (/
        (*
         u
         (+
          0.6666666666666666
          (* u (+ (* u (+ 8.0 (* u -4.0))) -4.666666666666667))))
        (* v (* v v)))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v / (v / ((((1.0f - u) * (-2.0f + ((4.0f + ((1.0f - u) * -4.0f)) / (v / 0.5f)))) + (((1.0f - u) * (-8.0f + ((1.0f - u) * (24.0f + ((1.0f - u) * -16.0f))))) * (0.16666666666666666f / (v * v)))) + ((u * (0.6666666666666666f + (u * ((u * (8.0f + (u * -4.0f))) + -4.666666666666667f)))) / (v * (v * v))))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v / (v / ((((1.0e0 - u) * ((-2.0e0) + ((4.0e0 + ((1.0e0 - u) * (-4.0e0))) / (v / 0.5e0)))) + (((1.0e0 - u) * ((-8.0e0) + ((1.0e0 - u) * (24.0e0 + ((1.0e0 - u) * (-16.0e0)))))) * (0.16666666666666666e0 / (v * v)))) + ((u * (0.6666666666666666e0 + (u * ((u * (8.0e0 + (u * (-4.0e0)))) + (-4.666666666666667e0))))) / (v * (v * v))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v / Float32(v / Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(-2.0) + Float32(Float32(Float32(4.0) + Float32(Float32(Float32(1.0) - u) * Float32(-4.0))) / Float32(v / Float32(0.5))))) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(-8.0) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(24.0) + Float32(Float32(Float32(1.0) - u) * Float32(-16.0)))))) * Float32(Float32(0.16666666666666666) / Float32(v * v)))) + Float32(Float32(u * Float32(Float32(0.6666666666666666) + Float32(u * Float32(Float32(u * Float32(Float32(8.0) + Float32(u * Float32(-4.0)))) + Float32(-4.666666666666667))))) / Float32(v * Float32(v * v)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v / (v / ((((single(1.0) - u) * (single(-2.0) + ((single(4.0) + ((single(1.0) - u) * single(-4.0))) / (v / single(0.5))))) + (((single(1.0) - u) * (single(-8.0) + ((single(1.0) - u) * (single(24.0) + ((single(1.0) - u) * single(-16.0)))))) * (single(0.16666666666666666) / (v * v)))) + ((u * (single(0.6666666666666666) + (u * ((u * (single(8.0) + (u * single(-4.0)))) + single(-4.666666666666667))))) / (v * (v * v))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   7. Applied egg-rr 65.1%

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

   8. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\left(u \cdot \left(\frac{2}{3} + u \cdot \left(u \cdot \left(8 + -4 \cdot u\right) - \frac{14}{3}\right)\right)\right)}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{2}{3} + u \cdot \left(u \cdot \left(8 + -4 \cdot u\right) - \frac{14}{3}\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \left(u \cdot \left(u \cdot \left(8 + -4 \cdot u\right) - \frac{14}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(8 + -4 \cdot u\right) - \frac{14}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(8 + -4 \cdot u\right) + \left(\mathsf{neg}\left(\frac{14}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(8 + -4 \cdot u\right) + \frac{-14}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(u \cdot \left(8 + -4 \cdot u\right)\right), \frac{-14}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(8 + -4 \cdot u\right)\right), \frac{-14}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \left(-4 \cdot u\right)\right)\right), \frac{-14}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \left(u \cdot -4\right)\right)\right), \frac{-14}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

      10. *-lowering-*.f32 65.0%

          \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \mathsf{/.f32}\left(v, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right), \mathsf{/.f32}\left(v, \frac{1}{2}\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(-8, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(24, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -16\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, -4\right)\right)\right), \frac{-14}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), 1\right) \]

   10. Simplified 65.0%

       \[\leadsto \frac{v}{\frac{v}{\left(\left(1 - u\right) \cdot \left(-2 + \frac{\left(1 - u\right) \cdot -4 + 4}{\frac{v}{0.5}}\right) + \left(\left(1 - u\right) \cdot \left(-8 + \left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right)\right)\right) \cdot \frac{0.16666666666666666}{v \cdot v}\right) + \frac{\color{blue}{u \cdot \left(0.6666666666666666 + u \cdot \left(u \cdot \left(8 + u \cdot -4\right) + -4.666666666666667\right)\right)}}{v \cdot \left(v \cdot v\right)}}} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{v}{\frac{v}{\left(\left(1 - u\right) \cdot \left(-2 + \frac{4 + \left(1 - u\right) \cdot -4}{\frac{v}{0.5}}\right) + \left(\left(1 - u\right) \cdot \left(-8 + \left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right)\right)\right) \cdot \frac{0.16666666666666666}{v \cdot v}\right) + \frac{u \cdot \left(0.6666666666666666 + u \cdot \left(u \cdot \left(8 + u \cdot -4\right) + -4.666666666666667\right)\right)}{v \cdot \left(v \cdot v\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot \left(1 - u\right)\\ t_1 := 4 + \left(1 - u\right) \cdot -4\\ 1 + \frac{v}{v \cdot \left(\frac{\frac{\left(-8 + \left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right)\right) \cdot -0.041666666666666664}{t\_0} + \left(\frac{-0.03125 \cdot \left(t\_1 \cdot t\_1\right)}{t\_0} + -0.125 \cdot \frac{t\_1}{1 - u}\right)}{v} + \frac{0.5}{u + -1}\right)} \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* v (- 1.0 u))) (t_1 (+ 4.0 (* (- 1.0 u) -4.0))))
   (+
    1.0
    (/
     v
     (*
      v
      (+
       (/
        (+
         (/
          (*
           (+ -8.0 (* (- 1.0 u) (+ 24.0 (* (- 1.0 u) -16.0))))
           -0.041666666666666664)
          t_0)
         (+ (/ (* -0.03125 (* t_1 t_1)) t_0) (* -0.125 (/ t_1 (- 1.0 u)))))
        v)
       (/ 0.5 (+ u -1.0))))))))

