HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.5s

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. log-lowering-log.f32 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. exp-lowering-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f32 N/A

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

   16. --lowering--.f32 99.4

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

5. Simplified 99.4%

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

Alternative 2: 90.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. accelerator-lowering-expm1.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in v around -inf

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. --lowering--.f32 N/A

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

   8. Simplified 74.6%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 90.5%

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

4. Final simplification 89.5%

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

Alternative 3: 90.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) + Float32(fma(u, Float32(2.0), Float32(-2.0)) - Float32(fma(u, Float32(Float32(-1.3333333333333333) / v), Float32(Float32(-2.0) * u)) / 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 91.9%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. accelerator-lowering-expm1.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in v around -inf

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

   8. Simplified 73.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 90.5%

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

4. Final simplification 89.4%

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

Alternative 4: 90.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 76.7%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 72.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 90.5%

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

4. Final simplification 89.4%

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

Alternative 5: 90.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(Float32(0.5), Float32(Float32(u * Float32(4.0)) / v), fma(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 91.9%

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

   3. Taylor expanded in v around inf

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

      1. /-lowering-/.f32 N/A

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

   5. Simplified 69.2%

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

   6. Taylor expanded in v around inf

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 69.0%

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

   9. Taylor expanded in u around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 69.7

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

   11. Simplified 69.7%

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

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 90.5%

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

4. Final simplification 89.2%

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

Alternative 6: 90.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

   3. Taylor expanded in v around inf

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

      1. /-lowering-/.f32 N/A

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

   5. Simplified 69.2%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-lowering-+.f32 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f32 69.7

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

   8. Simplified 69.7%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 90.5%

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

4. Final simplification 89.2%

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

Alternative 7: 95.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(log(Float32(u + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)) / v))))), 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. frac-2neg N/A

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

   2. distribute-frac-neg2 N/A

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

   3. exp-neg N/A

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

   4. /-lowering-/.f32 N/A

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

   5. exp-lowering-exp.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. metadata-eval 99.3

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

4. Applied egg-rr 99.3%

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

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. --lowering--.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. /-lowering-/.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. Simplified 95.2%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

9. Applied egg-rr 95.3%

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

Alternative 8: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(v, \log u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, -16, 24\right), \mathsf{fma}\left(-8, -u, -8\right)\right)}{v \cdot v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (fma v (log u) 1.0)
   (fma
    0.5
    (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
    (fma
     0.16666666666666666
     (/
      (fma
       (* (- 1.0 u) (- 1.0 u))
       (fma (- 1.0 u) -16.0 24.0)
       (fma -8.0 (- u) -8.0))
      (* v v))
     (fma -2.0 (- 1.0 u) 1.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = fmaf(v, logf(u), 1.0f);
	} else {
		tmp = fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), fmaf(0.16666666666666666f, (fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), -16.0f, 24.0f), fmaf(-8.0f, -u, -8.0f)) / (v * v)), fmaf(-2.0f, (1.0f - u), 1.0f)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = fma(v, log(u), Float32(1.0));
	else
		tmp = fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), fma(Float32(0.16666666666666666), Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(-16.0), Float32(24.0)), fma(Float32(-8.0), Float32(-u), Float32(-8.0))) / Float32(v * v)), fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. exp-neg N/A

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

      4. /-lowering-/.f32 N/A

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

      5. exp-lowering-exp.f32 N/A

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

      6. /-lowering-/.f32 N/A

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

      7. 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 egg-rr 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. --lowering--.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. /-lowering-/.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. Simplified 98.1%

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

   8. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1 + v \cdot \log u} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{v \cdot \log u + 1} \]

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. log-lowering-log.f32 99.6

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

   10. Simplified 99.6%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 75.6%

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

Alternative 9: 91.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, -16, 24\right), \mathsf{fma}\left(-8, -u, -8\right)\right)}{v \cdot v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (fma
    0.5
    (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
    (fma
     0.16666666666666666
     (/
      (fma
       (* (- 1.0 u) (- 1.0 u))
       (fma (- 1.0 u) -16.0 24.0)
       (fma -8.0 (- u) -8.0))
      (* v v))
     (fma -2.0 (- 1.0 u) 1.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), fmaf(0.16666666666666666f, (fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), -16.0f, 24.0f), fmaf(-8.0f, -u, -8.0f)) / (v * v)), fmaf(-2.0f, (1.0f - u), 1.0f)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), fma(Float32(0.16666666666666666), Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(-16.0), Float32(24.0)), fma(Float32(-8.0), Float32(-u), Float32(-8.0))) / Float32(v * v)), fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 75.6%

