HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 10.5s

Alternatives: 12

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

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

FPCore

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

Derivation

1. Initial program 99.4%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(-6 \cdot \left(1 - u\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (+ (* (log (cbrt (E))) (* -6.0 (- 1.0 u))) 1.0)
   1.0))

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 92.8%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 92.9

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

   4. Applied rewrites 92.9%

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

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

      4. pow3 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-E.f32 N/A

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

      8. lower-cbrt.f32 N/A

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

      9. lower-*.f32 91.6

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

   6. Applied rewrites 91.6%

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

   7. Taylor expanded in v around inf

      \[\leadsto 1 + \color{blue}{-6 \cdot \left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(1 - u\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + -6 \cdot \color{blue}{\left(\left(1 - u\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto 1 + \color{blue}{\left(-6 \cdot \left(1 - u\right)\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto 1 + \color{blue}{\left(-6 \cdot \left(1 - u\right)\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto 1 + \color{blue}{\left(-6 \cdot \left(1 - u\right)\right)} \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \]

      5. lower--.f32 N/A

         \[\leadsto 1 + \left(-6 \cdot \color{blue}{\left(1 - u\right)}\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \]

      6. lower-log.f32 N/A

         \[\leadsto 1 + \left(-6 \cdot \left(1 - u\right)\right) \cdot \color{blue}{\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

      7. lower-cbrt.f32 N/A

         \[\leadsto 1 + \left(-6 \cdot \left(1 - u\right)\right) \cdot \log \color{blue}{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

      8. lower-E.f32 54.4

         \[\leadsto 1 + \left(-6 \cdot \left(1 - u\right)\right) \cdot \log \left(\sqrt[3]{\color{blue}{\mathsf{E}\left(\right)}}\right) \]

   9. Applied rewrites 54.4%

      \[\leadsto 1 + \color{blue}{\left(-6 \cdot \left(1 - u\right)\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(-6 \cdot \left(1 - u\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float t_0 = (2.0f - (2.0f / u)) * u;
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = (1.0f / (1.0f - t_0)) * (1.0f - powf(t_0, 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) :: t_0
    real(4) :: tmp
    t_0 = (2.0e0 - (2.0e0 / u)) * u
    if ((log(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * v) <= (-1.0e0)) then
        tmp = (1.0e0 / (1.0e0 - t_0)) * (1.0e0 - (t_0 ** 2.0e0))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	t_0 = (single(2.0) - (single(2.0) / u)) * u;
	tmp = single(0.0);
	if ((log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) <= single(-1.0))
		tmp = (single(1.0) / (single(1.0) - t_0)) * (single(1.0) - (t_0 ^ 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 92.8%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 54.1%

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

      1. lift-+.f32 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f32 N/A

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

   10. Applied rewrites 54.2%

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

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.7%

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

4. Final simplification 90.1%

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

Alternative 4: 89.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float t_0 = -2.0f * (1.0f - u);
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = (1.0f / (1.0f - t_0)) * (1.0f - powf(t_0, 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) :: t_0
    real(4) :: tmp
    t_0 = (-2.0e0) * (1.0e0 - u)
    if ((log(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * v) <= (-1.0e0)) then
        tmp = (1.0e0 / (1.0e0 - t_0)) * (1.0e0 - (t_0 ** 2.0e0))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	t_0 = single(-2.0) * (single(1.0) - u);
	tmp = single(0.0);
	if ((log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) <= single(-1.0))
		tmp = (single(1.0) / (single(1.0) - t_0)) * (single(1.0) - (t_0 ^ 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 92.8%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 54.1

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

   5. Applied rewrites 54.1%

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

      1. lift-+.f32 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f32 N/A

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

   7. Applied rewrites 54.2%

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

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.7%

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

4. Final simplification 90.1%

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

Alternative 5: 89.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) <= single(-1.0))
		tmp = (((single(-2.0) / u) * u) + (single(2.0) * u)) + single(1.0);
	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 92.8%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 54.1%

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

   10. Applied rewrites 54.1%

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

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.7%

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

4. Final simplification 90.1%

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

Alternative 6: 89.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) <= single(-1.0))
		tmp = ((single(2.0) - (single(2.0) / u)) * u) + single(1.0);
	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 92.8%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 54.1%

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

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.7%

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

4. Final simplification 90.1%

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

Alternative 7: 89.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) <= single(-1.0))
		tmp = (single(-2.0) * (single(1.0) - u)) + single(1.0);
	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 92.8%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 54.1

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

   5. Applied rewrites 54.1%

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

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.7%

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

4. Final simplification 90.1%

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

Alternative 8: 38.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\log \left(\mathsf{fma}\left(-u, e^{\frac{-2}{v}}, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(-6 \cdot \left(1 - u\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ (* (log (fma (- u) (exp (/ -2.0 v)) u)) v) 1.0)
   (+ (* (log (cbrt (E))) (* -6.0 (- 1.0 u))) 1.0)))

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-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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in u around inf

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. associate-*r/ N/A

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

      6. log-E N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. 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. 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 93.4%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 93.5

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

   4. Applied rewrites 93.5%

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

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

      4. pow3 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-E.f32 N/A

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

      8. lower-cbrt.f32 N/A

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

      9. lower-*.f32 92.5

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

   6. Applied rewrites 92.5%

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

   7. Taylor expanded in v around inf

      \[\leadsto 1 + \color{blue}{-6 \cdot \left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(1 - u\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + -6 \cdot \color{blue}{\left(\left(1 - u\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto 1 + \color{blue}{\left(-6 \cdot \left(1 - u\right)\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto 1 + \color{blue}{\left(-6 \cdot \left(1 - u\right)\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto 1 + \color{blue}{\left(-6 \cdot \left(1 - u\right)\right)} \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \]

      5. lower--.f32 N/A

         \[\leadsto 1 + \left(-6 \cdot \color{blue}{\left(1 - u\right)}\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \]

      6. lower-log.f32 N/A

         \[\leadsto 1 + \left(-6 \cdot \left(1 - u\right)\right) \cdot \color{blue}{\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

      7. lower-cbrt.f32 N/A

         \[\leadsto 1 + \left(-6 \cdot \left(1 - u\right)\right) \cdot \log \color{blue}{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]

      8. lower-E.f32 48.1

         \[\leadsto 1 + \left(-6 \cdot \left(1 - u\right)\right) \cdot \log \left(\sqrt[3]{\color{blue}{\mathsf{E}\left(\right)}}\right) \]

   9. Applied rewrites 48.1%

      \[\leadsto 1 + \color{blue}{\left(-6 \cdot \left(1 - u\right)\right) \cdot \log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\log \left(\mathsf{fma}\left(-u, e^{\frac{-2}{v}}, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(-6 \cdot \left(1 - u\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 10: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

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 96.0%

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

8. Final simplification 96.0%

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

Alternative 11: 86.8% 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 99.4%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 86.7%

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

Alternative 12: 5.8% 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 99.4%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.0%

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

Reproduce

herbie shell --seed 2024244 
(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))))))))