HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.8s

Alternatives: 18

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: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot 16\\ t_1 := \left(1 - u\right) \cdot \left(1 - u\right)\\ \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + t\_1 \cdot \left(t\_0 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(t\_0 + t\_1 \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) 16.0)) (t_1 (* (- 1.0 u) (- 1.0 u))))
   (if (<= v 0.4699999988079071)
     (+ 1.0 (* v (log (* (expm1 (/ -2.0 v)) (- u)))))
     (+
      (- -1.0 (* u -2.0))
      (/
       (+
        (/
         (+
          (* (+ (* (- 1.0 u) 8.0) (* t_1 (+ t_0 -24.0))) -0.16666666666666666)
          (/
           (*
            0.041666666666666664
            (+
             (* -96.0 (pow (- 1.0 u) 4.0))
             (+ t_0 (* t_1 (+ -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) * 16.0f;
	float t_1 = (1.0f - u) * (1.0f - u);
	float tmp;
	if (v <= 0.4699999988079071f) {
		tmp = 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
	} else {
		tmp = (-1.0f - (u * -2.0f)) + ((((((((1.0f - u) * 8.0f) + (t_1 * (t_0 + -24.0f))) * -0.16666666666666666f) + ((0.041666666666666664f * ((-96.0f * powf((1.0f - u), 4.0f)) + (t_0 + (t_1 * (-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(16.0))
	t_1 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	tmp = Float32(0.0)
	if (v <= Float32(0.4699999988079071))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u)))));
	else
		tmp = Float32(Float32(Float32(-1.0) - Float32(u * Float32(-2.0))) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(t_1 * Float32(t_0 + 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_0 + Float32(t_1 * 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 < 0.469999999

   1. Initial program 100.0%

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

   3. Taylor expanded in u around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. expm1-define N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

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

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

      18. associate-*r/ N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-frac N/A

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

      21. metadata-eval N/A

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

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

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

      23. neg-mul-1 N/A

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

   5. Simplified 99.3%

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

   if 0.469999999 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 82.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\\ \end{array} \]
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: 91.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := \left(1 - u\right) \cdot 16\\ \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + t\_0 \cdot \left(t\_1 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(t\_1 + t\_0 \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u))) (t_1 (* (- 1.0 u) 16.0)))
   (if (<= v 0.4699999988079071)
     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 <= 0.4699999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = (-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 <= 0.4699999988079071e0) then
        tmp = 1.0e0
    else
        tmp = ((-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(0.4699999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(-1.0) - Float32(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(0.4699999988079071))
		tmp = single(1.0);
	else
		tmp = (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 < 0.469999999

   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 92.9%

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

   if 0.469999999 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 82.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(u + -1\right)\\ \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) + \frac{-0.5 \cdot \left(-4 \cdot t\_0 + 4 \cdot \left(u + -1\right)\right) + \frac{0.16666666666666666 \cdot \left(\left(8 \cdot \left(u + -1\right) + -24 \cdot t\_0\right) + 16 \cdot \left(\left(1 - u\right) \cdot t\_0\right)\right)}{v}}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (+ u -1.0))))
   (if (<= v 0.15000000596046448)
     1.0
     (+
      (+ 1.0 (* (- 1.0 u) -2.0))
      (/
       (+
        (* -0.5 (+ (* -4.0 t_0) (* 4.0 (+ u -1.0))))
        (/
         (*
          0.16666666666666666
          (+ (+ (* 8.0 (+ u -1.0)) (* -24.0 t_0)) (* 16.0 (* (- 1.0 u) t_0))))
         v))
       v)))))

C

float code(float u, float v) {
	float t_0 = (1.0f - u) * (u + -1.0f);
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = (1.0f + ((1.0f - u) * -2.0f)) + (((-0.5f * ((-4.0f * t_0) + (4.0f * (u + -1.0f)))) + ((0.16666666666666666f * (((8.0f * (u + -1.0f)) + (-24.0f * t_0)) + (16.0f * ((1.0f - u) * t_0)))) / v)) / 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) :: tmp
    t_0 = (1.0e0 - u) * (u + (-1.0e0))
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (1.0e0 + ((1.0e0 - u) * (-2.0e0))) + ((((-0.5e0) * (((-4.0e0) * t_0) + (4.0e0 * (u + (-1.0e0))))) + ((0.16666666666666666e0 * (((8.0e0 * (u + (-1.0e0))) + ((-24.0e0) * t_0)) + (16.0e0 * ((1.0e0 - u) * t_0)))) / v)) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(u + Float32(-1.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0))) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(-4.0) * t_0) + Float32(Float32(4.0) * Float32(u + Float32(-1.0))))) + Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(Float32(Float32(8.0) * Float32(u + Float32(-1.0))) + Float32(Float32(-24.0) * t_0)) + Float32(Float32(16.0) * Float32(Float32(Float32(1.0) - u) * t_0)))) / v)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = (single(1.0) - u) * (u + single(-1.0));
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = (single(1.0) + ((single(1.0) - u) * single(-2.0))) + (((single(-0.5) * ((single(-4.0) * t_0) + (single(4.0) * (u + single(-1.0))))) + ((single(0.16666666666666666) * (((single(8.0) * (u + single(-1.0))) + (single(-24.0) * t_0)) + (single(16.0) * ((single(1.0) - u) * t_0)))) / v)) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

