HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.6s

Alternatives: 20

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.3

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

5. Applied rewrites 99.3%

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

Alternative 2: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \frac{\mathsf{fma}\left(v, \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.6666666666666665, 4\right), -1.3333333333333333\right)\right), \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right), -0.6666666666666666\right)\right)}{v \cdot v}}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5)
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (*
      u
      (/
       (fma
        v
        (fma
         v
         (fma u 2.0 -2.0)
         (fma u (fma u -2.6666666666666665 4.0) -1.3333333333333333))
        (fma u (fma u (fma u 4.0 -8.0) 4.666666666666667) -0.6666666666666666))
       (* v v)))
     v))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5f) {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - ((u * (fmaf(v, fmaf(v, fmaf(u, 2.0f, -2.0f), fmaf(u, fmaf(u, -2.6666666666666665f, 4.0f), -1.3333333333333333f)), fmaf(u, fmaf(u, fmaf(u, 4.0f, -8.0f), 4.666666666666667f), -0.6666666666666666f)) / (v * v))) / v);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(Float32(u * Float32(fma(v, fma(v, fma(u, Float32(2.0), Float32(-2.0)), fma(u, fma(u, Float32(-2.6666666666666665), Float32(4.0)), Float32(-1.3333333333333333))), fma(u, fma(u, fma(u, Float32(4.0), Float32(-8.0)), Float32(4.666666666666667)), Float32(-0.6666666666666666))) / Float32(v * v))) / v));
	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)))))) < -0.5

   1. Initial program 92.4%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(-1 \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right) + 4 \cdot \frac{u}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}{v} \]

   8. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]

   9. Taylor expanded in v around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \frac{\left(u \cdot \left(\frac{14}{3} + u \cdot \left(4 \cdot u - 8\right)\right) + v \cdot \left(\left(u \cdot \left(4 + \frac{-8}{3} \cdot u\right) + v \cdot \left(2 \cdot u - 2\right)\right) - \frac{4}{3}\right)\right) - \frac{2}{3}}{{v}^{2}}}{v} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.8%

       \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \frac{\mathsf{fma}\left(v, \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.6666666666666665, 4\right), -1.3333333333333333\right)\right), \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right), -0.6666666666666666\right)\right)}{v \cdot v}}{v} \]

   if -0.5 < (*.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 91.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \frac{\mathsf{fma}\left(v, \mathsf{fma}\left(v, \mathsf{fma}\left(u, 2, -2\right), \mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.6666666666666665, 4\right), -1.3333333333333333\right)\right), \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right), -0.6666666666666666\right)\right)}{v \cdot v}}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(0.16666666666666666, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 16, -24\right), 8\right)}{v \cdot v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5)
   (fma
    0.5
    (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
    (fma
     0.16666666666666666
     (/ (* u (fma u (fma u 16.0 -24.0) 8.0)) (* v v))
     (fma -2.0 (- 1.0 u) 1.0)))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5f) {
		tmp = fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), fmaf(0.16666666666666666f, ((u * fmaf(u, fmaf(u, 16.0f, -24.0f), 8.0f)) / (v * v)), fmaf(-2.0f, (1.0f - u), 1.0f)));
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5))
		tmp = fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), fma(Float32(0.16666666666666666), Float32(Float32(u * fma(u, fma(u, Float32(16.0), Float32(-24.0)), Float32(8.0))) / Float32(v * v)), fma(Float32(-2.0), Float32(Float32(1.0) - u), 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)))))) < -0.5

   1. Initial program 92.4%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + \left(\frac{1}{6} \cdot \frac{-16 \cdot {\left(1 - u\right)}^{3} + \left(-8 \cdot \left(1 - u\right) + 24 \cdot {\left(1 - u\right)}^{2}\right)}{{v}^{2}} + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right)} \]

