HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.5s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) 16.0)) (t_1 (* (- 1.0 u) (- 1.0 u))))
   (if (<= v 0.5)
     (+ 1.0 (* v (log (* (expm1 (/ -2.0 v)) (- u)))))
     (+
      1.0
      (+
       (* (- 1.0 u) -2.0)
       (/
        (+
         (/
          (+
           (* (+ (* (- 1.0 u) 8.0) (* t_1 (+ -24.0 t_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 <= 0.5f) {
		tmp = 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((((((((1.0f - u) * 8.0f) + (t_1 * (-24.0f + t_0))) * -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(0.5))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u)))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(t_1 * Float32(Float32(-24.0) + t_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 < 0.5

   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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(\left(0 - e^{\frac{-2}{v}}\right) + 1\right) \cdot u\right)\right)\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(0 - \left(e^{\frac{-2}{v}} - 1\right)\right) \cdot u\right)\right)\right)\right) \]

      6. neg-sub0 N/A

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

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

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

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

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

      9. neg-mul-1 N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. accelerator-lowering-expm1.f32 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f32 N/A

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

      21. neg-mul-1 N/A

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

      22. neg-lowering-neg.f32 99.4%

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

   5. Simplified 99.4%

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

   if 0.5 < v

   1. Initial program 93.4%

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

   3. Taylor expanded in v around -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 83.3%

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

4. Final simplification 98.4%

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

Alternative 3: 91.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := \left(1 - u\right) \cdot 16\\ \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + t\_0 \cdot \left(-24 + t\_1\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(t\_1 + t\_0 \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u))) (t_1 (* (- 1.0 u) 16.0)))
   (if (<= v 0.25)
     1.0
     (+
      1.0
      (+
       (* (- 1.0 u) -2.0)
       (/
        (+
         (/
          (+
           (* (+ (* (- 1.0 u) 8.0) (* t_0 (+ -24.0 t_1))) -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 <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((((((((1.0f - u) * 8.0f) + (t_0 * (-24.0f + t_1))) * -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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((((((1.0e0 - u) * 8.0e0) + (t_0 * ((-24.0e0) + t_1))) * (-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(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(t_0 * Float32(Float32(-24.0) + t_1))) * 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(0.25))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((((((((single(1.0) - u) * single(8.0)) + (t_0 * (single(-24.0) + t_1))) * 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 < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 77.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-24 + \left(1 - u\right) \cdot 16\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{u \cdot \left(u \cdot \left(2 + \left(\frac{4}{v} + \frac{u \cdot -2.6666666666666665}{v}\right)\right) + \left(-2 + \frac{-1.3333333333333333}{v}\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (-
    (+ 1.0 (* (- 1.0 u) -2.0))
    (/
     (*
      u
      (+
       (* u (+ 2.0 (+ (/ 4.0 v) (/ (* u -2.6666666666666665) v))))
       (+ -2.0 (/ -1.3333333333333333 v))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = (1.0f + ((1.0f - u) * -2.0f)) - ((u * ((u * (2.0f + ((4.0f / v) + ((u * -2.6666666666666665f) / v)))) + (-2.0f + (-1.3333333333333333f / v)))) / v);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (1.0e0 + ((1.0e0 - u) * (-2.0e0))) - ((u * ((u * (2.0e0 + ((4.0e0 / v) + ((u * (-2.6666666666666665e0)) / v)))) + ((-2.0e0) + ((-1.3333333333333333e0) / v)))) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0))) - Float32(Float32(u * Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(4.0) / v) + Float32(Float32(u * Float32(-2.6666666666666665)) / v)))) + Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)))) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = (single(1.0) + ((single(1.0) - u) * single(-2.0))) - ((u * ((u * (single(2.0) + ((single(4.0) / v) + ((u * single(-2.6666666666666665)) / v)))) + (single(-2.0) + (single(-1.3333333333333333) / v)))) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 72.4%

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

   6. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

   8. Simplified 72.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{u \cdot \left(u \cdot \left(2 + \left(\frac{4}{v} + \frac{u \cdot -2.6666666666666665}{v}\right)\right) + \left(-2 + \frac{-1.3333333333333333}{v}\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(2 + \frac{1.3333333333333333}{v}\right) - u \cdot \left(\frac{4}{v} + \left(2 + \frac{u \cdot -2.6666666666666665}{v}\right)\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (*
       u
       (-
        (+ 2.0 (/ 1.3333333333333333 v))
        (* u (+ (/ 4.0 v) (+ 2.0 (/ (* u -2.6666666666666665) v))))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * ((2.0f + (1.3333333333333333f / v)) - (u * ((4.0f / v) + (2.0f + ((u * -2.6666666666666665f) / v)))))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * ((2.0e0 + (1.3333333333333333e0 / v)) - (u * ((4.0e0 / v) + (2.0e0 + ((u * (-2.6666666666666665e0)) / v)))))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) - Float32(u * Float32(Float32(Float32(4.0) / v) + Float32(Float32(2.0) + Float32(Float32(u * Float32(-2.6666666666666665)) / v)))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((u * ((single(2.0) + (single(1.3333333333333333) / v)) - (u * ((single(4.0) / v) + (single(2.0) + ((u * single(-2.6666666666666665)) / v)))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 72.3%

