HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.3s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. /-lowering-/.f32 99.4%

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

4. Applied egg-rr 99.4%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := \left(1 - u\right) \cdot 16\\ \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + t\_0 \cdot \left(t\_1 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(t\_1 + t\_0 \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u))) (t_1 (* (- 1.0 u) 16.0)))
   (if (<= v 0.4000000059604645)
     (+ 1.0 (* v (log (* (expm1 (/ -2.0 v)) (- u)))))
     (+
      1.0
      (+
       (* (- 1.0 u) -2.0)
       (/
        (+
         (/
          (+
           (* (+ (* (- 1.0 u) 8.0) (* t_0 (+ t_1 -24.0))) -0.16666666666666666)
           (/
            (*
             0.041666666666666664
             (+
              (* -96.0 (pow (- 1.0 u) 4.0))
              (+ t_1 (* t_0 (+ -112.0 (* (- 1.0 u) 192.0))))))
            v))
          v)
         (* -0.5 (* (- 1.0 u) (- (* -4.0 (+ u -1.0)) 4.0))))
        v))))))

C

float code(float u, float v) {
	float t_0 = (1.0f - u) * (1.0f - u);
	float t_1 = (1.0f - u) * 16.0f;
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((((((((1.0f - u) * 8.0f) + (t_0 * (t_1 + -24.0f))) * -0.16666666666666666f) + ((0.041666666666666664f * ((-96.0f * powf((1.0f - u), 4.0f)) + (t_1 + (t_0 * (-112.0f + ((1.0f - u) * 192.0f)))))) / v)) / v) + (-0.5f * ((1.0f - u) * ((-4.0f * (u + -1.0f)) - 4.0f)))) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	t_1 = Float32(Float32(Float32(1.0) - u) * Float32(16.0))
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u)))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(t_0 * Float32(t_1 + Float32(-24.0)))) * Float32(-0.16666666666666666)) + Float32(Float32(Float32(0.041666666666666664) * Float32(Float32(Float32(-96.0) * (Float32(Float32(1.0) - u) ^ Float32(4.0))) + Float32(t_1 + Float32(t_0 * Float32(Float32(-112.0) + Float32(Float32(Float32(1.0) - u) * Float32(192.0))))))) / v)) / v) + Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(-4.0) * Float32(u + Float32(-1.0))) - Float32(4.0))))) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

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

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

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

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

      10. neg-mul-1 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

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

      24. neg-lowering-neg.f32 99.5%

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

   5. Simplified 99.5%

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

   if 0.400000006 < v

   1. Initial program 90.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 87.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \]
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 99.4%

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

Alternative 4: 91.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := \left(1 - u\right) \cdot 16\\ \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + t\_0 \cdot \left(t\_1 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(t\_1 + t\_0 \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u))) (t_1 (* (- 1.0 u) 16.0)))
   (if (<= v 0.05000000074505806)
     1.0
     (+
      1.0
      (+
       (* (- 1.0 u) -2.0)
       (/
        (+
         (/
          (+
           (* (+ (* (- 1.0 u) 8.0) (* t_0 (+ t_1 -24.0))) -0.16666666666666666)
           (/
            (*
             0.041666666666666664
             (+
              (* -96.0 (pow (- 1.0 u) 4.0))
              (+ t_1 (* t_0 (+ -112.0 (* (- 1.0 u) 192.0))))))
            v))
          v)
         (* -0.5 (* (- 1.0 u) (- (* -4.0 (+ u -1.0)) 4.0))))
        v))))))