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, -16, 24\right), \mathsf{fma}\left(-8, -u, -8\right)\right)}{v \cdot v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(\frac{1}{6}, \frac{u \cdot \left(8 + u \cdot \left(16 \cdot u - 24\right)\right)}{v \cdot v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 74.3%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(0.16666666666666666, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 16, -24\right), 8\right)}{v \cdot v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)\right) \]

   if -0.5 < (*.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 91.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(0.16666666666666666, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 16, -24\right), 8\right)}{v \cdot v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\mathsf{fma}\left(u, 2, -2\right) - \frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, 2.6666666666666665, -4\right), 1.3333333333333333\right)}{v}\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5)
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (*
      u
      (-
       (fma u 2.0 -2.0)
       (/ (fma u (fma u 2.6666666666666665 -4.0) 1.3333333333333333) v)))
     v))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5f) {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - ((u * (fmaf(u, 2.0f, -2.0f) - (fmaf(u, fmaf(u, 2.6666666666666665f, -4.0f), 1.3333333333333333f) / v))) / v);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(Float32(u * Float32(fma(u, Float32(2.0), Float32(-2.0)) - Float32(fma(u, fma(u, Float32(2.6666666666666665), Float32(-4.0)), Float32(1.3333333333333333)) / v))) / v));
	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)))))) < -0.5

   1. Initial program 92.4%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(-1 \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right) + 4 \cdot \frac{u}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}{v} \]

   8. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]

   9. Taylor expanded in v around -inf

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2\right)}{v} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.2%

       \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, 2.6666666666666665, -4\right), 1.3333333333333333\right)}{-v} + \mathsf{fma}\left(u, 2, -2\right)\right)}{v} \]

   if -0.5 < (*.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 91.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\mathsf{fma}\left(u, 2, -2\right) - \frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, 2.6666666666666665, -4\right), 1.3333333333333333\right)}{v}\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(u, \mathsf{fma}\left(\frac{1}{v}, 2 + \frac{1.3333333333333333}{v}, 2\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5)
   (fma u (fma (/ 1.0 v) (+ 2.0 (/ 1.3333333333333333 v)) 2.0) -1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5f) {
		tmp = fmaf(u, fmaf((1.0f / v), (2.0f + (1.3333333333333333f / v)), 2.0f), -1.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5))
		tmp = fma(u, fma(Float32(Float32(1.0) / v), Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)), 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)))))) < -0.5

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 65.3%

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

   if -0.5 < (*.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 91.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(u, \mathsf{fma}\left(\frac{1}{v}, 2 + \frac{1.3333333333333333}{v}, 2\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5))
		tmp = Float32(fma(u, fma(Float32(Float32(-2.0) / v), Float32(u + Float32(-1.0)), 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)))))) < -0.5

   1. Initial program 92.4%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower--.f32 N/A

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

      13. lower-fma.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lower-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 64.6%

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

   if -0.5 < (*.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 91.0%

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

4. Final simplification 88.5%

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

Alternative 7: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.2%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 44.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.8%

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

4. Final simplification 86.6%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (fma v (log (* (expm1 (/ -2.0 v)) (- u))) 1.0)
   (-
    (fma u 2.0 -1.0)
    (/
     (*
      u
      (fma
       u
       (+
        (fma
         u
         (+ (fma 4.0 (/ u (* v v)) (/ -2.6666666666666665 v)) (/ -8.0 (* v v)))
         (/ 4.666666666666667 (* v v)))
        (+ 2.0 (/ 4.0 v)))
       (- -2.0 (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = fmaf(v, logf((expm1f((-2.0f / v)) * -u)), 1.0f);
	} else {
		tmp = fmaf(u, 2.0f, -1.0f) - ((u * fmaf(u, (fmaf(u, (fmaf(4.0f, (u / (v * v)), (-2.6666666666666665f / v)) + (-8.0f / (v * v))), (4.666666666666667f / (v * v))) + (2.0f + (4.0f / v))), (-2.0f - ((1.3333333333333333f + (0.6666666666666666f / v)) / v)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = fma(v, log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u))), Float32(1.0));
	else
		tmp = Float32(fma(u, Float32(2.0), Float32(-1.0)) - Float32(Float32(u * fma(u, Float32(fma(u, Float32(fma(Float32(4.0), Float32(u / Float32(v * v)), Float32(Float32(-2.6666666666666665) / v)) + Float32(Float32(-8.0) / Float32(v * v))), Float32(Float32(4.666666666666667) / Float32(v * v))) + Float32(Float32(2.0) + Float32(Float32(4.0) / v))), Float32(Float32(-2.0) - Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v)))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 99.2%

