HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 9.7s

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: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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 + (log(((u - (u * t_0)) + t_0)) * v)
end function

Julia

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

MATLAB

function tmp = code(u, v)
	t_0 = exp((single(-2.0) / v));
	tmp = single(1.0) + (log(((u - (u * t_0)) + t_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

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

   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 99.5

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

4. Applied rewrites 99.5%

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

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lower-+.f32 N/A

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

   6. lower-+.f32 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

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

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 90.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((log((u - ((u - single(1.0)) * exp((single(-2.0) / v))))) * v) <= single(-1.0))
		tmp = single(1.0) + (((single(-2.0) / v) - ((((single(2.0) / u) - single(2.0)) - (single(2.0) / v)) / u)) * (u * u));
	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 93.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 48.5

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

   5. Applied rewrites 48.5%

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

   7. Applied rewrites 48.5%

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

   8. Taylor expanded in u around -inf

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

   10. Applied rewrites 56.9%

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

   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 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 94.0%

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

4. Final simplification 91.3%

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

Alternative 3: 90.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((log((u - ((u - single(1.0)) * exp((single(-2.0) / v))))) * v) <= single(-1.0))
		tmp = single(1.0) + ((single(4.0) - ((u * u) * single(4.0))) / (single(-2.0) - (single(2.0) * u)));
	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 93.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 48.5

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

   5. Applied rewrites 48.5%

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

   7. Applied rewrites 48.5%

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

   8. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 40.2

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

   10. Applied rewrites 40.2%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(2, u, -2\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 48.5%

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

   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 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 94.0%

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

4. Final simplification 90.6%

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

Alternative 4: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((logf((u - ((u - 1.0f) * expf((-2.0f / v))))) * v) <= -1.0f) {
		tmp = (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((u - ((u - 1.0e0) * exp(((-2.0e0) / v))))) * v) <= (-1.0e0)) then
        tmp = (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(u - Float32(Float32(u - Float32(1.0)) * exp(Float32(Float32(-2.0) / v))))) * v) <= Float32(-1.0))
		tmp = Float32(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((u - ((u - single(1.0)) * exp((single(-2.0) / v))))) * v) <= single(-1.0))
		tmp = (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 93.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 93.2

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

   4. Applied rewrites 93.2%

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

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f32 N/A

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

      6. lower-+.f32 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-neg.f32 N/A

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

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

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

      5. unsub-neg N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 93.5

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

   8. Applied rewrites 93.5%

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

   9. Taylor expanded in v around inf

      \[\leadsto \color{blue}{1 + -1 \cdot \left(2 + -2 \cdot u\right)} \]
   10. 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. lower-*.f32 48.5

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

   11. Applied rewrites 48.5%

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

   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 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 94.0%

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

4. Final simplification 90.6%

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

Alternative 5: 50.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1 + \log \left(\mathsf{fma}\left(-u, e^{\frac{-2}{v}}, u\right)\right) \cdot v\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot u\right) \cdot \left(\left(\left(\frac{\frac{2}{u}}{v} + \frac{2}{u}\right) - \frac{2}{v}\right) - \frac{\frac{2}{u}}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   (+ 1.0 (* (log (fma (- u) (exp (/ -2.0 v)) u)) v))
   (+
    1.0
    (*
     (* u u)
     (- (- (+ (/ (/ 2.0 u) v) (/ 2.0 u)) (/ 2.0 v)) (/ (/ 2.0 u) u))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f + (logf(fmaf(-u, expf((-2.0f / v)), u)) * v);
	} else {
		tmp = 1.0f + ((u * u) * (((((2.0f / u) / v) + (2.0f / u)) - (2.0f / v)) - ((2.0f / u) / u)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(Float32(1.0) + Float32(log(fma(Float32(-u), exp(Float32(Float32(-2.0) / v)), u)) * v));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * u) * Float32(Float32(Float32(Float32(Float32(Float32(2.0) / u) / v) + Float32(Float32(2.0) / u)) - Float32(Float32(2.0) / v)) - Float32(Float32(Float32(2.0) / u) / u))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   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 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.0500000007 < 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. 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 46.2

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 54.5%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1 + \log \left(\mathsf{fma}\left(-u, e^{\frac{-2}{v}}, u\right)\right) \cdot v\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot u\right) \cdot \left(\left(\left(\frac{\frac{2}{u}}{v} + \frac{2}{u}\right) - \frac{2}{v}\right) - \frac{\frac{2}{u}}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (log((u - ((u - single(1.0)) * exp((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. Final simplification 99.5%

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

Alternative 7: 90.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.0%

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

   if 0.200000003 < v

   1. Initial program 93.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 48.5

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 57.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot u\right) \cdot \left(\left(\left(\frac{\frac{2}{u}}{v} + \frac{2}{u}\right) - \frac{2}{v}\right) - \frac{\frac{2}{u}}{u}\right)\\ \end{array} \]
5. 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 99.5%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 87.3%

   \[\leadsto \color{blue}{1} \]
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 99.5%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.0%

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

Reproduce

herbie shell --seed 2024296 
(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))))))))