HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.3s

Alternatives: 19

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

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(2.71828182845904523536) ^ (single(-2.0) / v))))));
end

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 2: 91.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(-0.5), Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), fma(Float32(8.0), Float32(-u), Float32(8.0))) * Float32(Float32(0.16666666666666666) / 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)))))) < -1

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 79.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

Alternative 3: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, -4, 4\right) \cdot -0.5, \frac{\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(-8, u, 8\right)\right)}{v \cdot 6}\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (-
    (fma (- 1.0 u) -2.0 1.0)
    (/
     (fma
      (- 1.0 u)
      (* (fma (- 1.0 u) -4.0 4.0) -0.5)
      (/
       (fma
        (- 1.0 u)
        (* (- 1.0 u) (fma (- 1.0 u) 16.0 -24.0))
        (fma -8.0 u 8.0))
       (* v 6.0)))
     v))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf((1.0f - u), -2.0f, 1.0f) - (fmaf((1.0f - u), (fmaf((1.0f - u), -4.0f, 4.0f) * -0.5f), (fmaf((1.0f - u), ((1.0f - u) * fmaf((1.0f - u), 16.0f, -24.0f)), fmaf(-8.0f, u, 8.0f)) / (v * 6.0f))) / v);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(fma(Float32(Float32(1.0) - u), Float32(-2.0), Float32(1.0)) - Float32(fma(Float32(Float32(1.0) - u), Float32(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)) * Float32(-0.5)), Float32(fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0))), fma(Float32(-8.0), u, Float32(8.0))) / Float32(v * Float32(6.0)))) / 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)))))) < -1

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   7. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - u, -2, 1\right) - \frac{\mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, -4, 4\right) \cdot -0.5, \frac{\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(-8, u, 8\right)\right)}{v \cdot 6}\right)}{v}} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

Alternative 4: 91.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   6. Taylor expanded in u around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

   8. Simplified 78.7%

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

   9. Taylor expanded in u around 0

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

   11. Simplified 79.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

Alternative 5: 91.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), ((u * fmaf(u, ((4.0f / v) + fmaf((u / v), -2.6666666666666665f, 2.0f)), (-2.0f + (-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(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(u * fma(u, Float32(Float32(Float32(4.0) / v) + fma(Float32(u / v), Float32(-2.6666666666666665), Float32(2.0))), Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)))) / Float32(-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)))))) < -1

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   6. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lower-fma.f32 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lower-+.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f32 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. neg-mul-1 N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f32 N/A

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

   8. Simplified 78.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

4. Final simplification 93.7%

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

Alternative 6: 91.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 78.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

Alternative 7: 91.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(-1.0) / u))) + Float32(Float32(u * Float32(Float32(fma(u, Float32(-4.0), Float32(1.3333333333333333)) / v) - fma(u, Float32(2.0), Float32(-2.0)))) / 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)))))) < -1

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   6. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{\color{blue}{u \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - \left(2 + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}}{v}\right)\right) \]

      2. associate--r+ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) - \frac{4}{3} \cdot \frac{1}{v}\right)}}{v}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      4. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \color{blue}{-2}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\mathsf{fma}\left(u, 2 + 4 \cdot \frac{1}{v}, -2\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      8. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, \color{blue}{2 + 4 \cdot \frac{1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4 \cdot 1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{\color{blue}{4}}{v}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      11. lower-/.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

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

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \frac{1}{v}}\right)}{v}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{-4}{3}} \cdot \frac{1}{v}\right)}{v}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{\frac{-4}{3} \cdot 1}{v}}\right)}{v}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \frac{\color{blue}{\frac{-4}{3}}}{v}\right)}{v}\right)\right) \]

      16. lower-/.f32 77.6

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

   8. Simplified 77.6%

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

   9. Taylor expanded in v around -inf

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

       1. associate-+r+ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \left(1 + -2 \cdot \left(1 - u\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - -1 \cdot \left(u \cdot \left(2 \cdot u - 2\right)\right)}{v}\right)\right)} \]