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

   if 0.400000006 < v

   1. Initial program 92.2%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(-1 \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right) + 4 \cdot \frac{u}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}{v} \]

   8. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]

   9. Taylor expanded in u around 0

      \[\leadsto \left(2 \cdot u - 1\right) - \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{\frac{-8}{3}}{v}\right) + \frac{-8}{v \cdot v}, \frac{\frac{14}{3}}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{\frac{4}{3} + \frac{\frac{2}{3}}{v}}{v}\right)}}{v} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.4%

       \[\leadsto \mathsf{fma}\left(u, 2, -1\right) - \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}}{v} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 95.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return fmaf(v, logf(fmaf((1.0f / (1.0f - ((-2.0f + ((-2.0f + (-1.3333333333333333f / v)) / v)) / v))), (1.0f - u), u)), 1.0f);
}

Julia

function code(u, v)
	return fma(v, log(fma(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)) / v))), Float32(Float32(1.0) - u), u)), Float32(1.0))
end

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.3

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in v around -inf

   \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\frac{1}{1 + -1 \cdot \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}, 1 - u, u\right)\right), 1\right) \]
9. Step-by-step derivation

10. Applied rewrites 93.8%

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

11. Final simplification 93.8%

    \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\frac{1}{1 - \frac{-2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}}{v}}, 1 - u, u\right)\right), 1\right) \]
12. Add Preprocessing

Alternative 10: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\frac{1}{\left(\frac{2}{v} + \frac{2}{v \cdot v}\right) + 1}, 1 - u, u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (fma
    v
    (log (fma (/ 1.0 (+ (+ (/ 2.0 v) (/ 2.0 (* v v))) 1.0)) (- 1.0 u) u))
    1.0)
   (-
    (fma u 2.0 -1.0)
    (/
     (*
      u
      (fma
       u
       (+
        (fma
         u
         (+ (fma 4.0 (/ u (* v v)) (/ -2.6666666666666665 v)) (/ -8.0 (* v v)))
         (/ 4.666666666666667 (* v v)))
        (+ 2.0 (/ 4.0 v)))
       (- -2.0 (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = fmaf(v, logf(fmaf((1.0f / (((2.0f / v) + (2.0f / (v * v))) + 1.0f)), (1.0f - u), u)), 1.0f);
	} else {
		tmp = fmaf(u, 2.0f, -1.0f) - ((u * fmaf(u, (fmaf(u, (fmaf(4.0f, (u / (v * v)), (-2.6666666666666665f / v)) + (-8.0f / (v * v))), (4.666666666666667f / (v * v))) + (2.0f + (4.0f / v))), (-2.0f - ((1.3333333333333333f + (0.6666666666666666f / v)) / v)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = fma(v, log(fma(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(2.0) / v) + Float32(Float32(2.0) / Float32(v * v))) + Float32(1.0))), Float32(Float32(1.0) - u), u)), Float32(1.0));
	else
		tmp = Float32(fma(u, Float32(2.0), Float32(-1.0)) - Float32(Float32(u * fma(u, Float32(fma(u, Float32(fma(Float32(4.0), Float32(u / Float32(v * v)), Float32(Float32(-2.6666666666666665) / v)) + Float32(Float32(-8.0) / Float32(v * v))), Float32(Float32(4.666666666666667) / Float32(v * v))) + Float32(Float32(2.0) + Float32(Float32(4.0) / v))), Float32(Float32(-2.0) - Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v)))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 96.4%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(-1 \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right) + 4 \cdot \frac{u}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}{v} \]