C

float code(float u, float v) {
	float t_0 = v * (1.0f - u);
	float t_1 = 4.0f + ((1.0f - u) * -4.0f);
	return 1.0f + (v / (v * ((((((-8.0f + ((1.0f - u) * (24.0f + ((1.0f - u) * -16.0f)))) * -0.041666666666666664f) / t_0) + (((-0.03125f * (t_1 * t_1)) / t_0) + (-0.125f * (t_1 / (1.0f - u))))) / v) + (0.5f / (u + -1.0f)))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    t_0 = v * (1.0e0 - u)
    t_1 = 4.0e0 + ((1.0e0 - u) * (-4.0e0))
    code = 1.0e0 + (v / (v * (((((((-8.0e0) + ((1.0e0 - u) * (24.0e0 + ((1.0e0 - u) * (-16.0e0))))) * (-0.041666666666666664e0)) / t_0) + ((((-0.03125e0) * (t_1 * t_1)) / t_0) + ((-0.125e0) * (t_1 / (1.0e0 - u))))) / v) + (0.5e0 / (u + (-1.0e0))))))
end function

Julia

function code(u, v)
	t_0 = Float32(v * Float32(Float32(1.0) - u))
	t_1 = Float32(Float32(4.0) + Float32(Float32(Float32(1.0) - u) * Float32(-4.0)))
	return Float32(Float32(1.0) + Float32(v / Float32(v * Float32(Float32(Float32(Float32(Float32(Float32(Float32(-8.0) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(24.0) + Float32(Float32(Float32(1.0) - u) * Float32(-16.0))))) * Float32(-0.041666666666666664)) / t_0) + Float32(Float32(Float32(Float32(-0.03125) * Float32(t_1 * t_1)) / t_0) + Float32(Float32(-0.125) * Float32(t_1 / Float32(Float32(1.0) - u))))) / v) + Float32(Float32(0.5) / Float32(u + Float32(-1.0)))))))
end

MATLAB

function tmp = code(u, v)
	t_0 = v * (single(1.0) - u);
	t_1 = single(4.0) + ((single(1.0) - u) * single(-4.0));
	tmp = single(1.0) + (v / (v * ((((((single(-8.0) + ((single(1.0) - u) * (single(24.0) + ((single(1.0) - u) * single(-16.0))))) * single(-0.041666666666666664)) / t_0) + (((single(-0.03125) * (t_1 * t_1)) / t_0) + (single(-0.125) * (t_1 / (single(1.0) - u))))) / v) + (single(0.5) / (u + single(-1.0))))));
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around inf

   \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

5. Simplified 11.3%

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

7. Applied egg-rr 11.3%

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

8. Taylor expanded in v around -inf

   \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(-1 \cdot \frac{\left(\frac{-1}{24} \cdot \frac{\left(24 + -16 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right) - 8}{v \cdot \left(1 - u\right)} + \frac{-1}{32} \cdot \frac{{\left(4 + -4 \cdot \left(1 - u\right)\right)}^{2}}{v \cdot \left(1 - u\right)}\right) - \frac{1}{8} \cdot \frac{4 + -4 \cdot \left(1 - u\right)}{1 - u}}{v} + \frac{1}{2} \cdot \frac{1}{1 - u}\right)\right)\right)}\right), 1\right) \]

10. Simplified 91.5%

    \[\leadsto \frac{v}{\color{blue}{\left(\frac{0.5}{1 - u} - \frac{\frac{-0.041666666666666664 \cdot \left(\left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right) + -8\right)}{v \cdot \left(1 - u\right)} + \left(\frac{-0.03125 \cdot \left(\left(4 + \left(1 - u\right) \cdot -4\right) \cdot \left(4 + \left(1 - u\right) \cdot -4\right)\right)}{v \cdot \left(1 - u\right)} + -0.125 \cdot \frac{4 + \left(1 - u\right) \cdot -4}{1 - u}\right)}{v}\right) \cdot \left(-v\right)}} + 1 \]