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

Alternative 10: 91.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - u, 0.5 \cdot \frac{\mathsf{fma}\left(1 - u, -4, 4\right)}{v}, 1 + \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(u, -8, 8\right)\right)}{v \cdot v}, \mathsf{fma}\left(u, 2, -2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (fma
    (- 1.0 u)
    (* 0.5 (/ (fma (- 1.0 u) -4.0 4.0) v))
    (+
     1.0
     (fma
      -0.16666666666666666
      (/
       (fma
        (- 1.0 u)
        (* (- 1.0 u) (fma (- 1.0 u) 16.0 -24.0))
        (fma u -8.0 8.0))
       (* v v))
      (fma u 2.0 -2.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf((1.0f - u), (0.5f * (fmaf((1.0f - u), -4.0f, 4.0f) / v)), (1.0f + fmaf(-0.16666666666666666f, (fmaf((1.0f - u), ((1.0f - u) * fmaf((1.0f - u), 16.0f, -24.0f)), fmaf(u, -8.0f, 8.0f)) / (v * v)), fmaf(u, 2.0f, -2.0f))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(Float32(1.0) - u), Float32(Float32(0.5) * Float32(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)) / v)), Float32(Float32(1.0) + fma(Float32(-0.16666666666666666), Float32(fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0))), fma(u, Float32(-8.0), Float32(8.0))) / Float32(v * v)), fma(u, Float32(2.0), Float32(-2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 75.2%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 75.2%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

   10. Applied egg-rr 75.4%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - u, 0.5 \cdot \frac{\mathsf{fma}\left(1 - u, -4, 4\right)}{v}, 1 + \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(u, -8, 8\right)\right)}{v \cdot v}, \mathsf{fma}\left(u, 2, -2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{0.16666666666666666}{v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (fma
      (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
      -0.5
      (*
       (fma
        (* (- 1.0 u) (- 1.0 u))
        (fma (- 1.0 u) 16.0 -24.0)
        (fma 8.0 (- u) 8.0))
       (/ 0.16666666666666666 v)))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, (fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), 16.0f, -24.0f), fmaf(8.0f, -u, 8.0f)) * (0.16666666666666666f / v))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(-0.5), Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), fma(Float32(8.0), Float32(-u), Float32(8.0))) * Float32(Float32(0.16666666666666666) / v))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 75.3%

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

Alternative 12: 91.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(1 - u, -4, 4\right), v \cdot -2\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (/
     (fma
      v
      (* (- 1.0 u) (fma 0.5 (fma (- 1.0 u) -4.0 4.0) (* v -2.0)))
      (fma
       -0.16666666666666666
       (fma (* (- 1.0 u) (- 1.0 u)) (fma (- 1.0 u) 16.0 -24.0) (* u -8.0))
       -1.3333333333333333))
     (* v v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (fmaf(v, ((1.0f - u) * fmaf(0.5f, fmaf((1.0f - u), -4.0f, 4.0f), (v * -2.0f))), fmaf(-0.16666666666666666f, fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), 16.0f, -24.0f), (u * -8.0f)), -1.3333333333333333f)) / (v * v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(fma(v, Float32(Float32(Float32(1.0) - u) * fma(Float32(0.5), fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)), Float32(v * Float32(-2.0)))), fma(Float32(-0.16666666666666666), fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), Float32(u * Float32(-8.0))), Float32(-1.3333333333333333))) / Float32(v * v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 75.2%

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

   6. Taylor expanded in v around 0

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

      1. /-lowering-/.f32 N/A

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

   8. Simplified 75.3%

      \[\leadsto 1 + \color{blue}{\frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(1 - u, -4, 4\right), -2 \cdot v\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{v \cdot v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(1 - u, -4, 4\right), v \cdot -2\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), u \cdot -8\right), -1.3333333333333333\right)\right)}{v \cdot v}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(-0.16666666666666666, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -16, 24\right), -8\right)}{v \cdot v}, \mathsf{fma}\left(u, 2, -2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (fma
     0.5
     (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
     (fma
      -0.16666666666666666
      (/ (* u (fma u (fma u -16.0 24.0) -8.0)) (* v v))
      (fma u 2.0 -2.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), fmaf(-0.16666666666666666f, ((u * fmaf(u, fmaf(u, -16.0f, 24.0f), -8.0f)) / (v * v)), fmaf(u, 2.0f, -2.0f)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), fma(Float32(-0.16666666666666666), Float32(Float32(u * fma(u, fma(u, Float32(-16.0), Float32(24.0)), Float32(-8.0))) / Float32(v * v)), fma(u, Float32(2.0), Float32(-2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 75.2%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 75.2%

