HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 14.8s

Alternatives: 26

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\left(-1 + u \cdot 2\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(1 - u, 8, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right)}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(u + \frac{1 - u}{\frac{2}{v} + 1}\right), v, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (-
    (+ -1.0 (* u 2.0))
    (/
     (fma
      0.16666666666666666
      (/
       (fma
        (- 1.0 u)
        8.0
        (* (* (- 1.0 u) (- 1.0 u)) (fma (- 1.0 u) 16.0 -24.0)))
       v)
      (* -0.5 (* (- 1.0 u) (fma -4.0 (- 1.0 u) 4.0))))
     v))
   (fma (log (+ u (/ (- 1.0 u) (+ (/ 2.0 v) 1.0)))) v 1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.4000000059604645f) {
		tmp = (-1.0f + (u * 2.0f)) - (fmaf(0.16666666666666666f, (fmaf((1.0f - u), 8.0f, (((1.0f - u) * (1.0f - u)) * fmaf((1.0f - u), 16.0f, -24.0f))) / v), (-0.5f * ((1.0f - u) * fmaf(-4.0f, (1.0f - u), 4.0f)))) / v);
	} else {
		tmp = fmaf(logf((u + ((1.0f - u) / ((2.0f / v) + 1.0f)))), v, 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(-0.4000000059604645))
		tmp = Float32(Float32(Float32(-1.0) + Float32(u * Float32(2.0))) - Float32(fma(Float32(0.16666666666666666), Float32(fma(Float32(Float32(1.0) - u), Float32(8.0), Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)) * fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)))) / v), Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0))))) / v));
	else
		tmp = fma(log(Float32(u + Float32(Float32(Float32(1.0) - u) / Float32(Float32(Float32(2.0) / v) + Float32(1.0))))), v, Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.400000006

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 70.9%

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

   if -0.400000006 < (*.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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in v around inf

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

      1. lower-+.f32 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f32 98.0

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

   7. Applied rewrites 98.0%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 98.0

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

      6. lift-*.f32 N/A

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

      7. lift-/.f32 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f32 98.0

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

   9. Applied rewrites 98.0%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\left(-1 + u \cdot 2\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(1 - u, 8, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right)}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(u + \frac{1 - u}{\frac{2}{v} + 1}\right), v, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\left(-1 + u \cdot 2\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(1 - u, 8, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right)}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (-
    (+ -1.0 (* u 2.0))
    (/
     (fma
      0.16666666666666666
      (/
       (fma
        (- 1.0 u)
        8.0
        (* (* (- 1.0 u) (- 1.0 u)) (fma (- 1.0 u) 16.0 -24.0)))
       v)
      (* -0.5 (* (- 1.0 u) (fma -4.0 (- 1.0 u) 4.0))))
     v))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.4000000059604645f) {
		tmp = (-1.0f + (u * 2.0f)) - (fmaf(0.16666666666666666f, (fmaf((1.0f - u), 8.0f, (((1.0f - u) * (1.0f - u)) * fmaf((1.0f - u), 16.0f, -24.0f))) / v), (-0.5f * ((1.0f - u) * fmaf(-4.0f, (1.0f - u), 4.0f)))) / 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(-0.4000000059604645))
		tmp = Float32(Float32(Float32(-1.0) + Float32(u * Float32(2.0))) - Float32(fma(Float32(0.16666666666666666), Float32(fma(Float32(Float32(1.0) - u), Float32(8.0), Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)) * fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)))) / v), Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0))))) / 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)))))) < -0.400000006

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 70.9%

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

   if -0.400000006 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\left(-1 + u \cdot 2\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(1 - u, 8, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right)}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right)\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (*
      u
      (fma
       u
       (+ (/ 4.0 v) (fma -2.6666666666666665 (/ u v) 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)))))) <= -0.4000000059604645f) {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - ((u * fmaf(u, ((4.0f / v) + fmaf(-2.6666666666666665f, (u / v), 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(-0.4000000059604645))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(Float32(u * fma(u, Float32(Float32(Float32(4.0) / v) + fma(Float32(-2.6666666666666665), Float32(u / v), Float32(2.0))), Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)))) / v));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.400000006