11. Final simplification 91.5%

    \[\leadsto 1 + \frac{v}{v \cdot \left(\frac{\frac{\left(-8 + \left(1 - u\right) \cdot \left(24 + \left(1 - u\right) \cdot -16\right)\right) \cdot -0.041666666666666664}{v \cdot \left(1 - u\right)} + \left(\frac{-0.03125 \cdot \left(\left(4 + \left(1 - u\right) \cdot -4\right) \cdot \left(4 + \left(1 - u\right) \cdot -4\right)\right)}{v \cdot \left(1 - u\right)} + -0.125 \cdot \frac{4 + \left(1 - u\right) \cdot -4}{1 - u}\right)}{v} + \frac{0.5}{u + -1}\right)} \]
12. Add Preprocessing

Alternative 6: 91.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot \left(v \cdot v\right)\\ \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \left(\frac{2}{v} + \frac{0.6666666666666666}{t\_0}\right)\right)\right) + u \cdot \left(\left(u \cdot \left(\frac{2.6666666666666665}{v \cdot v} + \frac{8}{t\_0}\right) - \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right) - \frac{4.666666666666667}{t\_0}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* v (* v v))))
   (if (<= v 0.30000001192092896)
     1.0
     (+
      (*
       u
       (+
        (+
         2.0
         (+
          (/ 1.3333333333333333 (* v v))
          (+ (/ 2.0 v) (/ 0.6666666666666666 t_0))))
        (*
         u
         (-
          (-
           (* u (+ (/ 2.6666666666666665 (* v v)) (/ 8.0 t_0)))
           (+ (/ 2.0 v) (/ 4.0 (* v v))))
          (/ 4.666666666666667 t_0)))))
      -1.0))))

C

float code(float u, float v) {
	float t_0 = v * (v * v);
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = (u * ((2.0f + ((1.3333333333333333f / (v * v)) + ((2.0f / v) + (0.6666666666666666f / t_0)))) + (u * (((u * ((2.6666666666666665f / (v * v)) + (8.0f / t_0))) - ((2.0f / v) + (4.0f / (v * v)))) - (4.666666666666667f / t_0))))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = v * (v * v)
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (u * ((2.0e0 + ((1.3333333333333333e0 / (v * v)) + ((2.0e0 / v) + (0.6666666666666666e0 / t_0)))) + (u * (((u * ((2.6666666666666665e0 / (v * v)) + (8.0e0 / t_0))) - ((2.0e0 / v) + (4.0e0 / (v * v)))) - (4.666666666666667e0 / t_0))))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(v * Float32(v * v))
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(Float32(2.0) / v) + Float32(Float32(0.6666666666666666) / t_0)))) + Float32(u * Float32(Float32(Float32(u * Float32(Float32(Float32(2.6666666666666665) / Float32(v * v)) + Float32(Float32(8.0) / t_0))) - Float32(Float32(Float32(2.0) / v) + Float32(Float32(4.0) / Float32(v * v)))) - Float32(Float32(4.666666666666667) / t_0))))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = v * (v * v);
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = (u * ((single(2.0) + ((single(1.3333333333333333) / (v * v)) + ((single(2.0) / v) + (single(0.6666666666666666) / t_0)))) + (u * (((u * ((single(2.6666666666666665) / (v * v)) + (single(8.0) / t_0))) - ((single(2.0) / v) + (single(4.0) / (v * v)))) - (single(4.666666666666667) / t_0))))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{3}} + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(u \cdot \left(\frac{8}{3} \cdot \frac{1}{{v}^{2}} + 8 \cdot \frac{1}{{v}^{3}}\right) - \left(2 \cdot \frac{1}{v} + \left(4 \cdot \frac{1}{{v}^{2}} + \frac{14}{3} \cdot \frac{1}{{v}^{3}}\right)\right)\right)\right)\right)\right)\right) - 1} \]

   8. Simplified 63.5%

      \[\leadsto \color{blue}{u \cdot \left(\left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \left(\frac{2}{v} + \frac{0.6666666666666666}{v \cdot \left(v \cdot v\right)}\right)\right)\right) + u \cdot \left(\left(u \cdot \left(\frac{2.6666666666666665}{v \cdot v} + \frac{8}{v \cdot \left(v \cdot v\right)}\right) - \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right) - \frac{4.666666666666667}{v \cdot \left(v \cdot v\right)}\right)\right) + -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * (((((1.0e0 - u) * (-2.0e0)) + (0.5e0 * ((4.0e0 + ((1.0e0 - u) * (-4.0e0))) * ((1.0e0 - u) / v)))) + ((0.16666666666666666e0 * (((1.0e0 - u) * (-8.0e0)) + ((24.0e0 + ((1.0e0 - u) * (-16.0e0))) * ((1.0e0 - u) * (1.0e0 - u))))) / (v * v))) / v))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * (((((single(1.0) - u) * single(-2.0)) + (single(0.5) * ((single(4.0) + ((single(1.0) - u) * single(-4.0))) * ((single(1.0) - u) / v)))) + ((single(0.16666666666666666) * (((single(1.0) - u) * single(-8.0)) + ((single(24.0) + ((single(1.0) - u) * single(-16.0))) * ((single(1.0) - u) * (single(1.0) - u))))) / (v * v))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

