HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.4s

Alternatives: 21

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1 + v \cdot \log \left(\left(u + \frac{1}{e^{\frac{2}{v}}}\right) - \frac{u}{e^{\frac{2}{v}}}\right)} \]
6. 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\left(2 \cdot u + -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{1 - u}{1 - \frac{\mathsf{fma}\left(-2, v, -1.3333333333333333\right)}{\left(v \cdot v\right) \cdot v}} + u\right), v, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (-
    (+ (* 2.0 u) -1.0)
    (/
     (fma
      (/ (fma (* -24.0 (- 1.0 u)) (- 1.0 u) (fma -56.0 u 24.0)) v)
      0.16666666666666666
      (* (* (fma -4.0 (- 1.0 u) 4.0) (- 1.0 u)) -0.5))
     v))
   (fma
    (log
     (+
      (/ (- 1.0 u) (- 1.0 (/ (fma -2.0 v -1.3333333333333333) (* (* v v) v))))
      u))
    v
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = ((2.0f * u) + -1.0f) - (fmaf((fmaf((-24.0f * (1.0f - u)), (1.0f - u), fmaf(-56.0f, u, 24.0f)) / v), 0.16666666666666666f, ((fmaf(-4.0f, (1.0f - u), 4.0f) * (1.0f - u)) * -0.5f)) / v);
	} else {
		tmp = fmaf(logf((((1.0f - u) / (1.0f - (fmaf(-2.0f, v, -1.3333333333333333f) / ((v * v) * v)))) + u)), v, 1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(Float32(Float32(Float32(2.0) * u) + Float32(-1.0)) - Float32(fma(Float32(fma(Float32(Float32(-24.0) * Float32(Float32(1.0) - u)), Float32(Float32(1.0) - u), fma(Float32(-56.0), u, Float32(24.0))) / v), Float32(0.16666666666666666), Float32(Float32(fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0)) * Float32(Float32(1.0) - u)) * Float32(-0.5))) / v));
	else
		tmp = fma(log(Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) - Float32(fma(Float32(-2.0), v, Float32(-1.3333333333333333)) / Float32(Float32(v * v) * v)))) + u)), v, Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in v around -inf

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in u around 0

      \[\leadsto \left(-1 + 2 \cdot u\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, 24 + -56 \cdot u\right)}{v}, \frac{1}{6}, \frac{-1}{2} \cdot \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right)\right)}{v} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

       \[\leadsto \left(-1 + 2 \cdot u\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, -0.5 \cdot \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right)\right)}{v} \]

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

   1. Initial program 100.0%

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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 99.3%

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{\frac{\frac{-1.3333333333333333}{v} + -2}{v} - 2}{v}}}\right) \]
   8. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 99.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\left(2 \cdot u + -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{1 - u}{1 - \frac{\mathsf{fma}\left(-2, v, -1.3333333333333333\right)}{\left(v \cdot v\right) \cdot v}} + u\right), v, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\left(2 \cdot u + -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{1 - u}{1 - \frac{-1.3333333333333333}{\left(v \cdot v\right) \cdot v}} + u\right), v, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (-
    (+ (* 2.0 u) -1.0)
    (/
     (fma
      (/ (fma (* -24.0 (- 1.0 u)) (- 1.0 u) (fma -56.0 u 24.0)) v)
      0.16666666666666666
      (* (* (fma -4.0 (- 1.0 u) 4.0) (- 1.0 u)) -0.5))
     v))
   (fma
    (log (+ (/ (- 1.0 u) (- 1.0 (/ -1.3333333333333333 (* (* v v) v)))) u))
    v
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = ((2.0f * u) + -1.0f) - (fmaf((fmaf((-24.0f * (1.0f - u)), (1.0f - u), fmaf(-56.0f, u, 24.0f)) / v), 0.16666666666666666f, ((fmaf(-4.0f, (1.0f - u), 4.0f) * (1.0f - u)) * -0.5f)) / v);
	} else {
		tmp = fmaf(logf((((1.0f - u) / (1.0f - (-1.3333333333333333f / ((v * v) * v)))) + u)), v, 1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(Float32(Float32(Float32(2.0) * u) + Float32(-1.0)) - Float32(fma(Float32(fma(Float32(Float32(-24.0) * Float32(Float32(1.0) - u)), Float32(Float32(1.0) - u), fma(Float32(-56.0), u, Float32(24.0))) / v), Float32(0.16666666666666666), Float32(Float32(fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0)) * Float32(Float32(1.0) - u)) * Float32(-0.5))) / v));
	else
		tmp = fma(log(Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) - Float32(Float32(-1.3333333333333333) / Float32(Float32(v * v) * v)))) + u)), v, Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in v around -inf