   8. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]

   9. Taylor expanded in u around 0

      \[\leadsto \left(2 \cdot u - 1\right) - \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{\frac{-8}{3}}{v}\right) + \frac{-8}{v \cdot v}, \frac{\frac{14}{3}}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{\frac{4}{3} + \frac{\frac{2}{3}}{v}}{v}\right)}}{v} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.2%

       \[\leadsto \mathsf{fma}\left(u, 2, -1\right) - \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\frac{1}{\left(\frac{2}{v} + \frac{2}{v \cdot v}\right) + 1}, 1 - u, u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\frac{1}{\frac{2}{v} + 1}, 1 - u, u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (fma v (log (fma (/ 1.0 (+ (/ 2.0 v) 1.0)) (- 1.0 u) u)) 1.0)
   (-
    (fma u 2.0 -1.0)
    (/
     (*
      u
      (fma
       u
       (+
        (fma
         u
         (+ (fma 4.0 (/ u (* v v)) (/ -2.6666666666666665 v)) (/ -8.0 (* v v)))
         (/ 4.666666666666667 (* v v)))
        (+ 2.0 (/ 4.0 v)))
       (- -2.0 (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = fmaf(v, logf(fmaf((1.0f / ((2.0f / v) + 1.0f)), (1.0f - u), u)), 1.0f);
	} else {
		tmp = fmaf(u, 2.0f, -1.0f) - ((u * fmaf(u, (fmaf(u, (fmaf(4.0f, (u / (v * v)), (-2.6666666666666665f / v)) + (-8.0f / (v * v))), (4.666666666666667f / (v * v))) + (2.0f + (4.0f / v))), (-2.0f - ((1.3333333333333333f + (0.6666666666666666f / v)) / v)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = fma(v, log(fma(Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / v) + Float32(1.0))), Float32(Float32(1.0) - u), u)), Float32(1.0));
	else
		tmp = Float32(fma(u, Float32(2.0), Float32(-1.0)) - Float32(Float32(u * fma(u, Float32(fma(u, Float32(fma(Float32(4.0), Float32(u / Float32(v * v)), Float32(Float32(-2.6666666666666665) / v)) + Float32(Float32(-8.0) / Float32(v * v))), Float32(Float32(4.666666666666667) / Float32(v * v))) + Float32(Float32(2.0) + Float32(Float32(4.0) / v))), Float32(Float32(-2.0) - Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v)))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 94.4%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(-1 \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right) + 4 \cdot \frac{u}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}{v} \]

   8. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]

   9. Taylor expanded in u around 0

      \[\leadsto \left(2 \cdot u - 1\right) - \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{\frac{-8}{3}}{v}\right) + \frac{-8}{v \cdot v}, \frac{\frac{14}{3}}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{\frac{4}{3} + \frac{\frac{2}{3}}{v}}{v}\right)}}{v} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.2%