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

   6. Taylor expanded in u around 0

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

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. associate-*r/ N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

   8. Simplified 72.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(2 + \frac{1.3333333333333333}{v}\right) - u \cdot \left(\frac{4}{v} + \left(2 + \frac{u \cdot -2.6666666666666665}{v}\right)\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{u \cdot \left(\left(-2 + \frac{-1.3333333333333333}{v}\right) + u \cdot \left(2 + \frac{4}{v}\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (-
    (+ 1.0 (* (- 1.0 u) -2.0))
    (/
     (* u (+ (+ -2.0 (/ -1.3333333333333333 v)) (* u (+ 2.0 (/ 4.0 v)))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = (1.0f + ((1.0f - u) * -2.0f)) - ((u * ((-2.0f + (-1.3333333333333333f / v)) + (u * (2.0f + (4.0f / v))))) / v);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (1.0e0 + ((1.0e0 - u) * (-2.0e0))) - ((u * (((-2.0e0) + ((-1.3333333333333333e0) / v)) + (u * (2.0e0 + (4.0e0 / v))))) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0))) - Float32(Float32(u * Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) + Float32(u * Float32(Float32(2.0) + Float32(Float32(4.0) / v))))) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = (single(1.0) + ((single(1.0) - u) * single(-2.0))) - ((u * ((single(-2.0) + (single(-1.3333333333333333) / v)) + (u * (single(2.0) + (single(4.0) / v))))) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 72.4%

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

   6. Taylor expanded in u around 0

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

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

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

      2. sub-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. neg-mul-1 N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f32 70.8%

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

   8. Simplified 70.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 - u\right) \cdot -2\right) - \frac{u \cdot \left(\left(-2 + \frac{-1.3333333333333333}{v}\right) + u \cdot \left(2 + \frac{4}{v}\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{\frac{u \cdot 0.6666666666666666}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (-
       (/ (+ (/ (* u 0.6666666666666666) v) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) + ((((((u * 0.6666666666666666f) / v) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + ((((((u * 0.6666666666666666e0) / v) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(0.6666666666666666)) / v) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) + ((((((u * single(0.6666666666666666)) / v) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. rec-exp N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. accelerator-lowering-expm1.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f32 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in v around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

   8. Simplified 66.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{\frac{u \cdot 0.6666666666666666}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v \cdot v} + \left(-1 + u \cdot \left(2 + \frac{2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+
    (/ (* u (+ 1.3333333333333333 (/ 0.6666666666666666 v))) (* v v))
    (+ -1.0 (* u (+ 2.0 (/ 2.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = ((u * (1.3333333333333333f + (0.6666666666666666f / v))) / (v * v)) + (-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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = ((u * (1.3333333333333333e0 + (0.6666666666666666e0 / v))) / (v * v)) + ((-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(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(u * Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v))) / Float32(v * v)) + 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(0.25))
		tmp = single(1.0);
	else
		tmp = ((u * (single(1.3333333333333333) + (single(0.6666666666666666) / v))) / (v * v)) + (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 < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. rec-exp N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. accelerator-lowering-expm1.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f32 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

   8. Simplified 66.5%

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

   9. Taylor expanded in v around inf

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

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

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

       2. associate-*r/ N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. distribute-lft-out N/A

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

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

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

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

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f32 66.5%

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

   11. Simplified 66.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{u \cdot \left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v \cdot v} + \left(-1 + u \cdot \left(2 + \frac{2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + u \cdot \left(2 + \frac{2}{v}\right)\right) + u \cdot \frac{1.3333333333333333}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+ (+ -1.0 (* u (+ 2.0 (/ 2.0 v)))) (* u (/ 1.3333333333333333 (* v v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = (-1.0f + (u * (2.0f + (2.0f / v)))) + (u * (1.3333333333333333f / (v * v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = ((-1.0e0) + (u * (2.0e0 + (2.0e0 / v)))) + (u * (1.3333333333333333e0 / (v * v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) / v)))) + Float32(u * Float32(Float32(1.3333333333333333) / Float32(v * v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = (single(-1.0) + (u * (single(2.0) + (single(2.0) / v)))) + (u * (single(1.3333333333333333) / (v * v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. rec-exp N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. accelerator-lowering-expm1.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f32 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

   8. Simplified 66.5%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f32 65.6%

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

   11. Simplified 65.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + u \cdot \left(2 + \frac{2}{v}\right)\right) + u \cdot \frac{1.3333333333333333}{v \cdot v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.6% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(2 + \frac{1.3333333333333333}{v}\right) \cdot \frac{1}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+ -1.0 (* u (+ 2.0 (* (+ 2.0 (/ 1.3333333333333333 v)) (/ 1.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (1.3333333333333333f / v)) * (1.0f / 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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (1.3333333333333333e0 / v)) * (1.0e0 / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) * Float32(Float32(1.0) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (single(1.3333333333333333) / v)) * (single(1.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. rec-exp N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. accelerator-lowering-expm1.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f32 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

