HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.0s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. /-lowering-/.f32 99.5%

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

4. Applied egg-rr 99.5%

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

Alternative 2: 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 99.5%

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

Alternative 3: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. /-lowering-/.f32 96.6%

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

5. Simplified 96.6%

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

Alternative 4: 91.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := \left(1 - u\right) \cdot 16\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + t\_0 \cdot \left(t\_1 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(t\_1 + t\_0 \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u))) (t_1 (* (- 1.0 u) 16.0)))
   (if (<= v 0.10000000149011612)
     1.0
     (+
      (- -1.0 (* u -2.0))
      (/
       (+
        (/
         (+
          (* (+ (* (- 1.0 u) 8.0) (* t_0 (+ t_1 -24.0))) -0.16666666666666666)
          (/
           (*
            0.041666666666666664
            (+
             (* -96.0 (pow (- 1.0 u) 4.0))
             (+ t_1 (* t_0 (+ -112.0 (* (- 1.0 u) 192.0))))))
           v))
         v)
        (* -0.5 (* (- 1.0 u) (- (* -4.0 (+ u -1.0)) 4.0))))
       v)))))

C

float code(float u, float v) {
	float t_0 = (1.0f - u) * (1.0f - u);
	float t_1 = (1.0f - u) * 16.0f;
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (-1.0f - (u * -2.0f)) + ((((((((1.0f - u) * 8.0f) + (t_0 * (t_1 + -24.0f))) * -0.16666666666666666f) + ((0.041666666666666664f * ((-96.0f * powf((1.0f - u), 4.0f)) + (t_1 + (t_0 * (-112.0f + ((1.0f - u) * 192.0f)))))) / v)) / v) + (-0.5f * ((1.0f - u) * ((-4.0f * (u + -1.0f)) - 4.0f)))) / v);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = (1.0e0 - u) * (1.0e0 - u)
    t_1 = (1.0e0 - u) * 16.0e0
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = ((-1.0e0) - (u * (-2.0e0))) + ((((((((1.0e0 - u) * 8.0e0) + (t_0 * (t_1 + (-24.0e0)))) * (-0.16666666666666666e0)) + ((0.041666666666666664e0 * (((-96.0e0) * ((1.0e0 - u) ** 4.0e0)) + (t_1 + (t_0 * ((-112.0e0) + ((1.0e0 - u) * 192.0e0)))))) / v)) / v) + ((-0.5e0) * ((1.0e0 - u) * (((-4.0e0) * (u + (-1.0e0))) - 4.0e0)))) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	t_1 = Float32(Float32(Float32(1.0) - u) * Float32(16.0))
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(-1.0) - Float32(u * Float32(-2.0))) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(8.0)) + Float32(t_0 * Float32(t_1 + Float32(-24.0)))) * Float32(-0.16666666666666666)) + Float32(Float32(Float32(0.041666666666666664) * Float32(Float32(Float32(-96.0) * (Float32(Float32(1.0) - u) ^ Float32(4.0))) + Float32(t_1 + Float32(t_0 * Float32(Float32(-112.0) + Float32(Float32(Float32(1.0) - u) * Float32(192.0))))))) / v)) / v) + Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(-4.0) * Float32(u + Float32(-1.0))) - Float32(4.0))))) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = (single(1.0) - u) * (single(1.0) - u);
	t_1 = (single(1.0) - u) * single(16.0);
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (single(-1.0) - (u * single(-2.0))) + ((((((((single(1.0) - u) * single(8.0)) + (t_0 * (t_1 + single(-24.0)))) * single(-0.16666666666666666)) + ((single(0.041666666666666664) * ((single(-96.0) * ((single(1.0) - u) ^ single(4.0))) + (t_1 + (t_0 * (single(-112.0) + ((single(1.0) - u) * single(192.0))))))) / v)) / v) + (single(-0.5) * ((single(1.0) - u) * ((single(-4.0) * (u + single(-1.0))) - single(4.0))))) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 88.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot 16 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(-2 + u \cdot 2\right) + \frac{\frac{-0.16666666666666666 \cdot \left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right)\right) + \left(\left(1 - u\right) \cdot \left(16 + \left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right) + t\_0 \cdot \left(t\_0 \cdot -96\right)\right) \cdot \frac{0.041666666666666664}{v}}{v} + \left(1 - u\right) \cdot \left(-0.5 \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u))))
   (if (<= v 0.10000000149011612)
     1.0
     (+
      1.0
      (+
       (+ -2.0 (* u 2.0))
       (/
        (+
         (/
          (+
           (*
            -0.16666666666666666
            (* (- 1.0 u) (+ 8.0 (* (- 1.0 u) (+ (* (- 1.0 u) 16.0) -24.0)))))
           (*
            (+
             (*
              (- 1.0 u)
              (+ 16.0 (* (- 1.0 u) (+ -112.0 (* (- 1.0 u) 192.0)))))
             (* t_0 (* t_0 -96.0)))
            (/ 0.041666666666666664 v)))
          v)
         (* (- 1.0 u) (* -0.5 (- (* -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 tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((-2.0f + (u * 2.0f)) + (((((-0.16666666666666666f * ((1.0f - u) * (8.0f + ((1.0f - u) * (((1.0f - u) * 16.0f) + -24.0f))))) + ((((1.0f - u) * (16.0f + ((1.0f - u) * (-112.0f + ((1.0f - u) * 192.0f))))) + (t_0 * (t_0 * -96.0f))) * (0.041666666666666664f / v))) / v) + ((1.0f - u) * (-0.5f * ((-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) :: tmp
    t_0 = (1.0e0 - u) * (1.0e0 - u)
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((-2.0e0) + (u * 2.0e0)) + ((((((-0.16666666666666666e0) * ((1.0e0 - u) * (8.0e0 + ((1.0e0 - u) * (((1.0e0 - u) * 16.0e0) + (-24.0e0)))))) + ((((1.0e0 - u) * (16.0e0 + ((1.0e0 - u) * ((-112.0e0) + ((1.0e0 - u) * 192.0e0))))) + (t_0 * (t_0 * (-96.0e0)))) * (0.041666666666666664e0 / v))) / v) + ((1.0e0 - u) * ((-0.5e0) * (((-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))
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(-2.0) + Float32(u * Float32(2.0))) + Float32(Float32(Float32(Float32(Float32(Float32(-0.16666666666666666) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(8.0) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(Float32(1.0) - u) * Float32(16.0)) + Float32(-24.0)))))) + Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(16.0) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(-112.0) + Float32(Float32(Float32(1.0) - u) * Float32(192.0)))))) + Float32(t_0 * Float32(t_0 * Float32(-96.0)))) * Float32(Float32(0.041666666666666664) / v))) / v) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(-0.5) * 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);
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(-2.0) + (u * single(2.0))) + (((((single(-0.16666666666666666) * ((single(1.0) - u) * (single(8.0) + ((single(1.0) - u) * (((single(1.0) - u) * single(16.0)) + single(-24.0)))))) + ((((single(1.0) - u) * (single(16.0) + ((single(1.0) - u) * (single(-112.0) + ((single(1.0) - u) * single(192.0)))))) + (t_0 * (t_0 * single(-96.0)))) * (single(0.041666666666666664) / v))) / v) + ((single(1.0) - u) * (single(-0.5) * ((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 < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 86.6%