       3. unsub-neg N/A

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

       4. lower--.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 N/A

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

       7. lower--.f32 N/A

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

       8. lower-/.f32 N/A

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

   11. Simplified 78.2%

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

   12. Taylor expanded in u around inf

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

       1. lower-*.f32 N/A

          \[\leadsto \color{blue}{u \cdot \left(2 - \frac{1}{u}\right)} - \frac{\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)}{v} \]

       2. sub-neg N/A

          \[\leadsto u \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(\frac{1}{u}\right)\right)\right)} - \frac{\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)}{v} \]

       3. lower-+.f32 N/A

          \[\leadsto u \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(\frac{1}{u}\right)\right)\right)} - \frac{\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)}{v} \]

       4. distribute-neg-frac N/A

          \[\leadsto u \cdot \left(2 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{u}}\right) - \frac{\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)}{v} \]

       5. metadata-eval N/A

          \[\leadsto u \cdot \left(2 + \frac{\color{blue}{-1}}{u}\right) - \frac{\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)}{v} \]

       6. lower-/.f32 78.3

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

   14. Simplified 78.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

4. Final simplification 93.7%

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

Alternative 8: 91.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf(((fmaf(u, -4.0f, 1.3333333333333333f) / v) - fmaf(2.0f, u, -2.0f)), (u / v), fmaf((1.0f - u), -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(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(Float32(Float32(fma(u, Float32(-4.0), Float32(1.3333333333333333)) / v) - fma(Float32(2.0), u, Float32(-2.0))), Float32(u / v), fma(Float32(Float32(1.0) - u), Float32(-2.0), Float32(1.0)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   6. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{\color{blue}{u \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - \left(2 + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}}{v}\right)\right) \]

      2. associate--r+ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) - \frac{4}{3} \cdot \frac{1}{v}\right)}}{v}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      4. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \color{blue}{-2}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\mathsf{fma}\left(u, 2 + 4 \cdot \frac{1}{v}, -2\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      8. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, \color{blue}{2 + 4 \cdot \frac{1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4 \cdot 1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{\color{blue}{4}}{v}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      11. lower-/.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

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

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \frac{1}{v}}\right)}{v}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{-4}{3}} \cdot \frac{1}{v}\right)}{v}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{\frac{-4}{3} \cdot 1}{v}}\right)}{v}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \frac{\color{blue}{\frac{-4}{3}}}{v}\right)}{v}\right)\right) \]

      16. lower-/.f32 77.6

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

   8. Simplified 77.6%

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

   9. Taylor expanded in v around -inf

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

       1. associate-+r+ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \left(1 + -2 \cdot \left(1 - u\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - -1 \cdot \left(u \cdot \left(2 \cdot u - 2\right)\right)}{v}\right)\right)} \]

       3. unsub-neg N/A

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

       4. lower--.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 N/A

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

       7. lower--.f32 N/A

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

       8. lower-/.f32 N/A

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

   11. Simplified 78.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{-u \cdot \left(\frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)}{v}} \]
   12. Step-by-step derivation

       1. lift--.f32 N/A

          \[\leadsto \left(-2 \cdot \color{blue}{\left(1 - u\right)} + 1\right) - \frac{\mathsf{neg}\left(u \cdot \left(\frac{u \cdot -4 + \frac{4}{3}}{v} - \left(u \cdot 2 + -2\right)\right)\right)}{v} \]

       2. lift-fma.f32 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right)} - \frac{\mathsf{neg}\left(u \cdot \left(\frac{u \cdot -4 + \frac{4}{3}}{v} - \left(u \cdot 2 + -2\right)\right)\right)}{v} \]

       3. lift-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{neg}\left(u \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}}{v} - \left(u \cdot 2 + -2\right)\right)\right)}{v} \]

