HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 9.7s

Alternatives: 10

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

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

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 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	t_0 = exp(Float32(Float32(-2.0) / v))
	tmp = Float32(0.0)
	if (Float32(log(Float32(Float32(t_0 * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(Float32(Float32(Float32(exp(Float32(Float32(2.0) / v)) * Float32(u * v)) + Float32(-2.0)) - Float32(u * v)) + Float32(1.0));
	else
		tmp = Float32(Float32(log(fma(t_0, Float32(Float32(1.0) - u), u)) * v) + Float32(1.0));
	end
	return 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 94.5%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 35.8

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

   5. Applied rewrites 35.8%

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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 67.6%

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

   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

   4. Applied rewrites 99.0%

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

4. Final simplification 95.5%

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

Alternative 3: 87.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (+
    (+ (fma u (/ (/ 1.3333333333333333 v) v) (* (+ (/ u v) u) 2.0)) -2.0)
    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 = (fmaf(u, ((1.3333333333333333f / v) / v), (((u / v) + u) * 2.0f)) + -2.0f) + 1.0f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

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(fma(u, Float32(Float32(Float32(1.3333333333333333) / v) / v), Float32(Float32(Float32(u / v) + u) * Float32(2.0))) + Float32(-2.0)) + Float32(1.0));
	else
		tmp = Float32(1.0);
	end
	return 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 94.5%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 35.8

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 40.8%

      \[\leadsto 1 + \mathsf{fma}\left(\frac{u}{v} + u, \color{blue}{2}, \mathsf{fma}\left(\frac{1.3333333333333333}{v}, \frac{u}{v}, -2\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.0%

       \[\leadsto 1 + \left(\mathsf{fma}\left(u, \frac{\frac{1.3333333333333333}{v}}{v}, \left(\frac{u}{v} + u\right) \cdot 2\right) + -2\right) \]

   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 93.7%

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

4. Final simplification 88.6%

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

Alternative 4: 89.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (+ (/ (- 4.0 (* (* u u) 4.0)) (- -2.0 (* 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 = ((4.0f - ((u * u) * 4.0f)) / (-2.0f - (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 = ((4.0e0 - ((u * u) * 4.0e0)) / ((-2.0e0) - (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(4.0) - Float32(Float32(u * u) * Float32(4.0))) / Float32(Float32(-2.0) - 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(4.0) - ((u * u) * single(4.0))) / (single(-2.0) - (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 94.5%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 35.8

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in v around inf

      \[\leadsto 1 + \left(2 \cdot u - 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 41.7%

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

   13. Applied rewrites 53.4%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.7%

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

4. Final simplification 90.7%

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

Alternative 5: 89.7% 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 94.5%

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

   3. Taylor expanded in v around 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 53.3

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

   5. Applied rewrites 53.3%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.7%

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

4. Final simplification 90.7%

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

Alternative 6: 51.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;\log \left(\mathsf{fma}\left(1 - u, 1 - \frac{2}{v}, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{\frac{2}{v}} \cdot \left(u \cdot v\right) + -2\right) - u \cdot v\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10599999874830246)
   (+ (* (log (fma (- 1.0 u) (- 1.0 (/ 2.0 v)) u)) v) 1.0)
   (+ (- (+ (* (exp (/ 2.0 v)) (* u v)) -2.0) (* u v)) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10599999874830246f) {
		tmp = (logf(fmaf((1.0f - u), (1.0f - (2.0f / v)), u)) * v) + 1.0f;
	} else {
		tmp = (((expf((2.0f / v)) * (u * v)) + -2.0f) - (u * v)) + 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10599999874830246))
		tmp = Float32(Float32(log(fma(Float32(Float32(1.0) - u), Float32(Float32(1.0) - Float32(Float32(2.0) / v)), u)) * v) + Float32(1.0));
	else
		tmp = Float32(Float32(Float32(Float32(exp(Float32(Float32(2.0) / v)) * Float32(u * v)) + Float32(-2.0)) - Float32(u * v)) + Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.105999999