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

   7. Applied egg-rr 86.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(-2 + u \cdot 2\right) + \frac{\frac{-0.16666666666666666 \cdot \left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right)\right) + \left(\left(1 - u\right) \cdot \left(16 + \left(1 - u\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right) + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot -96\right)\right) \cdot \frac{0.041666666666666664}{v}}{v} + \left(1 - u\right) \cdot \left(-0.5 \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(u \cdot \left(16 + u \cdot \left(-112 + u \cdot \left(192 + u \cdot -96\right)\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (+
       (/
        (+
         (*
          (+
           (* (- 1.0 u) 8.0)
           (* (* (- 1.0 u) (- 1.0 u)) (+ (* (- 1.0 u) 16.0) -24.0)))
          -0.16666666666666666)
         (/
          (*
           0.041666666666666664
           (* u (+ 16.0 (* u (+ -112.0 (* u (+ 192.0 (* u -96.0))))))))
          v))
        v)
       (* -0.5 (* (- 1.0 u) (- (* -4.0 (+ u -1.0)) 4.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((((((((1.0f - u) * 8.0f) + (((1.0f - u) * (1.0f - u)) * (((1.0f - u) * 16.0f) + -24.0f))) * -0.16666666666666666f) + ((0.041666666666666664f * (u * (16.0f + (u * (-112.0f + (u * (192.0f + (u * -96.0f)))))))) / 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) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((((((1.0e0 - u) * 8.0e0) + (((1.0e0 - u) * (1.0e0 - u)) * (((1.0e0 - u) * 16.0e0) + (-24.0e0)))) * (-0.16666666666666666e0)) + ((0.041666666666666664e0 * (u * (16.0e0 + (u * ((-112.0e0) + (u * (192.0e0 + (u * (-96.0e0))))))))) / 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)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		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(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)) * Float32(Float32(Float32(Float32(1.0) - u) * Float32(16.0)) + Float32(-24.0)))) * Float32(-0.16666666666666666)) + Float32(Float32(Float32(0.041666666666666664) * Float32(u * Float32(Float32(16.0) + Float32(u * Float32(Float32(-112.0) + Float32(u * Float32(Float32(192.0) + Float32(u * Float32(-96.0))))))))) / 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)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((((((((single(1.0) - u) * single(8.0)) + (((single(1.0) - u) * (single(1.0) - u)) * (((single(1.0) - u) * single(16.0)) + single(-24.0)))) * single(-0.16666666666666666)) + ((single(0.041666666666666664) * (u * (single(16.0) + (u * (single(-112.0) + (u * (single(192.0) + (u * single(-96.0))))))))) / 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 < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 86.6%