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in u around 0

      \[\leadsto \left(-1 + 2 \cdot u\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, 24 + -56 \cdot u\right)}{v}, \frac{1}{6}, \frac{-1}{2} \cdot \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right)\right)}{v} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

       \[\leadsto \left(-1 + 2 \cdot u\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, -0.5 \cdot \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right)\right)}{v} \]

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

   1. Initial program 100.0%

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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 99.3%

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{\frac{\frac{-1.3333333333333333}{v} + -2}{v} - 2}{v}}}\right) \]
   8. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 99.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\left(2 \cdot u + -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{1 - u}{1 - \frac{-1.3333333333333333}{\left(v \cdot v\right) \cdot v}} + u\right), v, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\left(2 \cdot u + -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{1 - u}{\frac{2}{v} + 1} + u\right), v, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (-
    (+ (* 2.0 u) -1.0)
    (/
     (fma
      (/ (fma (* -24.0 (- 1.0 u)) (- 1.0 u) (fma -56.0 u 24.0)) v)
      0.16666666666666666
      (* (* (fma -4.0 (- 1.0 u) 4.0) (- 1.0 u)) -0.5))
     v))
   (fma (log (+ (/ (- 1.0 u) (+ (/ 2.0 v) 1.0)) u)) v 1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = ((2.0f * u) + -1.0f) - (fmaf((fmaf((-24.0f * (1.0f - u)), (1.0f - u), fmaf(-56.0f, u, 24.0f)) / v), 0.16666666666666666f, ((fmaf(-4.0f, (1.0f - u), 4.0f) * (1.0f - u)) * -0.5f)) / v);
	} else {
		tmp = fmaf(logf((((1.0f - u) / ((2.0f / v) + 1.0f)) + u)), v, 1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(Float32(Float32(Float32(2.0) * u) + Float32(-1.0)) - Float32(fma(Float32(fma(Float32(Float32(-24.0) * Float32(Float32(1.0) - u)), Float32(Float32(1.0) - u), fma(Float32(-56.0), u, Float32(24.0))) / v), Float32(0.16666666666666666), Float32(Float32(fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0)) * Float32(Float32(1.0) - u)) * Float32(-0.5))) / v));
	else
		tmp = fma(log(Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(Float32(2.0) / v) + Float32(1.0))) + u)), v, Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in v around -inf

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in u around 0

      \[\leadsto \left(-1 + 2 \cdot u\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, 24 + -56 \cdot u\right)}{v}, \frac{1}{6}, \frac{-1}{2} \cdot \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right)\right)}{v} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

       \[\leadsto \left(-1 + 2 \cdot u\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, -0.5 \cdot \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right)\right)}{v} \]

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

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing
   3. 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 0