C

float code(float u, float v) {
	float t_0 = (1.0f - u) * (1.0f - u);
	float t_1 = (1.0f - u) * 16.0f;
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((((((((1.0f - u) * 8.0f) + (t_0 * (t_1 + -24.0f))) * -0.16666666666666666f) + ((0.041666666666666664f * ((-96.0f * powf((1.0f - u), 4.0f)) + (t_1 + (t_0 * (-112.0f + ((1.0f - u) * 192.0f)))))) / v)) / v) + (-0.5f * ((1.0f - u) * ((-4.0f * (u + -1.0f)) - 4.0f)))) / v));
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = (1.0e0 - u) * (1.0e0 - u)
    t_1 = (1.0e0 - u) * 16.0e0
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((((((1.0e0 - u) * 8.0e0) + (t_0 * (t_1 + (-24.0e0)))) * (-0.16666666666666666e0)) + ((0.041666666666666664e0 * (((-96.0e0) * ((1.0e0 - u) ** 4.0e0)) + (t_1 + (t_0 * ((-112.0e0) + ((1.0e0 - u) * 192.0e0)))))) / v)) / v) + ((-0.5e0) * ((1.0e0 - u) * (((-4.0e0) * (u + (-1.0e0))) - 4.0e0)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	t_1 = Float32(Float32(Float32(1.0) - u) * Float32(16.0))
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(t_0 * Float32(t_1 + Float32(-24.0)))) * Float32(-0.16666666666666666)) + Float32(Float32(Float32(0.041666666666666664) * Float32(Float32(Float32(-96.0) * (Float32(Float32(1.0) - u) ^ Float32(4.0))) + Float32(t_1 + Float32(t_0 * Float32(Float32(-112.0) + Float32(Float32(Float32(1.0) - u) * Float32(192.0))))))) / v)) / v) + Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(-4.0) * Float32(u + Float32(-1.0))) - Float32(4.0))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = (single(1.0) - u) * (single(1.0) - u);
	t_1 = (single(1.0) - u) * single(16.0);
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((((((((single(1.0) - u) * single(8.0)) + (t_0 * (t_1 + single(-24.0)))) * single(-0.16666666666666666)) + ((single(0.041666666666666664) * ((single(-96.0) * ((single(1.0) - u) ^ single(4.0))) + (t_1 + (t_0 * (single(-112.0) + ((single(1.0) - u) * single(192.0))))))) / v)) / v) + (single(-0.5) * ((single(1.0) - u) * ((single(-4.0) * (u + single(-1.0))) - single(4.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 77.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(u \cdot \left(u \cdot \left(\left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \frac{u \cdot 4}{v \cdot v}\right) - \left(\left(2 + \frac{4}{v}\right) + \frac{4.666666666666667}{v \cdot v}\right)\right) + \left(\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (*
       u
       (+
        (*
         u
         (-
          (*
           u
           (-
            (+ (/ 2.6666666666666665 v) (/ 8.0 (* v v)))
            (/ (* u 4.0) (* v v))))
          (+ (+ 2.0 (/ 4.0 v)) (/ 4.666666666666667 (* v v)))))
        (+ (+ 2.0 (/ 0.6666666666666666 (* v v))) (/ 1.3333333333333333 v))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * ((u * ((u * (((2.6666666666666665f / v) + (8.0f / (v * v))) - ((u * 4.0f) / (v * v)))) - ((2.0f + (4.0f / v)) + (4.666666666666667f / (v * v))))) + ((2.0f + (0.6666666666666666f / (v * v))) + (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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * ((u * ((u * (((2.6666666666666665e0 / v) + (8.0e0 / (v * v))) - ((u * 4.0e0) / (v * v)))) - ((2.0e0 + (4.0e0 / v)) + (4.666666666666667e0 / (v * v))))) + ((2.0e0 + (0.6666666666666666e0 / (v * v))) + (1.3333333333333333e0 / v)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(u * Float32(Float32(u * Float32(Float32(Float32(Float32(2.6666666666666665) / v) + Float32(Float32(8.0) / Float32(v * v))) - Float32(Float32(u * Float32(4.0)) / Float32(v * v)))) - Float32(Float32(Float32(2.0) + Float32(Float32(4.0) / v)) + Float32(Float32(4.666666666666667) / Float32(v * v))))) + Float32(Float32(Float32(2.0) + Float32(Float32(0.6666666666666666) / Float32(v * v))) + Float32(Float32(1.3333333333333333) / v)))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((u * ((u * ((u * (((single(2.6666666666666665) / v) + (single(8.0) / (v * v))) - ((u * single(4.0)) / (v * v)))) - ((single(2.0) + (single(4.0) / v)) + (single(4.666666666666667) / (v * v))))) + ((single(2.0) + (single(0.6666666666666666) / (v * v))) + (single(1.3333333333333333) / v)))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 77.1%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 77.