4. Final simplification 91.6%

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

Alternative 8: 91.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * ((-2.0f + (u * (2.0f + (((1.3333333333333333f / (v * v)) + (2.0f / v)) + (u * (((u * 2.6666666666666665f) / (v * v)) - ((2.0f / v) + (4.0f / (v * v))))))))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * (((-2.0e0) + (u * (2.0e0 + (((1.3333333333333333e0 / (v * v)) + (2.0e0 / v)) + (u * (((u * 2.6666666666666665e0) / (v * v)) - ((2.0e0 / v) + (4.0e0 / (v * v))))))))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(Float32(Float32(-2.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(2.0) / v)) + Float32(u * Float32(Float32(Float32(u * Float32(2.6666666666666665)) / Float32(v * v)) - Float32(Float32(Float32(2.0) / v) + Float32(Float32(4.0) / Float32(v * v))))))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * ((single(-2.0) + (u * (single(2.0) + (((single(1.3333333333333333) / (v * v)) + (single(2.0) / v)) + (u * (((u * single(2.6666666666666665)) / (v * v)) - ((single(2.0) / v) + (single(4.0) / (v * v))))))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

   9. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\color{blue}{\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}, v\right)\right)\right) \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) + \left(\mathsf{neg}\left(2\right)\right)\right), v\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) + -2\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), -2\right), v\right)\right)\right) \]

   11. Simplified 60.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{-2 + u \cdot \left(2 + \left(\left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right) + u \cdot \left(\frac{u \cdot 2.6666666666666665}{v \cdot v} - \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right)\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.1% accurate, 5.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + (((1.3333333333333333f / (v * v)) + (2.0f / v)) + ((u * u) * ((2.6666666666666665f / (v * v)) - (((2.0f / v) + (4.0f / (v * v))) / u))))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((1.3333333333333333e0 / (v * v)) + (2.0e0 / v)) + ((u * u) * ((2.6666666666666665e0 / (v * v)) - (((2.0e0 / v) + (4.0e0 / (v * v))) / u))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(2.0) / v)) + Float32(Float32(u * u) * Float32(Float32(Float32(2.6666666666666665) / Float32(v * v)) - Float32(Float32(Float32(Float32(2.0) / v) + Float32(Float32(4.0) / Float32(v * v))) / u)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (((single(1.3333333333333333) / (v * v)) + (single(2.0) / v)) + ((u * u) * ((single(2.6666666666666665) / (v * v)) - (((single(2.0) / v) + (single(4.0) / (v * v))) / u))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

          \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), \color{blue}{-1}\right) \]

   11. Simplified 60.2%

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

   12. Taylor expanded in u around inf

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \color{blue}{\left({u}^{2} \cdot \left(-1 \cdot \frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u} + \frac{8}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}\right)\right)\right), -1\right) \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\left({u}^{2}\right), \left(-1 \cdot \frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u} + \frac{8}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right), -1\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\left(u \cdot u\right), \left(-1 \cdot \frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u} + \frac{8}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \left(-1 \cdot \frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u} + \frac{8}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right), -1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \left(\frac{8}{3} \cdot \frac{1}{{v}^{2}} + -1 \cdot \frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right), -1\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \left(\frac{8}{3} \cdot \frac{1}{{v}^{2}} + \left(\mathsf{neg}\left(\frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right)\right)\right), -1\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \left(\frac{8}{3} \cdot \frac{1}{{v}^{2}} - \frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{\_.f32}\left(\left(\frac{8}{3} \cdot \frac{1}{{v}^{2}}\right), \left(\frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right)\right), -1\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{\_.f32}\left(\left(\frac{\frac{8}{3} \cdot 1}{{v}^{2}}\right), \left(\frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right)\right), -1\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{\_.f32}\left(\left(\frac{\frac{8}{3}}{{v}^{2}}\right), \left(\frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right)\right), -1\right) \]

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

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{8}{3}, \left({v}^{2}\right)\right), \left(\frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right)\right), -1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{8}{3}, \left(v \cdot v\right)\right), \left(\frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right)\right), -1\right) \]

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

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{8}{3}, \mathsf{*.f32}\left(v, v\right)\right), \left(\frac{2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}}{u}\right)\right)\right)\right)\right)\right), -1\right) \]

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

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{8}{3}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{/.f32}\left(\left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right), u\right)\right)\right)\right)\right)\right), -1\right) \]

