HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.9s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 96.0% 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 99.6%

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

3. Taylor expanded in u around 0

   \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \color{blue}{\left(e^{\frac{-2}{v}}\right)}\right)\right)\right)\right) \]
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 95.4%

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

5. Simplified 95.4%

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

Alternative 3: 91.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(2 + \left(\frac{0.6666666666666666}{v \cdot v} + \frac{1.3333333333333333}{v}\right)\right) + u \cdot \left(\left(u \cdot \left(\left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \frac{u \cdot 4}{v \cdot v}\right) - \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - 2\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (*
       u
       (+
        (+ 2.0 (+ (/ 0.6666666666666666 (* v v)) (/ 1.3333333333333333 v)))
        (*
         u
         (-
          (-
           (*
            u
            (-
             (+ (/ 2.6666666666666665 v) (/ 8.0 (* v v)))
             (/ (* u 4.0) (* v v))))
           (+ (/ 4.0 v) (/ 4.666666666666667 (* v v))))
          2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((u * ((2.0f + ((0.6666666666666666f / (v * v)) + (1.3333333333333333f / v))) + (u * (((u * (((2.6666666666666665f / v) + (8.0f / (v * v))) - ((u * 4.0f) / (v * v)))) - ((4.0f / v) + (4.666666666666667f / (v * v)))) - 2.0f)))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * ((2.0e0 + ((0.6666666666666666e0 / (v * v)) + (1.3333333333333333e0 / v))) + (u * (((u * (((2.6666666666666665e0 / v) + (8.0e0 / (v * v))) - ((u * 4.0e0) / (v * v)))) - ((4.0e0 / v) + (4.666666666666667e0 / (v * v)))) - 2.0e0)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(Float32(0.6666666666666666) / Float32(v * v)) + Float32(Float32(1.3333333333333333) / v))) + Float32(u * Float32(Float32(Float32(u * Float32(Float32(Float32(Float32(2.6666666666666665) / v) + Float32(Float32(8.0) / Float32(v * v))) - Float32(Float32(u * Float32(4.0)) / Float32(v * v)))) - Float32(Float32(Float32(4.0) / v) + Float32(Float32(4.666666666666667) / Float32(v * v)))) - Float32(2.0))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((u * ((single(2.0) + ((single(0.6666666666666666) / (v * v)) + (single(1.3333333333333333) / v))) + (u * (((u * (((single(2.6666666666666665) / v) + (single(8.0) / (v * v))) - ((u * single(4.0)) / (v * v)))) - ((single(4.0) / v) + (single(4.666666666666667) / (v * v)))) - single(2.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

   if 0.100000001 < v

   1. Initial program 94.8%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 72.8%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 72.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(\left(2 + \left(\frac{0.6666666666666666}{v \cdot v} + \frac{1.3333333333333333}{v}\right)\right) + u \cdot \left(\left(u \cdot \left(\left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \frac{u \cdot 4}{v \cdot v}\right) - \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - 2\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(u \cdot \left(\left(u \cdot \left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - 2\right) + \left(2 + \left(\frac{0.6666666666666666}{v \cdot v} + \frac{1.3333333333333333}{v}\right)\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (*
       u
       (+
        (*
         u
         (-
          (-
           (* u (+ (/ 2.6666666666666665 v) (/ 8.0 (* v v))))
           (+ (/ 4.0 v) (/ 4.666666666666667 (* v v))))
          2.0))
        (+ 2.0 (+ (/ 0.6666666666666666 (* v v)) (/ 1.3333333333333333 v)))))
      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) + ((u * ((u * (((u * ((2.6666666666666665f / v) + (8.0f / (v * v)))) - ((4.0f / v) + (4.666666666666667f / (v * v)))) - 2.0f)) + (2.0f + ((0.6666666666666666f / (v * v)) + (1.3333333333333333f / v))))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((u * ((u * (((u * ((2.6666666666666665e0 / v) + (8.0e0 / (v * v)))) - ((4.0e0 / v) + (4.666666666666667e0 / (v * v)))) - 2.0e0)) + (2.0e0 + ((0.6666666666666666e0 / (v * v)) + (1.3333333333333333e0 / v))))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(u * Float32(Float32(u * Float32(Float32(Float32(u * Float32(Float32(Float32(2.6666666666666665) / v) + Float32(Float32(8.0) / Float32(v * v)))) - Float32(Float32(Float32(4.0) / v) + Float32(Float32(4.666666666666667) / Float32(v * v)))) - Float32(2.0))) + Float32(Float32(2.0) + Float32(Float32(Float32(0.6666666666666666) / Float32(v * v)) + Float32(Float32(1.3333333333333333) / v))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((u * ((u * (((u * ((single(2.6666666666666665) / v) + (single(8.0) / (v * v)))) - ((single(4.0) / v) + (single(4.666666666666667) / (v * v)))) - single(2.0))) + (single(2.0) + ((single(0.6666666666666666) / (v * v)) + (single(1.3333333333333333) / v))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