   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 inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate--r- N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. div-sub N/A

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

      14. lower--.f32 N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f32 44.2

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

   5. Applied rewrites 44.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 47.5

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

   7. Applied rewrites 48.8%

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

   8. Taylor expanded in v around inf

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

      1. lower--.f32 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f32 43.3

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

   10. Applied rewrites 45.9%

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

   if 0.105999999 < v

   1. Initial program 94.8%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 37.5

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 5.7%

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

   9. Applied rewrites 64.9%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;\log \left(\mathsf{fma}\left(1 - u, 1 - \frac{2}{v}, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{\frac{2}{v}} \cdot \left(u \cdot v\right) + -2\right) - u \cdot v\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;\log \left(\mathsf{fma}\left(1 - u, 1 - \frac{2}{v}, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{2}{v} + 2\right) + \frac{1.3333333333333333}{v \cdot v}\right) - \frac{2}{u}\right) \cdot u + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10599999874830246)
   (+ (* (log (fma (- 1.0 u) (- 1.0 (/ 2.0 v)) u)) v) 1.0)
   (+
    (* (- (+ (+ (/ 2.0 v) 2.0) (/ 1.3333333333333333 (* v v))) (/ 2.0 u)) u)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10599999874830246f) {
		tmp = (logf(fmaf((1.0f - u), (1.0f - (2.0f / v)), u)) * v) + 1.0f;
	} else {
		tmp = (((((2.0f / v) + 2.0f) + (1.3333333333333333f / (v * v))) - (2.0f / u)) * u) + 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10599999874830246))
		tmp = Float32(Float32(log(fma(Float32(Float32(1.0) - u), Float32(Float32(1.0) - Float32(Float32(2.0) / v)), u)) * v) + Float32(1.0));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(2.0) / v) + Float32(2.0)) + Float32(Float32(1.3333333333333333) / Float32(v * v))) - Float32(Float32(2.0) / u)) * u) + Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.105999999

   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 inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate--r- N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. div-sub N/A

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

      14. lower--.f32 N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f32 44.2

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

   5. Applied rewrites 44.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 44.9

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

   7. Applied rewrites 48.2%

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

   8. Taylor expanded in v around inf

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

      1. lower--.f32 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f32 47.1

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

   10. Applied rewrites 50.0%

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

   if 0.105999999 < v

   1. Initial program 94.8%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 42.8%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 63.2%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10599999874830246:\\ \;\;\;\;\log \left(\mathsf{fma}\left(1 - u, 1 - \frac{2}{v}, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{2}{v} + 2\right) + \frac{1.3333333333333333}{v \cdot v}\right) - \frac{2}{u}\right) \cdot u + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{2}{v} + 2\right) + \frac{1.3333333333333333}{v \cdot v}\right) - \frac{2}{u}\right) \cdot u + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    (* (- (+ (+ (/ 2.0 v) 2.0) (/ 1.3333333333333333 (* v v))) (/ 2.0 u)) u)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = (((((2.0f / v) + 2.0f) + (1.3333333333333333f / (v * v))) - (2.0f / u)) * u) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (((((2.0e0 / v) + 2.0e0) + (1.3333333333333333e0 / (v * v))) - (2.0e0 / u)) * u) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(2.0) / v) + Float32(2.0)) + Float32(Float32(1.3333333333333333) / Float32(v * v))) - Float32(Float32(2.0) / u)) * u) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = (((((single(2.0) / v) + single(2.0)) + (single(1.3333333333333333) / (v * v))) - (single(2.0) / u)) * u) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.7%

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

   if 0.150000006 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 35.8

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 42.1%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 65.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{2}{v} + 2\right) + \frac{1.3333333333333333}{v \cdot v}\right) - \frac{2}{u}\right) \cdot u + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 87.0%

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

Alternative 10: 5.9% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.3%

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

Reproduce

herbie shell --seed 2024325 
(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))))))))