   14. Simplified 60.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right) + \left(u \cdot u\right) \cdot \left(\frac{2.6666666666666665}{v \cdot v} - \frac{\frac{2}{v} + \frac{4}{v \cdot v}}{u}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + (((1.3333333333333333f / (v * v)) + (2.0f / v)) + (((u * -2.0f) + ((u * (-4.0f + (u * 2.6666666666666665f))) / v)) / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((1.3333333333333333e0 / (v * v)) + (2.0e0 / v)) + (((u * (-2.0e0)) + ((u * ((-4.0e0) + (u * 2.6666666666666665e0))) / v)) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(2.0) / v)) + Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(-4.0) + Float32(u * Float32(2.6666666666666665)))) / v)) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (((single(1.3333333333333333) / (v * v)) + (single(2.0) / v)) + (((u * single(-2.0)) + ((u * (single(-4.0) + (u * single(2.6666666666666665)))) / v)) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

          \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), \color{blue}{-1}\right) \]

   11. Simplified 60.2%

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

   12. Taylor expanded in v around inf

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \color{blue}{\left(\frac{-2 \cdot u + \frac{u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v}}{v}\right)}\right)\right)\right), -1\right) \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(-2 \cdot u + \frac{u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v}\right), v\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(-2 \cdot u\right), \left(\frac{u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v}\right)\right), v\right)\right)\right)\right), -1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot -2\right), \left(\frac{u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v}\right)\right), v\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \left(\frac{u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v}\right)\right), v\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\left(u \cdot \left(\frac{8}{3} \cdot u - 4\right)\right), v\right)\right), v\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{8}{3} \cdot u - 4\right)\right), v\right)\right), v\right)\right)\right)\right), -1\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{8}{3} \cdot u + \left(\mathsf{neg}\left(4\right)\right)\right)\right), v\right)\right), v\right)\right)\right)\right), -1\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{8}{3} \cdot u + -4\right)\right), v\right)\right), v\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(\frac{8}{3} \cdot u\right), -4\right)\right), v\right)\right), v\right)\right)\right)\right), -1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(u \cdot \frac{8}{3}\right), -4\right)\right), v\right)\right), v\right)\right)\right)\right), -1\right) \]

       11. *-lowering-*.f32 60.2%

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \frac{8}{3}\right), -4\right)\right), v\right)\right), v\right)\right)\right)\right), -1\right) \]

   14. Simplified 60.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right) + \frac{u \cdot -2 + \frac{u \cdot \left(-4 + u \cdot 2.6666666666666665\right)}{v}}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.1% accurate, 7.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + (((((1.3333333333333333f + (u * (-4.0f + (u * 2.6666666666666665f)))) / v) - (u * 2.0f)) - -2.0f) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((((1.3333333333333333e0 + (u * ((-4.0e0) + (u * 2.6666666666666665e0)))) / v) - (u * 2.0e0)) - (-2.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(u * Float32(Float32(-4.0) + Float32(u * Float32(2.6666666666666665))))) / v) - Float32(u * Float32(2.0))) - Float32(-2.0)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (((((single(1.3333333333333333) + (u * (single(-4.0) + (u * single(2.6666666666666665))))) / v) - (u * single(2.0))) - single(-2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

          \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), \color{blue}{-1}\right) \]

   11. Simplified 60.2%

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

   12. Taylor expanded in v around -inf

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right)}\right), -1\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 + \left(\mathsf{neg}\left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)\right)\right), -1\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 - \frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\left(\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2\right), v\right)\right)\right), -1\right) \]

   14. Simplified 60.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{\left(\frac{1.3333333333333333 + u \cdot \left(-4 + u \cdot 2.6666666666666665\right)}{v} - u \cdot 2\right) - -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.8% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((1.3333333333333333e0 / (v * v)) + (2.0e0 / v)) + ((u * (-2.0e0)) / v))))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (((single(1.3333333333333333) / (v * v)) + (single(2.0) / v)) + ((u * single(-2.0)) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

          \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), \color{blue}{-1}\right) \]

   11. Simplified 60.2%

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

   12. Taylor expanded in v around inf

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \color{blue}{\left(-2 \cdot \frac{u}{v}\right)}\right)\right)\right), -1\right) \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{-2 \cdot u}{v}\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(-2 \cdot u\right), v\right)\right)\right)\right), -1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(u \cdot -2\right), v\right)\right)\right)\right), -1\right) \]

       4. *-lowering-*.f32 55.3%

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, -2\right), v\right)\right)\right)\right), -1\right) \]

   14. Simplified 55.3%

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

4. Final simplification 91.2%

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

Alternative 13: 90.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * (((-2.0e0) + (u * ((1.3333333333333333e0 / (v * v)) + (2.0e0 + (2.0e0 / v))))) / v))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * ((single(-2.0) + (u * ((single(1.3333333333333333) / (v * v)) + (single(2.0) + (single(2.0) / v))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

   9. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\color{blue}{\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + 2 \cdot \frac{1}{v}\right)\right) - 2\right)}, v\right)\right)\right) \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + 2 \cdot \frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)\right), v\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + 2 \cdot \frac{1}{v}\right)\right) + -2\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + 2 \cdot \frac{1}{v}\right)\right)\right), -2\right), v\right)\right)\right) \]