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

   9. Taylor expanded in u around 0

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

       1. *-lowering-*.f32 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. accelerator-lowering-fma.f32 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f32 75.2

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

   11. Simplified 75.2%

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

Alternative 14: 90.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(u, -\mathsf{fma}\left(u, \frac{2}{v} + \frac{4}{v \cdot v}, -2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right), -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (fma
     u
     (-
      (fma
       u
       (+ (/ 2.0 v) (/ 4.0 (* v v)))
       (+ -2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v))))
     -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(u, -fmaf(u, ((2.0f / v) + (4.0f / (v * v))), (-2.0f + ((-2.0f + (-1.3333333333333333f / v)) / v))), -2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(u, Float32(-fma(u, Float32(Float32(Float32(2.0) / v) + Float32(Float32(4.0) / Float32(v * v))), Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)))), Float32(-2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 75.2%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 72.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(u, -\mathsf{fma}\left(u, \frac{2}{v} + \frac{4}{v \cdot v}, -2 + \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right), -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(u, 2 + \frac{1.3333333333333333}{v \cdot v}, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (fma
     0.5
     (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
     (fma u (+ 2.0 (/ 1.3333333333333333 (* v v))) -2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), fmaf(u, (2.0f + (1.3333333333333333f / (v * v))), -2.0f));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), fma(u, Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v))), Float32(-2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 75.2%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 75.2%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f32 N/A

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

       4. +-lowering-+.f32 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. /-lowering-/.f32 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f32 71.9

          \[\leadsto 1 + \mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(u, 2 + \frac{1.3333333333333333}{\color{blue}{v \cdot v}}, -2\right)\right) \]

   11. Simplified 71.9%

       \[\leadsto 1 + \mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \color{blue}{\mathsf{fma}\left(u, 2 + \frac{1.3333333333333333}{v \cdot v}, -2\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 90.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around inf

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

      1. /-lowering-/.f32 N/A

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

   5. Simplified 67.2%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. /-lowering-/.f32 N/A

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

      9. +-lowering-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f32 67.1

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

   8. Simplified 67.1%

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

Alternative 17: 90.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{u \cdot \mathsf{fma}\left(u, -4, 4\right)}{v}, \mathsf{fma}\left(u, 2, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (fma 0.5 (/ (* u (fma u -4.0 4.0)) v) (fma u 2.0 -1.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(0.5f, ((u * fmaf(u, -4.0f, 4.0f)) / v), fmaf(u, 2.0f, -1.0f));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(0.5), Float32(Float32(u * fma(u, Float32(-4.0), Float32(4.0))) / v), fma(u, Float32(2.0), Float32(-1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around inf

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

      1. /-lowering-/.f32 N/A

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

   5. Simplified 67.2%

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

   6. Taylor expanded in v around inf

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 67.1%

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

   9. Taylor expanded in u around 0

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

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt-out N/A

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

       7. distribute-lft-in N/A

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

       8. *-lowering-*.f32 N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-out N/A

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

       11. metadata-eval N/A

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

       12. *-commutative N/A

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

       13. distribute-lft-in N/A

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

       14. associate-*r* N/A

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

       15. metadata-eval N/A

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

       16. *-commutative N/A

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

       17. metadata-eval N/A

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

       18. accelerator-lowering-fma.f32 67.1

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

   11. Simplified 67.1%

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

Alternative 18: 89.9% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. accelerator-lowering-fma.f32 N/A

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

      3. --lowering--.f32 56.4

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

   5. Simplified 56.4%

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

Alternative 19: 89.9% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.0%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around inf

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

      1. /-lowering-/.f32 N/A

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

   5. Simplified 67.2%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 56.4

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

   8. Simplified 56.4%

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

Alternative 20: 87.0% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

5. Simplified 84.7%

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

Alternative 21: 5.7% 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. Simplified 6.1%

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

Reproduce

herbie shell --seed 2024199 
(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))))))))