   if 0.150000006 < v

   1. Initial program 93.0%

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

      1. +-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 93.4%

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in v around -inf

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

   7. Simplified 69.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) + \frac{-0.5 \cdot \left(-4 \cdot \left(\left(1 - u\right) \cdot \left(u + -1\right)\right) + 4 \cdot \left(u + -1\right)\right) + \frac{0.16666666666666666 \cdot \left(\left(8 \cdot \left(u + -1\right) + -24 \cdot \left(\left(1 - u\right) \cdot \left(u + -1\right)\right)\right) + 16 \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(u + -1\right)\right)\right)\right)}{v}}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(24 + u \cdot -16\right) + -8\right) + v \cdot \left(\left(-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4\right) \cdot 0.5 + v \cdot \left(1 + \left(1 - u\right) \cdot -2\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (/
    (+
     (* (* u -0.16666666666666666) (+ (* u (+ 24.0 (* u -16.0))) -8.0))
     (*
      v
      (+
       (* (+ (* -4.0 (* (- 1.0 u) (- 1.0 u))) (* (- 1.0 u) 4.0)) 0.5)
       (* v (+ 1.0 (* (- 1.0 u) -2.0))))))
    (* v v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = (((u * -0.16666666666666666f) * ((u * (24.0f + (u * -16.0f))) + -8.0f)) + (v * ((((-4.0f * ((1.0f - u) * (1.0f - u))) + ((1.0f - u) * 4.0f)) * 0.5f) + (v * (1.0f + ((1.0f - u) * -2.0f)))))) / (v * v);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (((u * (-0.16666666666666666e0)) * ((u * (24.0e0 + (u * (-16.0e0)))) + (-8.0e0))) + (v * (((((-4.0e0) * ((1.0e0 - u) * (1.0e0 - u))) + ((1.0e0 - u) * 4.0e0)) * 0.5e0) + (v * (1.0e0 + ((1.0e0 - u) * (-2.0e0))))))) / (v * v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(u * Float32(-0.16666666666666666)) * Float32(Float32(u * Float32(Float32(24.0) + Float32(u * Float32(-16.0)))) + Float32(-8.0))) + Float32(v * Float32(Float32(Float32(Float32(Float32(-4.0) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))) + Float32(Float32(Float32(1.0) - u) * Float32(4.0))) * Float32(0.5)) + Float32(v * Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0))))))) / Float32(v * v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = (((u * single(-0.16666666666666666)) * ((u * (single(24.0) + (u * single(-16.0)))) + single(-8.0))) + (v * ((((single(-4.0) * ((single(1.0) - u) * (single(1.0) - u))) + ((single(1.0) - u) * single(4.0))) * single(0.5)) + (v * (single(1.0) + ((single(1.0) - u) * single(-2.0))))))) / (v * v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