   11. Simplified 53.0%

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

4. Final simplification 91.0%

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

Alternative 14: 90.6% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * ((1.3333333333333333e0 / (v * v)) + (2.0e0 + (2.0e0 / v))))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((single(1.3333333333333333) / (v * v)) + (single(2.0) + (single(2.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

          \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + 2 \cdot \frac{1}{v}\right)\right)\right), \color{blue}{-1}\right) \]

   11. Simplified 52.9%

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

4. Final simplification 91.0%

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

Alternative 15: 90.6% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = u * (2.0e0 + (((-1.0e0) / u) + ((2.0e0 + (1.3333333333333333e0 / v)) / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = u * (single(2.0) + ((single(-1.0) / u) + ((single(2.0) + (single(1.3333333333333333) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right) - 2 \cdot \frac{1}{v}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right) + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{1}{v}}\right)\right)\right)\right)\right) \]

      4. rec-exp N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(e^{\mathsf{neg}\left(\frac{-2}{v}\right)} - 1\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{-2}{v}\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \color{blue}{\frac{1}{v}}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(\frac{2}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      10. expm1-lowering-expm1.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \color{blue}{\frac{1}{v}}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\left(\frac{2}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot 1}{v}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{v}\right)\right)\right)\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{v}}\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\frac{-2}{v}\right)\right)\right)\right) \]

      18. /-lowering-/.f32 58.8%

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(-2, \color{blue}{v}\right)\right)\right)\right) \]

   5. Simplified 58.8%

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

   6. Taylor expanded in v around -inf

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f32}\left(\left(1 + -1 \cdot \left(2 + -2 \cdot u\right)\right), \color{blue}{\left(\frac{-2 \cdot u + \frac{-4}{3} \cdot \frac{u}{v}}{v}\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f32}\left(\left(1 + \left(\mathsf{neg}\left(\left(2 + -2 \cdot u\right)\right)\right)\right), \left(\frac{-2 \cdot u + \color{blue}{\frac{-4}{3} \cdot \frac{u}{v}}}{v}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f32}\left(\left(1 - \left(2 + -2 \cdot u\right)\right), \left(\frac{\color{blue}{-2 \cdot u + \frac{-4}{3} \cdot \frac{u}{v}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 + -2 \cdot u\right)\right), \left(\frac{\color{blue}{-2 \cdot u + \frac{-4}{3} \cdot \frac{u}{v}}}{v}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 \cdot 1 + -2 \cdot u\right)\right), \left(\frac{-2 \cdot u + \color{blue}{\frac{-4}{3}} \cdot \frac{u}{v}}{v}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 \cdot 1 + \left(\mathsf{neg}\left(2\right)\right) \cdot u\right)\right), \left(\frac{-2 \cdot u + \frac{-4}{3} \cdot \frac{\color{blue}{u}}{v}}{v}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 \cdot 1 + \left(\mathsf{neg}\left(2 \cdot u\right)\right)\right)\right), \left(\frac{-2 \cdot u + \frac{-4}{3} \cdot \color{blue}{\frac{u}{v}}}{v}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 \cdot 1 + 2 \cdot \left(\mathsf{neg}\left(u\right)\right)\right)\right), \left(\frac{-2 \cdot u + \frac{-4}{3} \cdot \color{blue}{\frac{u}{v}}}{v}\right)\right) \]

      12. neg-mul-1 N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 \cdot 1 + 2 \cdot \left(-1 \cdot u\right)\right)\right), \left(\frac{-2 \cdot u + \frac{-4}{3} \cdot \frac{u}{\color{blue}{v}}}{v}\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 \cdot \left(1 + -1 \cdot u\right)\right)\right), \left(\frac{-2 \cdot u + \color{blue}{\frac{-4}{3} \cdot \frac{u}{v}}}{v}\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 \cdot \left(1 + \left(\mathsf{neg}\left(u\right)\right)\right)\right)\right), \left(\frac{-2 \cdot u + \frac{-4}{3} \cdot \frac{u}{\color{blue}{v}}}{v}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(2 \cdot \left(1 - u\right)\right)\right), \left(\frac{-2 \cdot u + \frac{-4}{3} \cdot \color{blue}{\frac{u}{v}}}{v}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \left(\left(1 - u\right) \cdot 2\right)\right), \left(\frac{-2 \cdot u + \color{blue}{\frac{-4}{3} \cdot \frac{u}{v}}}{v}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left(1 - u\right), 2\right)\right), \left(\frac{-2 \cdot u + \color{blue}{\frac{-4}{3} \cdot \frac{u}{v}}}{v}\right)\right) \]

      18. --lowering--.f32 N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 2\right)\right), \left(\frac{-2 \cdot u + \color{blue}{\frac{-4}{3}} \cdot \frac{u}{v}}{v}\right)\right) \]

      19. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 2\right)\right), \mathsf{/.f32}\left(\left(-2 \cdot u + \frac{-4}{3} \cdot \frac{u}{v}\right), \color{blue}{v}\right)\right) \]