      \[\leadsto \color{blue}{1 + v \cdot \log \left(\left(u + \frac{1}{e^{\frac{2}{v}}}\right) - \frac{u}{e^{\frac{2}{v}}}\right)} \]
   6. 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. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 96.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\left(2 \cdot u + -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{1 - u}{\frac{2}{v} + 1} + u\right), v, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\left(2 \cdot u + -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (-
    (+ (* 2.0 u) -1.0)
    (/
     (fma
      (/ (fma (* -24.0 (- 1.0 u)) (- 1.0 u) (fma -56.0 u 24.0)) v)
      0.16666666666666666
      (* (* (fma -4.0 (- 1.0 u) 4.0) (- 1.0 u)) -0.5))
     v))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = ((2.0f * u) + -1.0f) - (fmaf((fmaf((-24.0f * (1.0f - u)), (1.0f - u), fmaf(-56.0f, u, 24.0f)) / v), 0.16666666666666666f, ((fmaf(-4.0f, (1.0f - u), 4.0f) * (1.0f - u)) * -0.5f)) / v);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(Float32(Float32(Float32(2.0) * u) + Float32(-1.0)) - Float32(fma(Float32(fma(Float32(Float32(-24.0) * Float32(Float32(1.0) - u)), Float32(Float32(1.0) - u), fma(Float32(-56.0), u, Float32(24.0))) / v), Float32(0.16666666666666666), Float32(Float32(fma(Float32(-4.0), Float32(Float32(1.0) - u), Float32(4.0)) * Float32(Float32(1.0) - u)) * Float32(-0.5))) / v));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in v around -inf

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in u around 0

      \[\leadsto \left(-1 + 2 \cdot u\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, 24 + -56 \cdot u\right)}{v}, \frac{1}{6}, \frac{-1}{2} \cdot \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right)\right)}{v} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

       \[\leadsto \left(-1 + 2 \cdot u\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, -0.5 \cdot \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right)\right)}{v} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v \leq -1:\\ \;\;\;\;\left(2 \cdot u + -1\right) - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-24 \cdot \left(1 - u\right), 1 - u, \mathsf{fma}\left(-56, u, 24\right)\right)}{v}, 0.16666666666666666, \left(\mathsf{fma}\left(-4, 1 - u, 4\right) \cdot \left(1 - u\right)\right) \cdot -0.5\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in v around -inf

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 73.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.9%

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

Alternative 7: 90.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 72.6%

      \[\leadsto \left(2 \cdot u - \frac{-2 \cdot u - \frac{\mathsf{fma}\left(0.6666666666666666, \frac{u}{v}, 1.3333333333333333 \cdot u\right)}{v}}{v}\right) - \color{blue}{1} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.9%

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

Alternative 8: 90.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 71.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.8%

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

Alternative 9: 90.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 92.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 71.4%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 71.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.8%

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

Alternative 10: 90.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-0.4000000059604645))
		tmp = fma(Float32(Float32(fma(Float32(-4.0), u, Float32(4.0)) * u) / v), Float32(0.5), fma(Float32(Float32(1.0) - u), Float32(-2.0), Float32(1.0)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.4%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)} \]
   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. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower--.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 64.7%

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

   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 93.6%

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

4. Final simplification 91.5%

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

Alternative 11: 90.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + \frac{1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)} \]
   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. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower--.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 66.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.4%

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

Alternative 12: 90.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 92.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 65.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.4%

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

Alternative 13: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.0%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 56.4

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

   5. Applied rewrites 56.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 90.8%

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

Alternative 14: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 92.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 91.2

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

   4. Applied rewrites 91.2%

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

   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 56.4

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

   7. Applied rewrites 56.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 90.8%

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

Alternative 15: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.5%

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

Alternative 16: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u around 0

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-exp.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 96.5

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

5. Applied rewrites 96.5%

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

6. Final simplification 96.5%

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

Alternative 17: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1 + v \cdot \log \left(\left(u + \frac{1}{e^{\frac{2}{v}}}\right) - \frac{u}{e^{\frac{2}{v}}}\right)} \]
6. 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 96.5%

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

11. Final simplification 96.5%

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

Alternative 18: 95.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 95.9%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{\frac{\frac{-1.3333333333333333}{v} + -2}{v} - 2}{v}}}\right) \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 95.9

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

9. Applied rewrites 95.9%

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

Alternative 19: 95.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 95.9%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{\frac{\frac{-1.3333333333333333}{v} + -2}{v} - 2}{v}}}\right) \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 95.9

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

9. Applied rewrites 95.9%

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

10. Taylor expanded in v around 0

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

12. Applied rewrites 95.9%

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

Alternative 20: 86.5% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 87.0%

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

Alternative 21: 6.1% 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 99.4%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.2%

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

Reproduce

herbie shell --seed 2024236 
(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))))))))