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(u \cdot \left(u \cdot \left(\left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \frac{u \cdot 4}{v \cdot v}\right) - \left(\left(2 + \frac{4}{v}\right) + \frac{4.666666666666667}{v \cdot v}\right)\right) + \left(\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}\right) + u \cdot \left(\left(u \cdot \left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - 2\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (*
       u
       (+
        (+ (+ 2.0 (/ 0.6666666666666666 (* v v))) (/ 1.3333333333333333 v))
        (*
         u
         (-
          (-
           (* u (+ (/ 2.6666666666666665 v) (/ 8.0 (* v v))))
           (+ (/ 4.0 v) (/ 4.666666666666667 (* v v))))
          2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * (((2.0f + (0.6666666666666666f / (v * v))) + (1.3333333333333333f / v)) + (u * (((u * ((2.6666666666666665f / v) + (8.0f / (v * v)))) - ((4.0f / v) + (4.666666666666667f / (v * v)))) - 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * (((2.0e0 + (0.6666666666666666e0 / (v * v))) + (1.3333333333333333e0 / v)) + (u * (((u * ((2.6666666666666665e0 / v) + (8.0e0 / (v * v)))) - ((4.0e0 / v) + (4.666666666666667e0 / (v * v)))) - 2.0e0)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.6666666666666666) / Float32(v * v))) + Float32(Float32(1.3333333333333333) / v)) + Float32(u * Float32(Float32(Float32(u * Float32(Float32(Float32(2.6666666666666665) / v) + Float32(Float32(8.0) / Float32(v * v)))) - Float32(Float32(Float32(4.0) / v) + Float32(Float32(4.666666666666667) / Float32(v * v)))) - Float32(2.0))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((u * (((single(2.0) + (single(0.6666666666666666) / (v * v))) + (single(1.3333333333333333) / v)) + (u * (((u * ((single(2.6666666666666665) / v) + (single(8.0) / (v * v)))) - ((single(4.0) / v) + (single(4.666666666666667) / (v * v)))) - single(2.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 77.1%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 72.2%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}\right) + u \cdot \left(\left(u \cdot \left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - 2\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(u \cdot \left(\left(\frac{-2}{v} + 2.6666666666666665 \cdot \frac{u}{v \cdot v}\right) + \frac{-4}{v \cdot v}\right) + \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    (*
     u
     (+
      2.0
      (+
       (*
        u
        (+
         (+ (/ -2.0 v) (* 2.6666666666666665 (/ u (* v v))))
         (/ -4.0 (* v v))))
       (/ (+ 2.0 (/ 1.3333333333333333 v)) v))))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (2.0f + ((u * (((-2.0f / v) + (2.6666666666666665f * (u / (v * v)))) + (-4.0f / (v * v)))) + ((2.0f + (1.3333333333333333f / v)) / v)))) + -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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (u * (2.0e0 + ((u * ((((-2.0e0) / v) + (2.6666666666666665e0 * (u / (v * v)))) + ((-4.0e0) / (v * v)))) + ((2.0e0 + (1.3333333333333333e0 / v)) / v)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(u * Float32(Float32(Float32(Float32(-2.0) / v) + Float32(Float32(2.6666666666666665) * Float32(u / Float32(v * v)))) + Float32(Float32(-4.0) / Float32(v * v)))) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = (u * (single(2.0) + ((u * (((single(-2.0) / v) + (single(2.6666666666666665) * (u / (v * v)))) + (single(-4.0) / (v * v)))) + ((single(2.0) + (single(1.3333333333333333) / v)) / v)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 71.0%