   1. Initial program 95.4%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 70.5%

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

   if -0.400000006 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

Alternative 5: 91.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{0.16666666666666666}{v} \cdot \left(u \cdot \mathsf{fma}\left(u, 24, -8\right)\right)\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (fma
      (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
      -0.5
      (* (/ 0.16666666666666666 v) (* u (fma u 24.0 -8.0))))
     v))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.4000000059604645f) {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, ((0.16666666666666666f / v) * (u * fmaf(u, 24.0f, -8.0f)))) / 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(-0.4000000059604645))
		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(Float32(Float32(0.16666666666666666) / v) * Float32(u * fma(u, Float32(24.0), Float32(-8.0))))) / 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)))))) < -0.400000006

   1. Initial program 95.4%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 64.5%

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

   if -0.400000006 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{0.16666666666666666}{v} \cdot \left(u \cdot \mathsf{fma}\left(u, 24, -8\right)\right)\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{u \cdot 4}{v}, -1 + u \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (fma 0.5 (/ (* u 4.0) v) (+ -1.0 (* u 2.0)))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.4000000059604645f) {
		tmp = fmaf(0.5f, ((u * 4.0f) / v), (-1.0f + (u * 2.0f)));
	} 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(-0.4000000059604645))
		tmp = fma(Float32(0.5), Float32(Float32(u * Float32(4.0)) / v), Float32(Float32(-1.0) + Float32(u * Float32(2.0))));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.400000006

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

   8. Applied rewrites 60.8%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 57.7%

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

   if -0.400000006 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{u \cdot 4}{v}, -1 + u \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (+ -1.0 (* 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)))))) <= -0.4000000059604645f) {
		tmp = -1.0f + (u * (2.0f + (2.0f / v)));
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v)))))) <= (-0.4000000059604645e0)) then
        tmp = (-1.0e0) + (u * (2.0e0 + (2.0e0 / v)))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-0.4000000059604645))
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) / v))));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-0.4000000059604645))
		tmp = single(-1.0) + (u * (single(2.0) + (single(2.0) / v)));
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.400000006

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

   8. Applied rewrites 60.8%

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in u around 0

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

   13. Applied rewrites 57.7%

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

   if -0.400000006 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{2}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (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)))))) <= -0.4000000059604645f) {
		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(-0.4000000059604645))
		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)))))) < -0.400000006

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

   8. Applied rewrites 60.8%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 57.7%

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

   if -0.400000006 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

Alternative 9: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (fma -2.0 (- 1.0 u) 1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.4000000059604645f) {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-0.4000000059604645))
		tmp = fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.400000006

   1. Initial program 95.4%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 51.5

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

   5. Applied rewrites 51.5%

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

   if -0.400000006 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

Alternative 10: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(2, u, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -0.4000000059604645)
   (fma 2.0 u -1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -0.4000000059604645f) {
		tmp = fmaf(2.0f, u, -1.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-0.4000000059604645))
		tmp = fma(Float32(2.0), u, Float32(-1.0));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.400000006

   1. Initial program 95.4%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. metadata-eval 94.8

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f32 51.5

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

   7. Applied rewrites 51.5%

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

   if -0.400000006 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

Alternative 11: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

6. Final simplification 99.6%

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

Alternative 12: 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 99.7%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.6

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

5. Applied rewrites 99.6%

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

Alternative 13: 95.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in v around -inf

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

10. Applied rewrites 96.2%

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

11. Final simplification 96.2%

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

Alternative 14: 95.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 88.0%

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

8. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. associate--l+ N/A

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

   5. div-sub N/A

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

   6. lower-+.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. lower-exp.f32 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 99.6

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

10. Applied rewrites 99.6%

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

11. Taylor expanded in v around -inf

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

13. Applied rewrites 96.1%

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

14. Final simplification 96.1%

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

Alternative 15: 93.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 94.9

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

7. Applied rewrites 94.9%

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

8. Final simplification 94.9%

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

Alternative 16: 93.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in v around inf

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

10. Applied rewrites 94.9%

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

11. Final simplification 94.9%

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

Alternative 17: 93.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 88.0%

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

8. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. associate--l+ N/A

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

   5. div-sub N/A

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

   6. lower-+.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. lower-exp.f32 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 99.6