   if 0.100000001 < v

   1. Initial program 94.8%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 72.8%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 69.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{u \cdot \left(u \cdot \left(\left(u \cdot \left(\frac{2.6666666666666665}{v} + \frac{8}{v \cdot v}\right) - \left(\frac{4}{v} + \frac{4.666666666666667}{v \cdot v}\right)\right) - 2\right) + \left(2 + \left(\frac{0.6666666666666666}{v \cdot v} + \frac{1.3333333333333333}{v}\right)\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (u * (2.0e0 + (((2.0e0 / v) + (1.3333333333333333e0 / (v * v))) + (u * (((u * 2.6666666666666665e0) / (v * v)) + (((-2.0e0) / v) + ((-4.0e0) / (v * v)))))))) + (-1.0e0)
    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(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(2.0) / v) + 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)))))))) + Float32(-1.0));
	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 = (u * (single(2.0) + (((single(2.0) / v) + (single(1.3333333333333333) / (v * v))) + (u * (((u * single(2.6666666666666665)) / (v * v)) + ((single(-2.0) / v) + (single(-4.0) / (v * v)))))))) + single(-1.0);
	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 94.5%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 68.6%

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

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) - 1} \]
   7. 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

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

   8. Simplified 68.8%

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

4. Final simplification 92.3%

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

Alternative 6: 91.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{\left(\frac{1.3333333333333333 + \left(\frac{0.6666666666666666}{v} + u \cdot -4\right)}{v} - u \cdot 2\right) - -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (-
        (-
         (/ (+ 1.3333333333333333 (+ (/ 0.6666666666666666 v) (* u -4.0))) v)
         (* 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 + (((((1.3333333333333333f + ((0.6666666666666666f / v) + (u * -4.0f))) / v) - (u * 2.0f)) - -2.0f) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((((1.3333333333333333e0 + ((0.6666666666666666e0 / v) + (u * (-4.0e0)))) / v) - (u * 2.0e0)) - (-2.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(0.6666666666666666) / v) + Float32(u * Float32(-4.0)))) / v) - Float32(u * Float32(2.0))) - Float32(-2.0)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (((((single(1.3333333333333333) + ((single(0.6666666666666666) / v) + (u * single(-4.0)))) / v) - (u * single(2.0))) - single(-2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 70.3%

      \[\leadsto \color{blue}{-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \left({\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2} \cdot \frac{v}{e^{2 \cdot \frac{-2}{v}}}\right) \cdot \left(-0.5 \cdot u\right)\right)} \]

   6. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4}{v}\right)}, \mathsf{*.f32}\left(\frac{-1}{2}, u\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4}{v}\right)\right), \mathsf{*.f32}\left(\color{blue}{\frac{-1}{2}}, u\right)\right)\right)\right)\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\color{blue}{\frac{-1}{2}}, u\right)\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4}{-1 \cdot v}\right), \mathsf{*.f32}\left(\frac{-1}{2}, u\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4\right), \left(-1 \cdot v\right)\right), \mathsf{*.f32}\left(\color{blue}{\frac{-1}{2}}, u\right)\right)\right)\right)\right) \]

   8. Simplified 68.6%

      \[\leadsto -1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \color{blue}{\frac{\frac{8 + \left(\frac{8}{v \cdot v} + \frac{9.333333333333334}{v}\right)}{-v} + -4}{-v}} \cdot \left(-0.5 \cdot u\right)\right) \]

   9. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \color{blue}{\left(\frac{-4 \cdot \frac{u}{v} + -2 \cdot u}{v}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\left(-4 \cdot \frac{u}{v} + -2 \cdot u\right), \color{blue}{v}\right)\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\left(-2 \cdot u + -4 \cdot \frac{u}{v}\right), v\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(-2 \cdot u\right), \left(-4 \cdot \frac{u}{v}\right)\right), v\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot -2\right), \left(-4 \cdot \frac{u}{v}\right)\right), v\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \left(-4 \cdot \frac{u}{v}\right)\right), v\right)\right)\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \left(\frac{-4 \cdot u}{v}\right)\right), v\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\left(-4 \cdot u\right), v\right)\right), v\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\left(u \cdot -4\right), v\right)\right), v\right)\right)\right)\right) \]