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

   6. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. distribute-neg-in N/A

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

   8. Simplified 71.0%

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

   9. Taylor expanded in u around 0

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

   11. Simplified 71.5%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(u \cdot \left(\left(\frac{-2}{v} + 2.6666666666666665 \cdot \frac{u}{v \cdot v}\right) + \frac{-4}{v \cdot v}\right) + \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + u \cdot 2\right) - \frac{u \cdot \left(u \cdot \left(\left(2 + \frac{4}{v}\right) + \frac{u}{v} \cdot -2.6666666666666665\right) + \left(-2 + \frac{-1.3333333333333333}{v}\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (-
    (+ -1.0 (* u 2.0))
    (/
     (*
      u
      (+
       (* u (+ (+ 2.0 (/ 4.0 v)) (* (/ u v) -2.6666666666666665)))
       (+ -2.0 (/ -1.3333333333333333 v))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = (-1.0f + (u * 2.0f)) - ((u * ((u * ((2.0f + (4.0f / v)) + ((u / v) * -2.6666666666666665f))) + (-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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = ((-1.0e0) + (u * 2.0e0)) - ((u * ((u * ((2.0e0 + (4.0e0 / v)) + ((u / v) * (-2.6666666666666665e0)))) + ((-2.0e0) + ((-1.3333333333333333e0) / v)))) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(-1.0) + Float32(u * Float32(2.0))) - Float32(Float32(u * Float32(Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(4.0) / v)) + Float32(Float32(u / v) * Float32(-2.6666666666666665)))) + 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(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = (single(-1.0) + (u * single(2.0))) - ((u * ((u * ((single(2.0) + (single(4.0) / v)) + ((u / v) * single(-2.6666666666666665)))) + (single(-2.0) + (single(-1.3333333333333333) / v)))) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 71.0%

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

   6. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. distribute-neg-in N/A

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

   8. Simplified 71.0%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f32 71.4%

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

   11. Simplified 71.4%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + u \cdot 2\right) - \frac{u \cdot \left(u \cdot \left(\left(2 + \frac{4}{v}\right) + \frac{u}{v} \cdot -2.6666666666666665\right) + \left(-2 + \frac{-1.3333333333333333}{v}\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{u \cdot \left(\left(-2 + \frac{-1.3333333333333333}{v}\right) + u \cdot \left(2 + \frac{4}{v}\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (-
    (+ 1.0 (* (- 1.0 u) -2.0))
    (/
     (* u (+ (+ -2.0 (/ -1.3333333333333333 v)) (* u (+ 2.0 (/ 4.0 v)))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = (1.0f + ((1.0f - u) * -2.0f)) - ((u * ((-2.0f + (-1.3333333333333333f / v)) + (u * (2.0f + (4.0f / 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (1.0e0 + ((1.0e0 - u) * (-2.0e0))) - ((u * (((-2.0e0) + ((-1.3333333333333333e0) / v)) + (u * (2.0e0 + (4.0e0 / v))))) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0))) - Float32(Float32(u * Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) + Float32(u * Float32(Float32(2.0) + Float32(Float32(4.0) / v))))) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = (single(1.0) + ((single(1.0) - u) * single(-2.0))) - ((u * ((single(-2.0) + (single(-1.3333333333333333) / v)) + (u * (single(2.0) + (single(4.0) / v))))) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 71.0%

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

   6. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

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

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 65.2%

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

   8. Simplified 65.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{u \cdot \left(\left(-2 + \frac{-1.3333333333333333}{v}\right) + u \cdot \left(2 + \frac{4}{v}\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{u \cdot \left(u \cdot 2 + \left(-2 + \frac{-1.3333333333333333}{v}\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (-
    (+ 1.0 (* (- 1.0 u) -2.0))
    (/ (* u (+ (* u 2.0) (+ -2.0 (/ -1.3333333333333333 v)))) v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = (1.0f + ((1.0f - u) * -2.0f)) - ((u * ((u * 2.0f) + (-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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (1.0e0 + ((1.0e0 - u) * (-2.0e0))) - ((u * ((u * 2.0e0) + ((-2.0e0) + ((-1.3333333333333333e0) / v)))) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0))) - Float32(Float32(u * Float32(Float32(u * Float32(2.0)) + 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(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = (single(1.0) + ((single(1.0) - u) * single(-2.0))) - ((u * ((u * single(2.0)) + (single(-2.0) + (single(-1.3333333333333333) / v)))) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 71.0%