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

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \color{blue}{\left(u \cdot \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)\right)}\right), v\right)\right), v\right)\right), v\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \left(16 + u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \left(u \cdot \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(u, \left(u \cdot \left(192 + -96 \cdot u\right) - 112\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(u, \left(u \cdot \left(192 + -96 \cdot u\right) + \left(\mathsf{neg}\left(112\right)\right)\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(u, \left(u \cdot \left(192 + -96 \cdot u\right) + -112\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(u \cdot \left(192 + -96 \cdot u\right)\right), -112\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(192 + -96 \cdot u\right)\right), -112\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(192, \left(-96 \cdot u\right)\right)\right), -112\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(192, \left(u \cdot -96\right)\right)\right), -112\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

      10. *-lowering-*.f32 86.6%

          \[\leadsto \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), -4\right), 4\right)\right), \frac{-1}{2}\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 8\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), 16\right), -24\right)\right)\right), \frac{-1}{6}\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{24}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(16, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(192, \mathsf{*.f32}\left(u, -96\right)\right)\right), -112\right)\right)\right)\right)\right), v\right)\right), v\right)\right), v\right)\right)\right) \]

   8. Simplified 86.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{\frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(u \cdot \left(16 + u \cdot \left(-112 + u \cdot \left(192 + u \cdot -96\right)\right)\right)\right)}{v}}{v} + -0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{-0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right) + \frac{0.16666666666666666}{v} \cdot \left(8 \cdot \left(u + -1\right) + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(16 \cdot \left(u + -1\right) - -24\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    (- -1.0 (* u -2.0))
    (/
     (+
      (* -0.5 (* (- 1.0 u) (- (* -4.0 (+ u -1.0)) 4.0)))
      (*
       (/ 0.16666666666666666 v)
       (+
        (* 8.0 (+ u -1.0))
        (* (* (- 1.0 u) (- 1.0 u)) (- (* 16.0 (+ u -1.0)) -24.0)))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (-1.0f - (u * -2.0f)) + (((-0.5f * ((1.0f - u) * ((-4.0f * (u + -1.0f)) - 4.0f))) + ((0.16666666666666666f / v) * ((8.0f * (u + -1.0f)) + (((1.0f - u) * (1.0f - u)) * ((16.0f * (u + -1.0f)) - -24.0f))))) / v);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = ((-1.0e0) - (u * (-2.0e0))) + ((((-0.5e0) * ((1.0e0 - u) * (((-4.0e0) * (u + (-1.0e0))) - 4.0e0))) + ((0.16666666666666666e0 / v) * ((8.0e0 * (u + (-1.0e0))) + (((1.0e0 - u) * (1.0e0 - u)) * ((16.0e0 * (u + (-1.0e0))) - (-24.0e0)))))) / v)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(-1.0) - Float32(u * Float32(-2.0))) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(-4.0) * Float32(u + Float32(-1.0))) - Float32(4.0)))) + Float32(Float32(Float32(0.16666666666666666) / v) * Float32(Float32(Float32(8.0) * Float32(u + Float32(-1.0))) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)) * Float32(Float32(Float32(16.0) * Float32(u + Float32(-1.0))) - Float32(-24.0)))))) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (single(-1.0) - (u * single(-2.0))) + (((single(-0.5) * ((single(1.0) - u) * ((single(-4.0) * (u + single(-1.0))) - single(4.0)))) + ((single(0.16666666666666666) / v) * ((single(8.0) * (u + single(-1.0))) + (((single(1.0) - u) * (single(1.0) - u)) * ((single(16.0) * (u + single(-1.0))) - single(-24.0)))))) / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 83.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + \frac{-0.5 \cdot \left(\left(1 - u\right) \cdot \left(-4 \cdot \left(u + -1\right) - 4\right)\right) + \frac{0.16666666666666666}{v} \cdot \left(8 \cdot \left(u + -1\right) + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(16 \cdot \left(u + -1\right) - -24\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\left(\frac{2}{v} + \left(2 + \frac{1.3333333333333333}{v \cdot v}\right)\right) + u \cdot \left(\frac{u \cdot 2.6666666666666665}{v \cdot v} - \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (+
      (+ (/ 2.0 v) (+ 2.0 (/ 1.3333333333333333 (* v v))))
      (*
       u
       (-
        (/ (* u 2.6666666666666665) (* v v))
        (+ (/ 2.0 v) (/ 4.0 (* v v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (((2.0f / v) + (2.0f + (1.3333333333333333f / (v * v)))) + (u * (((u * 2.6666666666666665f) / (v * v)) - ((2.0f / v) + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (((2.0e0 / v) + (2.0e0 + (1.3333333333333333e0 / (v * v)))) + (u * (((u * 2.6666666666666665e0) / (v * v)) - ((2.0e0 / v) + (4.0e0 / (v * v)))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(Float32(2.0) / v) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v)))) + Float32(u * Float32(Float32(Float32(u * Float32(2.6666666666666665)) / Float32(v * v)) - Float32(Float32(Float32(2.0) / v) + Float32(Float32(4.0) / Float32(v * v))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (((single(2.0) / v) + (single(2.0) + (single(1.3333333333333333) / (v * v)))) + (u * (((u * single(2.6666666666666665)) / (v * v)) - ((single(2.0) / v) + (single(4.0) / (v * v)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 82.8%