       \[\leadsto \mathsf{fma}\left(u, 2, -1\right) - \frac{\color{blue}{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\frac{1}{\frac{2}{v} + 1}, 1 - u, u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\frac{1}{\frac{2}{v} + 1}, 1 - u, u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \left(2 + \frac{4}{v}\right) + \frac{\mathsf{fma}\left(v, u \cdot -2.6666666666666665, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right)\right)}{v \cdot v}, -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (fma v (log (fma (/ 1.0 (+ (/ 2.0 v) 1.0)) (- 1.0 u) u)) 1.0)
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (*
      u
      (fma
       u
       (+
        (+ 2.0 (/ 4.0 v))
        (/
         (fma
          v
          (* u -2.6666666666666665)
          (fma u (fma u 4.0 -8.0) 4.666666666666667))
         (* v v)))
       (- -2.0 (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = fmaf(v, logf(fmaf((1.0f / ((2.0f / v) + 1.0f)), (1.0f - u), u)), 1.0f);
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - ((u * fmaf(u, ((2.0f + (4.0f / v)) + (fmaf(v, (u * -2.6666666666666665f), fmaf(u, fmaf(u, 4.0f, -8.0f), 4.666666666666667f)) / (v * v))), (-2.0f - ((1.3333333333333333f + (0.6666666666666666f / v)) / v)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = fma(v, log(fma(Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / v) + Float32(1.0))), Float32(Float32(1.0) - u), u)), Float32(1.0));
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(Float32(u * fma(u, Float32(Float32(Float32(2.0) + Float32(Float32(4.0) / v)) + Float32(fma(v, Float32(u * Float32(-2.6666666666666665)), fma(u, fma(u, Float32(4.0), Float32(-8.0)), Float32(4.666666666666667))) / Float32(v * v))), Float32(Float32(-2.0) - Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v)))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 N/A

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

      16. lower--.f32 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 94.4%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(-1 \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right) + 4 \cdot \frac{u}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}{v} \]

   8. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]

   9. Taylor expanded in v around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \frac{\frac{14}{3} + \left(\frac{-8}{3} \cdot \left(u \cdot v\right) + u \cdot \left(4 \cdot u - 8\right)\right)}{{v}^{2}} + \left(2 + \frac{4}{v}\right), -2 - \frac{\frac{4}{3} + \frac{\frac{2}{3}}{v}}{v}\right)}{v} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.8%

       \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(v, u \cdot -2.6666666666666665, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right)\right)}{v \cdot v} + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\frac{1}{\frac{2}{v} + 1}, 1 - u, u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \left(2 + \frac{4}{v}\right) + \frac{\mathsf{fma}\left(v, u \cdot -2.6666666666666665, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right)\right)}{v \cdot v}, -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \left(2 + \frac{4}{v}\right) + \frac{\mathsf{fma}\left(v, u \cdot -2.6666666666666665, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right)\right)}{v \cdot v}, -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (*
      u
      (fma
       u
       (+
        (+ 2.0 (/ 4.0 v))
        (/
         (fma
          v
          (* u -2.6666666666666665)
          (fma u (fma u 4.0 -8.0) 4.666666666666667))
         (* v v)))
       (- -2.0 (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - ((u * fmaf(u, ((2.0f + (4.0f / v)) + (fmaf(v, (u * -2.6666666666666665f), fmaf(u, fmaf(u, 4.0f, -8.0f), 4.666666666666667f)) / (v * v))), (-2.0f - ((1.3333333333333333f + (0.6666666666666666f / v)) / v)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(Float32(u * fma(u, Float32(Float32(Float32(2.0) + Float32(Float32(4.0) / v)) + Float32(fma(v, Float32(u * Float32(-2.6666666666666665)), fma(u, fma(u, Float32(4.0), Float32(-8.0)), Float32(4.666666666666667))) / Float32(v * v))), Float32(Float32(-2.0) - Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v)))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.0%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(-1 \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right) + 4 \cdot \frac{u}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}{v} \]

   8. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]

   9. Taylor expanded in v around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \frac{\frac{14}{3} + \left(\frac{-8}{3} \cdot \left(u \cdot v\right) + u \cdot \left(4 \cdot u - 8\right)\right)}{{v}^{2}} + \left(2 + \frac{4}{v}\right), -2 - \frac{\frac{4}{3} + \frac{\frac{2}{3}}{v}}{v}\right)}{v} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.8%