       4. lift-/.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{neg}\left(u \cdot \left(\color{blue}{\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v}} - \left(u \cdot 2 + -2\right)\right)\right)}{v} \]

       5. lift-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \color{blue}{\mathsf{fma}\left(u, 2, -2\right)}\right)\right)}{v} \]

       6. lift--.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{neg}\left(u \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)}\right)}{v} \]

       7. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{neg}\left(\color{blue}{u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)}\right)}{v} \]

       8. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)\right)\right)\right)}\right)}{v} \]

       9. lift-neg.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)\right)}\right)\right)\right)}{v} \]

       10. distribute-frac-neg N/A

           \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)\right)\right)}{v}\right)\right)} \]

       11. distribute-frac-neg2 N/A

           \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)\right)\right)}{\mathsf{neg}\left(v\right)}} \]

       12. frac-2neg N/A

           \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \color{blue}{\frac{\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)}{v}} \]

       13. lift-/.f32 N/A

           \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \color{blue}{\frac{\mathsf{neg}\left(u \cdot \left(\frac{\mathsf{fma}\left(u, -4, \frac{4}{3}\right)}{v} - \mathsf{fma}\left(u, 2, -2\right)\right)\right)}{v}} \]

   13. Applied egg-rr 78.2%

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

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

   1. Initial program 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. Simplified 94.5%

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

Alternative 9: 91.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(u * Float32(Float32(Float32(-1.3333333333333333) / v) + fma(u, Float32(2.0), Float32(-2.0)))) / Float32(-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)))))) < -1

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   6. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{\color{blue}{u \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - \left(2 + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}}{v}\right)\right) \]

      2. associate--r+ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) - \frac{4}{3} \cdot \frac{1}{v}\right)}}{v}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      4. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \color{blue}{-2}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\mathsf{fma}\left(u, 2 + 4 \cdot \frac{1}{v}, -2\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      8. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, \color{blue}{2 + 4 \cdot \frac{1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4 \cdot 1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{\color{blue}{4}}{v}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      11. lower-/.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

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

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \frac{1}{v}}\right)}{v}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{-4}{3}} \cdot \frac{1}{v}\right)}{v}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{\frac{-4}{3} \cdot 1}{v}}\right)}{v}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \frac{\color{blue}{\frac{-4}{3}}}{v}\right)}{v}\right)\right) \]

      16. lower-/.f32 77.6

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

   8. Simplified 77.6%

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

   9. Taylor expanded in v around inf

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

       1. sub-neg N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\left(2 \cdot u + \left(\mathsf{neg}\left(2\right)\right)\right)} + \frac{\frac{-4}{3}}{v}\right)}{v}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\left(\color{blue}{u \cdot 2} + \left(\mathsf{neg}\left(2\right)\right)\right) + \frac{\frac{-4}{3}}{v}\right)}{v}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\left(u \cdot 2 + \color{blue}{-2}\right) + \frac{\frac{-4}{3}}{v}\right)}{v}\right)\right) \]

       4. lower-fma.f32 76.7

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

   11. Simplified 76.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

4. Final simplification 93.6%

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

Alternative 10: 91.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) + Float32(Float32(u * Float32(Float32(Float32(1.3333333333333333) / v) - fma(u, Float32(2.0), Float32(-2.0)))) / 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)))))) < -1

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   6. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{\color{blue}{u \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - \left(2 + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}}{v}\right)\right) \]

      2. associate--r+ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) - \frac{4}{3} \cdot \frac{1}{v}\right)}}{v}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      4. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \color{blue}{-2}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\mathsf{fma}\left(u, 2 + 4 \cdot \frac{1}{v}, -2\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      8. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, \color{blue}{2 + 4 \cdot \frac{1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4 \cdot 1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{\color{blue}{4}}{v}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      11. lower-/.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

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

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \frac{1}{v}}\right)}{v}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{-4}{3}} \cdot \frac{1}{v}\right)}{v}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{\frac{-4}{3} \cdot 1}{v}}\right)}{v}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \frac{\color{blue}{\frac{-4}{3}}}{v}\right)}{v}\right)\right) \]