   8. Simplified 66.5%

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

   9. Taylor expanded in v around inf

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

   11. Simplified 65.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(2 + \frac{1.3333333333333333}{v}\right) \cdot \frac{1}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+ -1.0 (* u (+ 2.0 (/ (+ 2.0 (/ 1.3333333333333333 v)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (1.3333333333333333f / v)) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (1.3333333333333333e0 / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (single(1.3333333333333333) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

      1. +-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f32 93.7%

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. associate-*r/ N/A

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

      18. accelerator-lowering-expm1.f32 N/A

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

      19. associate-*r/ N/A

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

   7. Simplified 70.5%

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

   8. Taylor expanded in v around inf

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

   10. Simplified 65.4%

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

       1. associate-+r+ N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

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

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

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

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

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

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

       7. associate-*l/ N/A

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

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

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

       9. *-lft-identity N/A

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

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

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

       11. /-lowering-/.f32 65.5%

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

   12. Applied egg-rr 65.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + u \cdot 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ -1.0 (* u (- 2.0 (/ (+ -2.0 (* u 2.0)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - ((-2.0f + (u * 2.0f)) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (((-2.0e0) + (u * 2.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(-2.0) + Float32(u * Float32(2.0))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - ((single(-2.0) + (u * single(2.0))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

      18. /-lowering-/.f32 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in u around 0

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

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

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. /-lowering-/.f32 62.7%

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

   8. Simplified 62.7%

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

   9. Taylor expanded in v around inf

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

       1. metadata-eval N/A

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

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

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

       3. neg-sub0 N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \left(2 + \left(\left(0 - 2 \cdot \frac{u}{v}\right) + \color{blue}{2} \cdot \frac{1}{v}\right)\right)\right)\right) \]

       4. associate--r- N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \left(2 + \left(0 - \color{blue}{\left(2 \cdot \frac{u}{v} - 2 \cdot \frac{1}{v}\right)}\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \left(2 + \left(0 - \left(\frac{2 \cdot u}{v} - \color{blue}{2} \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \left(2 + \left(0 - \left(\frac{2 \cdot u}{v} - \frac{2 \cdot 1}{\color{blue}{v}}\right)\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \left(2 + \left(0 - \left(\frac{2 \cdot u}{v} - \frac{2}{v}\right)\right)\right)\right)\right) \]

       8. div-sub N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \left(2 + \left(0 - \frac{2 \cdot u - 2}{\color{blue}{v}}\right)\right)\right)\right) \]

       9. neg-sub0 N/A

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

       10. unsub-neg N/A

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

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

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

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

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

       13. sub-neg N/A

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

       14. metadata-eval N/A

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

       15. +-commutative N/A

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

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

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

       17. *-commutative N/A

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

       18. *-lowering-*.f32 62.7%

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

   11. Simplified 62.7%

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

Alternative 13: 90.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + u \cdot \left(2 + \frac{2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ 1.0 (+ -2.0 (* u (+ 2.0 (/ 2.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f + (u * (2.0f + (2.0f / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((-2.0e0) + (u * (2.0e0 + (2.0e0 / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (single(-2.0) + (u * (single(2.0) + (single(2.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

      18. /-lowering-/.f32 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

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

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-in N/A

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

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

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

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

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f32 61.2%

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

   8. Simplified 61.2%

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

Alternative 14: 90.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ -1.0 (* u (+ 2.0 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		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 <= 0.25e0) 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(0.25))
		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(0.25))
		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 < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

      18. /-lowering-/.f32 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in u around 0

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

      1. distribute-rgt-in N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. associate--l+ N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-in N/A

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

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

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

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

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

      19. associate-*r/ N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f32 61.2%

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

   8. Simplified 61.2%

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

Alternative 15: 89.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (* u (+ 2.0 (/ -1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = u * (2.0f + (-1.0f / u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = u * (2.0e0 + ((-1.0e0) / u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(2.0) + Float32(Float32(-1.0) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = u * (single(2.0) + (single(-1.0) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. rec-exp N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. accelerator-lowering-expm1.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f32 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 51.6%

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

   8. Simplified 51.6%

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

   9. Taylor expanded in u around inf

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

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

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

       2. sub-neg N/A

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

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

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

       4. distribute-neg-frac N/A

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

       5. metadata-eval N/A

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

       6. /-lowering-/.f32 51.6%

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

   11. Simplified 51.6%

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

Alternative 16: 89.8% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.25) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		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 <= 0.25e0) 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(0.25))
		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(0.25))
		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 < 0.25

   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 91.7%

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

   if 0.25 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. rec-exp N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. accelerator-lowering-expm1.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f32 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 51.6%

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

   8. Simplified 51.6%

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

Alternative 17: 87.0% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

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

5. Simplified 85.6%

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

Alternative 18: 5.7% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 6.0%

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

Reproduce

herbie shell --seed 2024191 
(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))))))))