       \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(v, u \cdot -2.6666666666666665, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right)\right)}{v \cdot v} + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \left(2 + \frac{4}{v}\right) + \frac{\mathsf{fma}\left(v, u \cdot -2.6666666666666665, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 4, -8\right), 4.666666666666667\right)\right)}{v \cdot v}, -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \left(\frac{2}{v} + \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v \cdot v}\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (fma
    u
    (+
     2.0
     (+ (/ 2.0 v) (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) (* v v))))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, (2.0f + ((2.0f / v) + ((1.3333333333333333f + (0.6666666666666666f / v)) / (v * v)))), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = fma(u, Float32(Float32(2.0) + Float32(Float32(Float32(2.0) / v) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / Float32(v * v)))), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.0%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{\mathsf{fma}\left(0.041666666666666664, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -112 + \left(1 - u\right) \cdot 192, \mathsf{fma}\left(1 - u, 16, -96 \cdot {\left(1 - u\right)}^{4}\right)\right)}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot -0.16666666666666666\right)}{-v}\right)}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \left(\left(-1 \cdot \frac{\frac{4}{3} + \frac{2}{3} \cdot \frac{1}{v}}{v} + u \cdot \left(2 + \left(4 \cdot \frac{1}{v} + \left(\frac{14}{3} \cdot \frac{1}{{v}^{2}} + u \cdot \left(-1 \cdot \left(\frac{8}{3} \cdot \frac{1}{v} + 8 \cdot \frac{1}{{v}^{2}}\right) + 4 \cdot \frac{u}{{v}^{2}}\right)\right)\right)\right)\right) - 2\right)}{v} \]

   8. Applied rewrites 79.8%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(4, \frac{u}{v \cdot v}, \frac{-2.6666666666666665}{v}\right) + \frac{-8}{v \cdot v}, \frac{4.666666666666667}{v \cdot v}\right) + \left(2 + \frac{4}{v}\right), -2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v} \]

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 66.8%

       \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 + \left(\frac{2}{v} + \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v \cdot v}\right)}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 91.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{2 + \frac{1.3333333333333333}{v}}{v} - -2}{v}, v \cdot u, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (fma (/ (- (/ (+ 2.0 (/ 1.3333333333333333 v)) v) -2.0) v) (* v u) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(((((2.0f + (1.3333333333333333f / v)) / v) - -2.0f) / v), (v * u), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) - Float32(-2.0)) / v), Float32(v * u), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.0%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 65.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{2 + \frac{1.3333333333333333}{v}}{v} - -2}{v}, v \cdot u, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 90.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{2}{v}, -2\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896) 1.0 (+ (fma u (+ 2.0 (/ 2.0 v)) -2.0) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, (2.0f + (2.0f / v)), -2.0f) + 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(u, Float32(Float32(2.0) + Float32(Float32(2.0) / v)), Float32(-2.0)) + Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.0%

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

   if 0.300000012 < v

   1. Initial program 92.4%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower--.f32 N/A

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

      13. lower-fma.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lower-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 62.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{2}{v}, -2\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 90.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{2}{v}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896) 1.0 (fma u (+ 2.0 (/ 2.0 v)) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, (2.0f + (2.0f / v)), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = fma(u, Float32(Float32(2.0) + Float32(Float32(2.0) / v)), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.0%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 62.4%

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

Alternative 18: 90.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896) 1.0 (* u (+ 2.0 (/ -1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = u * (2.0f + (-1.0f / u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0
    else
        tmp = u * (2.0e0 + ((-1.0e0) / u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(2.0) + Float32(Float32(-1.0) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0);
	else
		tmp = u * (single(2.0) + (single(-1.0) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.0%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in v around inf

      \[\leadsto 2 \cdot u - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.2%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 53.3%

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

Alternative 19: 90.4% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896) 1.0 (fma u 2.0 -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, 2.0f, -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = fma(u, Float32(2.0), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.0%

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

   if 0.300000012 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in v around inf

      \[\leadsto 2 \cdot u - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.2%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{2}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 5.8% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.9%

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

Reproduce

herbie shell --seed 2024237 
(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))))))))