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

   6. Taylor expanded in u around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. associate--l+ N/A

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

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

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

   8. Simplified 82.7%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

   11. Simplified 83.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\left(\frac{2}{v} + \left(2 + \frac{1.3333333333333333}{v \cdot v}\right)\right) + u \cdot \left(\frac{u \cdot 2.6666666666666665}{v \cdot v} - \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\frac{2}{v} + \frac{\frac{1.3333333333333333}{v} - u \cdot 2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (* u (+ 2.0 (+ (/ 2.0 v) (/ (- (/ 1.3333333333333333 v) (* u 2.0)) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f / v) + (((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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 / v) + (((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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) / v) + Float32(Float32(Float32(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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) / v) + (((single(1.3333333333333333) / v) - (u * single(2.0))) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 82.8%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

   8. Simplified 80.3%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r/ N/A

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f32 80.6%

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

   11. Simplified 80.6%

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. metadata-eval N/A

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

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

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

   13. Applied egg-rr 81.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\frac{2}{v} + \frac{\frac{1.3333333333333333}{v} - u \cdot 2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{2}{v} + \left(2 + \frac{1.3333333333333333}{v \cdot v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (* u (+ (/ 2.0 v) (+ 2.0 (/ 1.3333333333333333 (* v v))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((2.0f / v) + (2.0f + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * ((2.0e0 / 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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(2.0) / v) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((single(2.0) / v) + (single(2.0) + (single(1.3333333333333333) / (v * v)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 82.8%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

   8. Simplified 80.3%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

   11. Simplified 78.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{2}{v} + \left(2 + \frac{1.3333333333333333}{v \cdot v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.6% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 82.8%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

   8. Simplified 80.3%

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

   9. Taylor expanded in v around inf

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

   11. Simplified 74.8%

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

4. Final simplification 91.9%

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

Alternative 12: 90.5% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

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

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

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

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

   5. Simplified 74.7%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f32 72.6%

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

   8. Simplified 72.6%

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

4. Final simplification 91.8%

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

Alternative 13: 89.9% accurate, 21.2× speedup

Math

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

FPCore

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

C

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.0%

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

   if 0.100000001 < v

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

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

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

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

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

   5. Simplified 74.7%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f32 62.1%

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

   8. Simplified 62.1%

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

4. Final simplification 91.1%

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

Alternative 14: 86.9% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0

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

5. Simplified 87.7%

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

Alternative 15: 5.8% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.9%

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

Reproduce

herbie shell --seed 2024161 
(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))))))))