   if 0.150000006 < v

   1. Initial program 93.0%

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

      1. +-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 93.4%

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 68.6%

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

   8. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f32 68.6%

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

   10. Simplified 68.6%

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

   11. Taylor expanded in v around 0

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

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

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

   13. Simplified 68.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(24 + u \cdot -16\right) + -8\right) + v \cdot \left(\left(-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4\right) \cdot 0.5 + v \cdot \left(1 + \left(1 - u\right) \cdot -2\right)\right)}{v \cdot v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot \left(u \cdot u\right)\right) \cdot \left(\frac{\left(\frac{-2}{v} + \frac{-4}{v \cdot v}\right) + \frac{\left(\left(2 + \frac{1.3333333333333333}{v \cdot v}\right) + \frac{2}{v}\right) + \frac{-1}{u}}{u}}{u} - \frac{-2.6666666666666665}{v \cdot v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (*
    (* u (* u u))
    (-
     (/
      (+
       (+ (/ -2.0 v) (/ -4.0 (* v v)))
       (/
        (+ (+ (+ 2.0 (/ 1.3333333333333333 (* v v))) (/ 2.0 v)) (/ -1.0 u))
        u))
      u)
     (/ -2.6666666666666665 (* v v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (u * u)) * (((((-2.0f / v) + (-4.0f / (v * v))) + ((((2.0f + (1.3333333333333333f / (v * v))) + (2.0f / v)) + (-1.0f / u)) / u)) / u) - (-2.6666666666666665f / (v * v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (u * (u * u)) * ((((((-2.0e0) / v) + ((-4.0e0) / (v * v))) + ((((2.0e0 + (1.3333333333333333e0 / (v * v))) + (2.0e0 / v)) + ((-1.0e0) / u)) / u)) / u) - ((-2.6666666666666665e0) / (v * v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(u * u)) * Float32(Float32(Float32(Float32(Float32(Float32(-2.0) / v) + Float32(Float32(-4.0) / Float32(v * v))) + Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v))) + Float32(Float32(2.0) / v)) + Float32(Float32(-1.0) / u)) / u)) / u) - Float32(Float32(-2.6666666666666665) / Float32(v * v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = (u * (u * u)) * (((((single(-2.0) / v) + (single(-4.0) / (v * v))) + ((((single(2.0) + (single(1.3333333333333333) / (v * v))) + (single(2.0) / v)) + (single(-1.0) / u)) / u)) / u) - (single(-2.6666666666666665) / (v * v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

   if 0.150000006 < v

   1. Initial program 93.0%

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

      1. +-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 93.4%

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 68.6%

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

   8. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f32 68.6%

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

   10. Simplified 68.6%

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

   11. Taylor expanded in u around -inf

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

   13. Simplified 68.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot \left(u \cdot u\right)\right) \cdot \left(\frac{\left(\frac{-2}{v} + \frac{-4}{v \cdot v}\right) + \frac{\left(\left(2 + \frac{1.3333333333333333}{v \cdot v}\right) + \frac{2}{v}\right) + \frac{-1}{u}}{u}}{u} - \frac{-2.6666666666666665}{v \cdot v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{u \cdot \left(\left(2 + \frac{1.3333333333333333}{v}\right) - u \cdot \left(2 + \left(\frac{4}{v} + -2.6666666666666665 \cdot \frac{u}{v}\right)\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    (- -1.0 (* u -2.0))
    (/
     (*
      u
      (-
       (+ 2.0 (/ 1.3333333333333333 v))
       (* u (+ 2.0 (+ (/ 4.0 v) (* -2.6666666666666665 (/ u v)))))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = (-1.0f - (u * -2.0f)) + ((u * ((2.0f + (1.3333333333333333f / v)) - (u * (2.0f + ((4.0f / v) + (-2.6666666666666665f * (u / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = ((-1.0e0) - (u * (-2.0e0))) + ((u * ((2.0e0 + (1.3333333333333333e0 / v)) - (u * (2.0e0 + ((4.0e0 / v) + ((-2.6666666666666665e0) * (u / v))))))) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(-1.0) - Float32(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(Float32(4.0) / v) + Float32(Float32(-2.6666666666666665) * Float32(u / v))))))) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = (single(-1.0) - (u * single(-2.0))) + ((u * ((single(2.0) + (single(1.3333333333333333) / v)) - (u * (single(2.0) + ((single(4.0) / v) + (single(-2.6666666666666665) * (u / v))))))) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

   if 0.150000006 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around -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 68.7%

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

   6. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 68.7%

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

   8. Simplified 68.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{u \cdot \left(\left(2 + \frac{1.3333333333333333}{v}\right) - u \cdot \left(2 + \left(\frac{4}{v} + -2.6666666666666665 \cdot \frac{u}{v}\right)\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\left(2 - u \cdot \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right) + \left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4699999988079071)
   1.0
   (+
    -1.0
    (*
     u
     (+
      (- 2.0 (* u (+ (/ 2.0 v) (/ 4.0 (* v v)))))
      (+ (/ 1.3333333333333333 (* v v)) (/ 2.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4699999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((2.0f - (u * ((2.0f / v) + (4.0f / (v * v))))) + ((1.3333333333333333f / (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 <= 0.4699999988079071e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * ((2.0e0 - (u * ((2.0e0 / v) + (4.0e0 / (v * v))))) + ((1.3333333333333333e0 / (v * v)) + (2.0e0 / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4699999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(2.0) - Float32(u * Float32(Float32(Float32(2.0) / v) + Float32(Float32(4.0) / Float32(v * v))))) + Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(2.0) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4699999988079071))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((single(2.0) - (u * ((single(2.0) / v) + (single(4.0) / (v * v))))) + ((single(1.3333333333333333) / (v * v)) + (single(2.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.469999999

