HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 11.0s

Alternatives: 9

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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-2}{v}}\\ 1 + v \cdot \log \left(\left(t\_0 + u\right) - t\_0 \cdot u\right) \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (exp (/ -2.0 v)))) (+ 1.0 (* v (log (- (+ t_0 u) (* t_0 u)))))))

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v));
	return 1.0f + (v * logf(((t_0 + u) - (t_0 * u))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = exp(((-2.0e0) / v))
    code = 1.0e0 + (v * log(((t_0 + u) - (t_0 * u))))
end function

Julia

function code(u, v)
	t_0 = exp(Float32(Float32(-2.0) / v))
	return Float32(Float32(1.0) + Float32(v * log(Float32(Float32(t_0 + u) - Float32(t_0 * u)))))
end

MATLAB

function tmp = code(u, v)
	t_0 = exp((single(-2.0) / v));
	tmp = single(1.0) + (v * log(((t_0 + u) - (t_0 * u))));
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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-lft-neg-out N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left(e^{3}\right)}^{\left(0.3333333333333333 \cdot \frac{-2}{v}\right)}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (*
   v
   (log
    (+ u (* (- 1.0 u) (pow (exp 3.0) (* 0.3333333333333333 (/ -2.0 v)))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * powf(expf(3.0f), (0.3333333333333333f * (-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(3.0e0) ** (0.3333333333333333e0 * ((-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(3.0)) ^ Float32(Float32(0.3333333333333333) * 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(3.0)) ^ (single(0.3333333333333333) * (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. Step-by-step derivation

   1. lift-exp.f32 N/A

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

   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. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-pow.f32 N/A

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

   2. lift-E.f32 N/A

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

   3. add-cbrt-cube N/A

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

   4. pow1/3 N/A

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

   5. metadata-eval N/A

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

   6. log-E N/A

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

   7. lift-E.f32 N/A

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

   8. log-pow N/A

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

   9. pow1/3 N/A

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

   10. lift-E.f32 N/A

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

   11. pow-pow N/A

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

   12. lower-pow.f32 N/A

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

   13. rem-cube-cbrt N/A

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

   14. add-cbrt-cube N/A

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

   15. e-exp-1 N/A

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

   16. pow-exp N/A

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

   17. metadata-eval N/A

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

   18. lower-exp.f32 N/A

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

   19. lower-*.f32 N/A

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

6. Applied rewrites 99.6%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{3}\right)}^{\left(0.3333333333333333 \cdot \frac{-2}{v}\right)}}\right) \]
7. Add Preprocessing

Alternative 3: 47.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{fma}\left(-u, e^{\frac{-2}{v}}, u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \left(\left(\left(\frac{4}{u} - 4\right) \cdot u\right) \cdot u\right) \cdot \frac{0.5}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (* v (log (fma (- u) (exp (/ -2.0 v)) u))))
   (+ 1.0 (+ (* (- 1.0 u) -2.0) (* (* (* (- (/ 4.0 u) 4.0) u) u) (/ 0.5 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v * logf(fmaf(-u, expf((-2.0f / v)), u)));
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((((4.0f / u) - 4.0f) * u) * u) * (0.5f / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v * log(fma(Float32(-u), exp(Float32(Float32(-2.0) / v)), u))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(4.0) / u) - Float32(4.0)) * u) * u) * Float32(Float32(0.5) / v))));
	end
	return 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. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in u around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. lower-exp.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f32 99.3

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

   7. Applied rewrites 99.3%

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

   if 0.100000001 < v

   1. Initial program 94.7%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-fma.f32 N/A

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

      11. lower--.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lower--.f32 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 45.2%

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

   10. Applied rewrites 56.3%

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

Alternative 4: 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 5: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

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. Step-by-step derivation

   1. lift-exp.f32 N/A

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

   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. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 96.7%

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

Alternative 6: 87.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-lft-neg-out N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. associate-*r/ N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. metadata-eval N/A

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

   7. distribute-rgt-in N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. mul-1-neg N/A

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

   14. sub-neg N/A

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

   15. associate-*r/ N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 N/A

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

9. Applied rewrites 81.0%

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

10. Taylor expanded in u around inf

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

12. Applied rewrites 86.7%

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

13. Final simplification 86.7%

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

Alternative 7: 90.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \left(\left(\left(\frac{4}{u} - 4\right) \cdot u\right) \cdot u\right) \cdot \frac{0.5}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ 1.0 (+ (* (- 1.0 u) -2.0) (* (* (* (- (/ 4.0 u) 4.0) u) u) (/ 0.5 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((((4.0f / u) - 4.0f) * u) * u) * (0.5f / 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 + (((1.0e0 - u) * (-2.0e0)) + (((((4.0e0 / u) - 4.0e0) * u) * u) * (0.5e0 / 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(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(Float32(Float32(4.0) / u) - Float32(4.0)) * u) * u) * Float32(Float32(0.5) / 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) + (((single(1.0) - u) * single(-2.0)) + (((((single(4.0) / u) - single(4.0)) * u) * u) * (single(0.5) / 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. Applied rewrites 91.3%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-fma.f32 N/A

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

      11. lower--.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lower--.f32 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 50.4%

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

   10. Applied rewrites 63.0%

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

Alternative 8: 86.7% 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 99.6%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 86.0%

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

Alternative 9: 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 99.6%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.7%

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

Reproduce

herbie shell --seed 2024322 
(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))))))))