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

10. Applied rewrites 99.6%

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

11. Taylor expanded in v around inf

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

13. Applied rewrites 94.9%

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

14. Final simplification 94.9%

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

Alternative 18: 91.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, u, -1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{0.16666666666666666}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (-
    (fma 2.0 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))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(2.0f, 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);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(2.0), 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));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

   if 0.100000001 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 70.9%

      \[\leadsto \mathsf{fma}\left(2, u, -1\right) - \frac{\color{blue}{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{0.16666666666666666}{v}\right)}}{v} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 91.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{0.16666666666666666}{v} \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -16, 24\right), -8\right)\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (fma
      (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
      -0.5
      (* (/ 0.16666666666666666 v) (* u (fma u (fma u -16.0 24.0) -8.0))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, ((0.16666666666666666f / v) * (u * fmaf(u, fmaf(u, -16.0f, 24.0f), -8.0f)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		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(Float32(Float32(0.16666666666666666) / v) * Float32(u * fma(u, fma(u, Float32(-16.0), Float32(24.0)), Float32(-8.0))))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

   if 0.100000001 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 70.7%

      \[\leadsto \mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -16, 24\right), -8\right)\right) \cdot \frac{0.16666666666666666}{v}\right)}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{0.16666666666666666}{v} \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -16, 24\right), -8\right)\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 91.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{u \cdot \mathsf{fma}\left(u, 2 + \frac{4}{v}, -2 + \frac{-1.3333333333333333}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/ (* u (fma u (+ 2.0 (/ 4.0 v)) (+ -2.0 (/ -1.3333333333333333 v)))) v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - ((u * fmaf(u, (2.0f + (4.0f / v)), (-2.0f + (-1.3333333333333333f / v)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(Float32(u * fma(u, Float32(Float32(2.0) + Float32(Float32(4.0) / v)), Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

   if 0.100000001 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 64.3%

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

Alternative 21: 90.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{0.5}{v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (fma
    (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
    (/ 0.5 v)
    (fma -2.0 (- 1.0 u) 1.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), (0.5f / v), fmaf(-2.0f, (1.0f - u), 1.0f));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(Float32(0.5) / v), fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

   if 0.100000001 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)} \]
   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. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{0.5}{v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 90.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \frac{\mathsf{fma}\left(1 - u, -4, 4\right)}{v}, u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (fma 0.5 (* (- 1.0 u) (/ (fma (- 1.0 u) -4.0 4.0) v)) (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + fmaf(0.5f, ((1.0f - u) * (fmaf((1.0f - u), -4.0f, 4.0f) / v)), (u * 2.0f));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + fma(Float32(0.5), Float32(Float32(Float32(1.0) - u) * Float32(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)) / v)), Float32(u * Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

   if 0.100000001 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

   8. Applied rewrites 60.8%

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

   10. Applied rewrites 60.8%

       \[\leadsto \mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \frac{\mathsf{fma}\left(1 - u, -4, 4\right)}{v}, 2 \cdot u\right) + \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \frac{\mathsf{fma}\left(1 - u, -4, 4\right)}{v}, u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 90.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \frac{u \cdot 4}{v}, u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (fma 0.5 (* (- 1.0 u) (/ (* u 4.0) v)) (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + fmaf(0.5f, ((1.0f - u) * ((u * 4.0f) / v)), (u * 2.0f));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + fma(Float32(0.5), Float32(Float32(Float32(1.0) - u) * Float32(Float32(u * Float32(4.0)) / v)), Float32(u * Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

   if 0.100000001 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

   8. Applied rewrites 60.8%

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

   10. Applied rewrites 60.8%

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

   11. Taylor expanded in u around 0

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

   13. Applied rewrites 60.8%

       \[\leadsto \mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \frac{u \cdot 4}{v}, 2 \cdot u\right) + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \frac{u \cdot 4}{v}, u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 90.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \mathsf{fma}\left(-2, \frac{u}{v}, \frac{2}{v}\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (fma u (+ 2.0 (fma -2.0 (/ u v) (/ 2.0 v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, (2.0f + fmaf(-2.0f, (u / v), (2.0f / v))), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = fma(u, Float32(Float32(2.0) + fma(Float32(-2.0), Float32(u / v), Float32(Float32(2.0) / v))), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.5%

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

   if 0.100000001 < v

   1. Initial program 95.4%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

   8. Applied rewrites 60.8%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 60.8%

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

Alternative 25: 86.8% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 88.0%

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

Alternative 26: 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 99.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.9%

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

Reproduce

herbie shell --seed 2024221 
(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))))))))