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

   6. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. distribute-neg-in N/A

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

   8. Simplified 71.0%

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

   9. Taylor expanded in v around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 63.7%

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

   11. Simplified 63.7%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{u \cdot \left(u \cdot 2 + \left(-2 + \frac{-1.3333333333333333}{v}\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \left(\frac{-2}{v} + \frac{\frac{-1}{u} - \left(-2 + \frac{-2}{v}\right)}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (* (* u u) (+ (/ -2.0 v) (/ (- (/ -1.0 u) (+ -2.0 (/ -2.0 v))) u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = (u * u) * ((-2.0f / v) + (((-1.0f / u) - (-2.0f + (-2.0f / 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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (u * u) * (((-2.0e0) / v) + ((((-1.0e0) / u) - ((-2.0e0) + ((-2.0e0) / v))) / u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * u) * Float32(Float32(Float32(-2.0) / v) + Float32(Float32(Float32(Float32(-1.0) / u) - Float32(Float32(-2.0) + Float32(Float32(-2.0) / v))) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = (u * u) * ((single(-2.0) / v) + (((single(-1.0) / u) - (single(-2.0) + (single(-2.0) / v))) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

      18. /-lowering-/.f32 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in u around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. sub-neg N/A

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

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

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

   8. Simplified 63.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \left(\frac{-2}{v} + \frac{\frac{-1}{u} - \left(-2 + \frac{-2}{v}\right)}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot \left(-2 + \frac{u \cdot 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+ 1.0 (* (- 1.0 u) (+ -2.0 (/ (* u 2.0) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((1.0f - u) * (-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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((1.0e0 - u) * ((-2.0e0) + ((u * 2.0e0) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(-2.0) + 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(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(1.0) - u) * (single(-2.0) + ((u * single(2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

      18. /-lowering-/.f32 63.0%

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

   5. Simplified 63.0%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot \frac{0.5}{v}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

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

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

   7. Applied egg-rr 63.0%

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

   8. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f32 63.0%

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

   10. Simplified 63.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot \left(-2 + \frac{u \cdot 2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+ -1.0 (* u (- 2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - ((-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 <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - 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(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 94.9%

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

   if 0.0500000007 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 71.0%

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

   6. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. distribute-neg-in N/A

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

   8. Simplified 71.0%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

   11. Simplified 60.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   1.0
   (+ 1.0 (+ (* (- 1.0 u) -2.0) (/ (* u 2.0) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * 2.0e0) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + 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(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((u * single(2.0)) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.9%

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

   if 0.400000006 < v

   1. Initial program 90.2%

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

   3. Taylor expanded in v around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

      18. /-lowering-/.f32 71.5%

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

   5. Simplified 71.5%

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

   6. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 64.4%

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

   8. Simplified 64.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot 2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645) 1.0 (+ -1.0 (* u (+ 2.0 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (2.0e0 / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + 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(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (single(2.0) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.9%

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

   if 0.400000006 < v

   1. Initial program 90.2%

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

   3. Taylor expanded in v around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

      18. /-lowering-/.f32 71.5%

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

   5. Simplified 71.5%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f32 64.4%

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

   8. Simplified 64.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 87.0% 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 99.4%

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

3. Taylor expanded in v around 0

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

5. Simplified 88.9%

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

Alternative 17: 6.0% 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 99.4%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.5%

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

Reproduce

herbie shell --seed 2024145 
(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))))))))