HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.4s

Alternatives: 11

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 99.5%

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

Alternative 2: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in v around inf

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 51.6

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

   8. Applied rewrites 51.6%

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

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   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. Applied rewrites 93.0%

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

Alternative 3: 95.9% 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. pow-neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u around 0

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

   1. rec-exp N/A

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

   2. log-E N/A

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

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

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

   4. lower-exp.f32 N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 95.4

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

7. Applied rewrites 95.4%

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

Alternative 4: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 + \frac{\frac{2 + \frac{1.3333333333333333}{v}}{v} - -2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (*
   v
   (log
    (+
     u
     (*
      (- 1.0 u)
      (/
       1.0
       (+ 1.0 (/ (- (/ (+ 2.0 (/ 1.3333333333333333 v)) v) -2.0) v)))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * (1.0f / (1.0f + ((((2.0f + (1.3333333333333333f / v)) / v) - -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) * (1.0e0 / (1.0e0 + ((((2.0e0 + (1.3333333333333333e0 / v)) / v) - (-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) * Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) - Float32(-2.0)) / v))))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(1.0) / (single(1.0) + ((((single(2.0) + (single(1.3333333333333333) / v)) / v) - 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. pow-neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 94.2%

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

8. Final simplification 94.2%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 + \frac{\frac{2 + \frac{1.3333333333333333}{v}}{v} - -2}{v}}\right) \]
9. Add Preprocessing

Alternative 5: 93.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(1.0) / (single(1.0) - ((single(-2.0) + (single(-2.0) / v)) / 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. pow-neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. log-E N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. log-E N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

   11. distribute-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. lower-/.f32 N/A

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

7. Applied rewrites 92.6%

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

Alternative 6: 91.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * (1.0f / (1.0f + (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) * (1.0e0 / (1.0e0 + (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) * Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(2.0) / v))))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(1.0) / (single(1.0) + (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. pow-neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around inf

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

   1. log-E N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 90.6

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

7. Applied rewrites 90.6%

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

Alternative 7: 90.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2}{v \cdot v}\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \left(\left(u \cdot u\right) \cdot \left(t\_0 + \frac{\frac{t\_0 + \frac{-2 + \frac{2}{v}}{v}}{u} - \left(\frac{-2}{v} + t\_0\right)}{u}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (/ -2.0 (* v v))))
   (if (<= v 0.10000000149011612)
     1.0
     (+
      1.0
      (*
       v
       (*
        (* u u)
        (+
         t_0
         (/
          (- (/ (+ t_0 (/ (+ -2.0 (/ 2.0 v)) v)) u) (+ (/ -2.0 v) t_0))
          u))))))))

C

float code(float u, float v) {
	float t_0 = -2.0f / (v * v);
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * ((u * u) * (t_0 + ((((t_0 + ((-2.0f + (2.0f / v)) / v)) / u) - ((-2.0f / v) + t_0)) / u))));
	}
	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 = (-2.0e0) / (v * v)
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * ((u * u) * (t_0 + ((((t_0 + (((-2.0e0) + (2.0e0 / v)) / v)) / u) - (((-2.0e0) / v) + t_0)) / u))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(-2.0) / Float32(v * v))
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(Float32(u * u) * Float32(t_0 + Float32(Float32(Float32(Float32(t_0 + Float32(Float32(Float32(-2.0) + Float32(Float32(2.0) / v)) / v)) / u) - Float32(Float32(Float32(-2.0) / v) + t_0)) / u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = single(-2.0) / (v * v);
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * ((u * u) * (t_0 + ((((t_0 + ((single(-2.0) + (single(2.0) / v)) / v)) / u) - ((single(-2.0) / v) + t_0)) / u))));
	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. Applied rewrites 94.1%

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

   if 0.100000001 < v

   1. Initial program 94.4%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 9.0%

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

   7. Applied rewrites 6.8%

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

      1. lift-fma.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. lift-fma.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f32 N/A

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

   9. Applied rewrites 6.9%

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

   10. Taylor expanded in u around -inf

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

   12. Applied rewrites 56.0%

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

4. Final simplification 90.6%

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

Alternative 8: 90.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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. Applied rewrites 94.1%

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

   if 0.100000001 < v

   1. Initial program 94.4%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in u around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f32 N/A

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

   8. Applied rewrites 55.9%

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

4. Final simplification 90.6%

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

Alternative 9: 90.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (u * u) * ((-2.0f / v) + ((((-1.0f / u) - (-2.0f / v)) - -2.0f) / 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (u * u) * (((-2.0e0) / v) + (((((-1.0e0) / u) - ((-2.0e0) / v)) - (-2.0e0)) / u))
    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 * u) * Float32(Float32(Float32(-2.0) / v) + Float32(Float32(Float32(Float32(Float32(-1.0) / u) - Float32(Float32(-2.0) / v)) - Float32(-2.0)) / u)));
	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 * u) * ((single(-2.0) / v) + ((((single(-1.0) / u) - (single(-2.0) / v)) - single(-2.0)) / u));
	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. Applied rewrites 94.1%

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

   if 0.100000001 < v

   1. Initial program 94.4%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 9.0%

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

      1. lift--.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 8.0

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

   7. Applied rewrites 7.0%

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

   8. Taylor expanded in u around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f32 N/A

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

   10. Applied rewrites 55.9%

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

4. Final simplification 90.6%

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

Alternative 10: 87.0% accurate, 231.0× 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. Applied rewrites 86.3%

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

Alternative 11: 5.9% accurate, 231.0× 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. Applied rewrites 6.2%

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

Reproduce

herbie shell --seed 2024216 
(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))))))))