   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 92.9%

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

   if 0.469999999 < v

   1. Initial program 92.1%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 92.4%

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

   4. Applied egg-rr 92.4%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 74.6%

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

   8. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

   10. Simplified 70.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\left(2 - u \cdot \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right) + \left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{u \cdot \left(\left(2 + \frac{1.3333333333333333}{v}\right) - u \cdot \left(2 + \frac{4}{v}\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4699999988079071)
   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 <= 0.4699999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = (-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 <= 0.4699999988079071e0) then
        tmp = 1.0e0
    else
        tmp = ((-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(0.4699999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(-1.0) - Float32(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(0.4699999988079071))
		tmp = single(1.0);
	else
		tmp = (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 < 0.469999999

   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 92.9%

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

   if 0.469999999 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 74.6%

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

   6. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 70.0%

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

   8. Simplified 70.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{u \cdot \left(\left(2 + \frac{1.3333333333333333}{v}\right) - u \cdot \left(2 + \frac{4}{v}\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.8% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(-2 + u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (- (* u 2.0) (/ (* u (+ -2.0 (* u 2.0))) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - ((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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - ((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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(u * Float32(Float32(-2.0) + Float32(u * Float32(2.0)))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - ((u * (single(-2.0) + (u * single(2.0)))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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 94.0%

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

   if 0.100000001 < v

   1. Initial program 93.3%

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

   3. Taylor expanded in v around inf

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

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

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

   5. Simplified 61.6%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. associate-+r+ N/A

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

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

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

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

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

      8. associate-*r/ N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f32 61.9%

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

   8. Simplified 61.9%

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

   9. Taylor expanded in v around -inf

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

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

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

       8. *-commutative N/A

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

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

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

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

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

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

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

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

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

       15. *-commutative N/A

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

       16. *-lowering-*.f32 62.1%

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

   11. Simplified 62.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(-2 + u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.8% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{2}{v} + \left(2 + \frac{u \cdot -2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (* u (+ (/ 2.0 v) (+ 2.0 (/ (* u -2.0) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((2.0f / v) + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * ((2.0e0 / v) + (2.0e0 + ((u * (-2.0e0)) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(2.0) / v) + 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((single(2.0) / v) + (single(2.0) + ((u * single(-2.0)) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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 94.0%

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

   if 0.100000001 < v

   1. Initial program 93.3%

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

   3. Taylor expanded in v around inf

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

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

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

   5. Simplified 61.6%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. associate-+r+ N/A

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

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

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

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

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

      8. associate-*r/ N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f32 62.1%

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

   8. Simplified 62.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{2}{v} + \left(2 + \frac{u \cdot -2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.9% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+ -1.0 (* u (+ 2.0 (+ (/ 1.3333333333333333 (* v v)) (/ 2.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((1.3333333333333333f / (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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((1.3333333333333333e0 / (v * v)) + (2.0e0 / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(2.0) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(1.3333333333333333) / (v * v)) + (single(2.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

   if 0.150000006 < v

   1. Initial program 93.0%

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

      1. +-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 93.4%

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 68.6%

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

   8. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f32 58.6%

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

   10. Simplified 58.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \frac{2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.6% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{-2 + u \cdot \left(2 + \frac{2}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ 1.0 (* v (/ (+ -2.0 (* u (+ 2.0 (/ 2.0 v)))) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * ((-2.0f + (u * (2.0f + (2.0f / v)))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * (((-2.0e0) + (u * (2.0e0 + (2.0e0 / v)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(Float32(Float32(-2.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) / v)))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * ((single(-2.0) + (u * (single(2.0) + (single(2.0) / v)))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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 94.0%

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

   if 0.100000001 < v

   1. Initial program 93.3%

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

   3. Taylor expanded in v around inf

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

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

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

   5. Simplified 61.6%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f32 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{-2 + u \cdot \left(2 + \frac{2}{v}\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ -1.0 (* u (+ 2.0 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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 < 0.100000001

   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 94.0%

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

   if 0.100000001 < v

   1. Initial program 93.3%

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

   3. Taylor expanded in v around inf

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

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

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

   5. Simplified 61.6%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      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 56.3%

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

   8. Simplified 56.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 89.8% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4699999988079071) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4699999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4699999988079071e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4699999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4699999988079071))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.469999999

   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 92.9%

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

   if 0.469999999 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf

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

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

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

   5. Simplified 67.5%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. associate-+r+ N/A

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

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

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

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

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

      8. associate-*r/ N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f32 67.9%

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

   8. Simplified 67.9%

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

   9. Taylor expanded in v around inf

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

          \[\leadsto 2 \cdot u + -1 \]

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f32 53.3%

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

   11. Simplified 53.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4699999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 86.8% 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 86.5%

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

Alternative 18: 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 6.3%

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

Reproduce

herbie shell --seed 2024288 
(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))))))))