   8. Simplified 52.9%

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

   9. Taylor expanded in u around inf

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

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

          \[\leadsto \mathsf{*.f32}\left(u, \color{blue}{\left(2 - \left(-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} + \frac{1}{u}\right)\right)}\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(u, \left(2 + \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} + \frac{1}{u}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} + \frac{1}{u}\right)\right)\right)}\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\left(\frac{1}{u} + -1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right)\right)\right)\right)\right) \]

       5. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\left(\mathsf{neg}\left(\frac{1}{u}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right)\right)}\right)\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{\mathsf{neg}\left(1\right)}{u} + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}}\right)\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{-1}{u} + \left(\mathsf{neg}\left(\color{blue}{-1} \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right)\right)\right)\right)\right) \]

       8. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{-1}{u} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right)\right)\right)\right)\right)\right)\right) \]

       9. remove-double-neg N/A

          \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{-1}{u} + \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{\color{blue}{v}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\left(\frac{-1}{u}\right), \color{blue}{\left(\frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, u\right), \left(\frac{\color{blue}{2 + \frac{4}{3} \cdot \frac{1}{v}}}{v}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, u\right), \mathsf{/.f32}\left(\left(2 + \frac{4}{3} \cdot \frac{1}{v}\right), \color{blue}{v}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, u\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(\frac{4}{3} \cdot \frac{1}{v}\right)\right), v\right)\right)\right)\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, u\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(\frac{\frac{4}{3} \cdot 1}{v}\right)\right), v\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, u\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(\frac{\frac{4}{3}}{v}\right)\right), v\right)\right)\right)\right) \]

       16. /-lowering-/.f32 52.9%

           \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, u\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{4}{3}, v\right)\right), v\right)\right)\right)\right) \]

   11. Simplified 52.9%

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

Alternative 16: 90.5% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (((-2.0e0) + (u * 2.0e0)) / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - ((single(-2.0) + (u * single(2.0))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{{v}^{3}} + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)}{v}\right)}\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)}{v}\right)}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-8 \cdot \left(1 - u\right) + \left(24 + -16 \cdot \left(1 - u\right)\right) \cdot {\left(1 - u\right)}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)}{v}\right)\right), \color{blue}{v}\right)\right)\right) \]

   8. Simplified 60.5%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

          \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), \color{blue}{-1}\right) \]

   11. Simplified 60.2%

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

   12. Taylor expanded in v around -inf

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \color{blue}{\left(2 + -1 \cdot \frac{2 \cdot u - 2}{v}\right)}\right), -1\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 + \left(\mathsf{neg}\left(\frac{2 \cdot u - 2}{v}\right)\right)\right)\right), -1\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 - \frac{2 \cdot u - 2}{v}\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \left(\frac{2 \cdot u - 2}{v}\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\left(2 \cdot u - 2\right), v\right)\right)\right), -1\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\left(2 \cdot u + \left(\mathsf{neg}\left(2\right)\right)\right), v\right)\right)\right), -1\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\left(2 \cdot u + -2\right), v\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(2 \cdot u\right), -2\right), v\right)\right)\right), -1\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot 2\right), -2\right), v\right)\right)\right), -1\right) \]

       9. *-lowering-*.f32 52.8%

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, 2\right), -2\right), v\right)\right)\right), -1\right) \]

   14. Simplified 52.8%

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

4. Final simplification 91.0%

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

Alternative 17: 90.4% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((u * 2.0e0) / v) + (2.0e0 * (u + (-1.0e0))))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((u * single(2.0)) / v) + (single(2.0) * (u + single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right) - 2 \cdot \frac{1}{v}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right) + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\left(u \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{1}{v}}\right)\right)\right)\right)\right) \]

      4. rec-exp N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(e^{\mathsf{neg}\left(\frac{-2}{v}\right)} - 1\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{-2}{v}\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \color{blue}{\frac{1}{v}}\right)\right)\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(\frac{2}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{expm1}\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      10. expm1-lowering-expm1.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \color{blue}{\frac{1}{v}}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\left(\frac{2}{v}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{\color{blue}{1}}{v}\right)\right)\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot 1}{v}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{v}\right)\right)\right)\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{v}}\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \left(\frac{-2}{v}\right)\right)\right)\right) \]

      18. /-lowering-/.f32 58.8%

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(-2, \color{blue}{v}\right)\right)\right)\right) \]

   5. Simplified 58.8%

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

   6. Taylor expanded in v around -inf

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

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

         \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-1 \cdot \left(2 + -2 \cdot u\right) + 2 \cdot \frac{u}{v}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(2 \cdot \frac{u}{v} + \color{blue}{-1 \cdot \left(2 + -2 \cdot u\right)}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(2 \cdot \frac{u}{v} + \left(\mathsf{neg}\left(\left(2 + -2 \cdot u\right)\right)\right)\right)\right) \]