      16. lower-/.f32 77.6

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

   8. Simplified 77.6%

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

   9. Taylor expanded in v around -inf

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

       1. associate-+r+ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \left(1 + -2 \cdot \left(1 - u\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{u \cdot \left(\frac{4}{3} + -4 \cdot u\right)}{v} - -1 \cdot \left(u \cdot \left(2 \cdot u - 2\right)\right)}{v}\right)\right)} \]

       3. unsub-neg N/A

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

       4. lower--.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 N/A

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

       7. lower--.f32 N/A

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

       8. lower-/.f32 N/A

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

   11. Simplified 78.2%

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

   12. Taylor expanded in u around 0

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

   14. Simplified 77.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

4. Final simplification 93.7%

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

Alternative 11: 90.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.7%

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

   6. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{\color{blue}{u \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - \left(2 + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}}{v}\right)\right) \]

      2. associate--r+ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) - \frac{4}{3} \cdot \frac{1}{v}\right)}}{v}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      4. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \color{blue}{\left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) - 2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}}{v}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right) + \color{blue}{-2}\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\color{blue}{\mathsf{fma}\left(u, 2 + 4 \cdot \frac{1}{v}, -2\right)} + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      8. lower-+.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, \color{blue}{2 + 4 \cdot \frac{1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4 \cdot 1}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{\color{blue}{4}}{v}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

      11. lower-/.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \color{blue}{\frac{4}{v}}, -2\right) + \left(\mathsf{neg}\left(\frac{4}{3} \cdot \frac{1}{v}\right)\right)\right)}{v}\right)\right) \]

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

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \frac{1}{v}}\right)}{v}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{-4}{3}} \cdot \frac{1}{v}\right)}{v}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \color{blue}{\frac{\frac{-4}{3} \cdot 1}{v}}\right)}{v}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \mathsf{neg}\left(\frac{u \cdot \left(\mathsf{fma}\left(u, 2 + \frac{4}{v}, -2\right) + \frac{\color{blue}{\frac{-4}{3}}}{v}\right)}{v}\right)\right) \]

      16. lower-/.f32 77.6

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

   8. Simplified 77.6%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

          \[\leadsto \color{blue}{u \cdot \left(2 - -1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

       2. metadata-eval N/A

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

       3. lower-fma.f32 N/A

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

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

          \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}}, -1\right) \]

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

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

       7. lower-+.f32 N/A

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

       8. lower-/.f32 N/A

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

       9. lower-+.f32 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(u, 2 + \frac{2 + \frac{\color{blue}{\frac{4}{3}}}{v}}{v}, -1\right) \]

       12. lower-/.f32 74.8

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

   11. Simplified 74.8%

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

   12. Taylor expanded in v around 0

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

       1. lower-/.f32 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f32 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f32 74.8

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

   14. Simplified 74.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

Alternative 12: 90.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Simplified 72.4%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 71.3

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

   8. Simplified 71.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

Alternative 13: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 14: 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 99.6%

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

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)} \]
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 99.5

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

5. Simplified 99.5%

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

Alternative 15: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.6

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

4. Applied egg-rr 99.6%

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

   1. lift--.f32 N/A

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

   2. lift-E.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 99.6

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in u around 0

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

9. Simplified 97.1%

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

Alternative 16: 90.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Simplified 72.4%

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

      1. lift--.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift--.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f32 N/A

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

   7. Applied egg-rr 72.2%

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

   8. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 72.2

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

   10. Simplified 72.2%

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

4. Final simplification 93.4%

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

Alternative 17: 90.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.5%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 71.6

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

   8. Simplified 71.6%

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

Alternative 18: 86.6% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

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

5. Simplified 89.9%

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

Alternative 19: 6.0% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.5%

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

Reproduce

herbie shell --seed 2024207 
(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))))))))