       9. *-lowering-*.f32 68.2%

          \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u, -2\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, -4\right), v\right)\right), v\right)\right)\right)\right) \]

   11. Simplified 68.2%

       \[\leadsto -1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \color{blue}{\frac{u \cdot -2 + \frac{u \cdot -4}{v}}{v}}\right) \]

   12. Taylor expanded in v around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

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

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

   14. Simplified 67.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{\left(\frac{1.3333333333333333 + \left(\frac{0.6666666666666666}{v} + u \cdot -4\right)}{v} - u \cdot 2\right) - -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 8.2× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/ (- (- (/ (+ 1.3333333333333333 (* u -4.0)) v) (* 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 + (((((1.3333333333333333f + (u * -4.0f)) / v) - (u * 2.0f)) - -2.0f) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((((1.3333333333333333e0 + (u * (-4.0e0))) / v) - (u * 2.0e0)) - (-2.0e0)) / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 70.3%

      \[\leadsto \color{blue}{-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \left({\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2} \cdot \frac{v}{e^{2 \cdot \frac{-2}{v}}}\right) \cdot \left(-0.5 \cdot u\right)\right)} \]

   6. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4}{v}\right)}, \mathsf{*.f32}\left(\frac{-1}{2}, u\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4}{v}\right)\right), \mathsf{*.f32}\left(\color{blue}{\frac{-1}{2}}, u\right)\right)\right)\right)\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\color{blue}{\frac{-1}{2}}, u\right)\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4}{-1 \cdot v}\right), \mathsf{*.f32}\left(\frac{-1}{2}, u\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(v, \mathsf{expm1.f32}\left(\mathsf{/.f32}\left(2, v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{8 + \left(\frac{28}{3} \cdot \frac{1}{v} + \frac{8}{{v}^{2}}\right)}{v} - 4\right), \left(-1 \cdot v\right)\right), \mathsf{*.f32}\left(\color{blue}{\frac{-1}{2}}, u\right)\right)\right)\right)\right) \]

   8. Simplified 68.6%

      \[\leadsto -1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \color{blue}{\frac{\frac{8 + \left(\frac{8}{v \cdot v} + \frac{9.333333333333334}{v}\right)}{-v} + -4}{-v}} \cdot \left(-0.5 \cdot u\right)\right) \]

   9. Taylor expanded in v around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

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

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

   11. Simplified 65.3%

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

4. Final simplification 92.0%

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

Alternative 8: 91.1% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * ((1.3333333333333333e0 / (v * v)) + (2.0e0 + (2.0e0 / v))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 68.6%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 61.4%

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

   8. Simplified 61.4%

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

4. Final simplification 91.6%

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

Alternative 9: 91.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + u \cdot 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ -1.0 (* u (- 2.0 (/ (+ -2.0 (* u 2.0)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   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 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 76.4%

      \[\leadsto \color{blue}{-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \left({\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2} \cdot \frac{v}{e^{2 \cdot \frac{-2}{v}}}\right) \cdot \left(-0.5 \cdot u\right)\right)} \]

   6. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f32 63.4%

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

   8. Simplified 63.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + u \cdot 2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* u (+ 2.0 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   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 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

   3. Taylor expanded in v around inf

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

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

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

   5. Simplified 63.4%

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

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + 2 \cdot \frac{1}{v}\right) - 1} \]
   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 60.7%

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

   8. Simplified 60.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.2% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (* u (+ 2.0 (/ -1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = u * (2.0f + (-1.0f / u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = u * (2.0e0 + ((-1.0e0) / u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(2.0) + Float32(Float32(-1.0) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = u * (single(2.0) + (single(-1.0) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 76.4%

      \[\leadsto \color{blue}{-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \left({\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2} \cdot \frac{v}{e^{2 \cdot \frac{-2}{v}}}\right) \cdot \left(-0.5 \cdot u\right)\right)} \]

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-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 52.5%

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

   8. Simplified 52.5%

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

   9. Taylor expanded in u around inf

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

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

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

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

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

       3. /-lowering-/.f32 52.5%

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

   11. Simplified 52.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \frac{-1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.2% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   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 93.8%

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

   if 0.200000003 < v

   1. Initial program 94.6%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 76.4%

      \[\leadsto \color{blue}{-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \left({\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2} \cdot \frac{v}{e^{2 \cdot \frac{-2}{v}}}\right) \cdot \left(-0.5 \cdot u\right)\right)} \]

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-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 52.5%

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

   8. Simplified 52.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.2% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

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

5. Simplified 86.9%

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

Alternative 14: 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 99.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 6.3%

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

Reproduce

herbie shell --seed 2024150 
(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))))))))