      4. unsub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(2 \cdot \frac{u}{v} - \color{blue}{\left(2 + -2 \cdot u\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\left(2 \cdot \frac{u}{v}\right), \color{blue}{\left(2 + -2 \cdot u\right)}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\left(\frac{2 \cdot u}{v}\right), \left(\color{blue}{2} + -2 \cdot u\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(2 \cdot u\right), v\right), \left(\color{blue}{2} + -2 \cdot u\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(u \cdot 2\right), v\right), \left(2 + -2 \cdot u\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 + -2 \cdot u\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 \cdot 1 + \color{blue}{-2} \cdot u\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 \cdot 1 + \left(\mathsf{neg}\left(2\right)\right) \cdot u\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 \cdot 1 + \left(\mathsf{neg}\left(2 \cdot u\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 \cdot 1 + 2 \cdot \color{blue}{\left(\mathsf{neg}\left(u\right)\right)}\right)\right)\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 \cdot 1 + 2 \cdot \left(-1 \cdot \color{blue}{u}\right)\right)\right)\right) \]

      15. distribute-lft-in N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 \cdot \color{blue}{\left(1 + -1 \cdot u\right)}\right)\right)\right) \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 \cdot \left(1 + \left(\mathsf{neg}\left(u\right)\right)\right)\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(2 \cdot \left(1 - \color{blue}{u}\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \left(\left(1 - u\right) \cdot \color{blue}{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \mathsf{*.f32}\left(\left(1 - u\right), \color{blue}{2}\right)\right)\right) \]

      20. --lowering--.f32 52.1%

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, 2\right), v\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 2\right)\right)\right) \]

   8. Simplified 52.1%

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

4. Final simplification 90.9%

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

Alternative 18: 90.4% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((-2.0e0) + (u * (2.0e0 + (2.0e0 / v))))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (single(-2.0) + (u * (single(2.0) + (single(2.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}}{v}\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{/.f32}\left(\left(-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right), \color{blue}{v}\right)\right)\right) \]

   5. Simplified 52.8%

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

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) - 2\right)}\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) + -2\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(u \cdot \left(2 + 2 \cdot \frac{1}{v}\right)\right), \color{blue}{-2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 + 2 \cdot \frac{1}{v}\right)\right), -2\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(2 \cdot \frac{1}{v}\right)\right)\right), -2\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{2 \cdot 1}{v}\right)\right)\right), -2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \left(\frac{2}{v}\right)\right)\right), -2\right)\right) \]

      8. /-lowering-/.f32 52.1%

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(2, v\right)\right)\right), -2\right)\right) \]

   8. Simplified 52.1%

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

4. Final simplification 90.9%

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

Alternative 19: 90.4% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (2.0e0 * (u + (u / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (single(2.0) * (u + (u / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.2%

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

   if 0.300000012 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 2\right)}\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) + -2\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \left(-2 + \color{blue}{u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \color{blue}{\left(u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right)\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \left(\left(u \cdot v\right) \cdot \color{blue}{\left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \left(\left(\frac{1}{e^{\frac{-2}{v}}} - 1\right) \cdot \color{blue}{\left(u \cdot v\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\left(\frac{1}{e^{\frac{-2}{v}}} - 1\right), \color{blue}{\left(u \cdot v\right)}\right)\right)\right) \]

      8. rec-exp N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\left(e^{\mathsf{neg}\left(\frac{-2}{v}\right)} - 1\right), \left(u \cdot v\right)\right)\right)\right) \]

      9. expm1-define N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{-2}{v}\right)\right)\right), \left(\color{blue}{u} \cdot v\right)\right)\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{\mathsf{neg}\left(-2\right)}{v}\right)\right), \left(u \cdot v\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{2}{v}\right)\right), \left(u \cdot v\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{2 \cdot 1}{v}\right)\right), \left(u \cdot v\right)\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\left(\mathsf{expm1}\left(2 \cdot \frac{1}{v}\right)\right), \left(u \cdot v\right)\right)\right)\right) \]

      14. expm1-lowering-expm1.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(2 \cdot \frac{1}{v}\right)\right), \left(\color{blue}{u} \cdot v\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\frac{2 \cdot 1}{v}\right)\right), \left(u \cdot v\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\frac{2}{v}\right)\right), \left(u \cdot v\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right), \left(u \cdot v\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right), \left(v \cdot \color{blue}{u}\right)\right)\right)\right) \]

      19. *-lowering-*.f32 59.0%

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{u}\right)\right)\right)\right) \]

   5. Simplified 59.0%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f32}\left(\left(2 \cdot u + 2 \cdot \frac{u}{v}\right), \color{blue}{-1}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f32}\left(\left(2 \cdot \left(u + \frac{u}{v}\right)\right), -1\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \left(u + \frac{u}{v}\right)\right), -1\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{+.f32}\left(u, \left(\frac{u}{v}\right)\right)\right), -1\right) \]

      7. /-lowering-/.f32 52.1%

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{+.f32}\left(u, \mathsf{/.f32}\left(u, v\right)\right)\right), -1\right) \]

   8. Simplified 52.1%

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

4. Final simplification 90.9%

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

Alternative 20: 86.7% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

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

5. Simplified 87.4%

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

Alternative 21: 5.9% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.9%

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

Reproduce

herbie shell --seed 2024158 
(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))))))))