HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.2s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 90.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

   8. Simplified 70.8%

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

   9. Taylor expanded in v around inf

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

   11. Simplified 70.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.5%

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

Alternative 3: 90.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.5%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 70.2%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 67.5

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

   8. Simplified 67.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.5%

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

Alternative 4: 90.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

   8. Simplified 70.8%

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

   9. Taylor expanded in v around inf

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

       1. sub-neg N/A

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

       2. +-commutative N/A

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

       3. associate-*r/ N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

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

       6. associate-*r/ N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

       11. distribute-lft-out N/A

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

       12. associate-+r+ N/A

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

       13. metadata-eval N/A

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

       14. lower-fma.f32 N/A

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

   11. Simplified 67.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.5%

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

Alternative 5: 90.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

   8. Simplified 70.8%

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

   9. Taylor expanded in v around inf

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

       1. sub-neg N/A

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

       2. +-commutative N/A

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

       3. associate-*r/ N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

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

       6. associate-*r/ N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-/l* N/A

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

       11. distribute-lft-out N/A

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

       12. associate-+r+ N/A

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

       13. metadata-eval N/A

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

       14. lower-fma.f32 N/A

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

   11. Simplified 67.5%

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

       1. lift-/.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. lift-+.f32 N/A

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

       4. lift-fma.f32 N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f32 67.5

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

       7. lift-fma.f32 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f32 N/A

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

       10. lift-/.f32 N/A

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

       11. associate-*l/ N/A

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

       12. lower-/.f32 N/A

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

       13. *-lft-identity 67.5

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

   13. Applied egg-rr 67.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.5%

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

Alternative 6: 90.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.5%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Simplified 64.4%

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

   6. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 65.6

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

   8. Simplified 65.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.5%

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

Alternative 7: 90.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.5%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Simplified 64.4%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 65.5

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

   8. Simplified 65.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.5%

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

Alternative 8: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(u, Float32(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 94.5%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 56.7

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

   8. Simplified 56.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 91.5%

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

Alternative 9: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 49.2%

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

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

   1. Initial program 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 91.5%

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

Alternative 10: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Simplified 99.6%

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

Alternative 11: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.6

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

5. Simplified 99.6%

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

Alternative 12: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((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 + exp(((-2.0e0) / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Simplified 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 99.6

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in u around 0

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

10. Simplified 96.4%

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

    1. lift-/.f32 N/A

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

    2. lift-exp.f32 N/A

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

    3. lift-fma.f32 N/A

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

    4. lift-log.f32 N/A

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

    5. lower-+.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 96.4

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

    8. lift-fma.f32 N/A

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

    9. +-commutative N/A

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

    10. lower-+.f32 N/A

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

    11. *-rgt-identity 96.4

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

12. Applied egg-rr 96.4%

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

13. Final simplification 96.4%

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

Alternative 13: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Simplified 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 99.6

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in u around 0

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

10. Simplified 96.4%

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

    1. lift-/.f32 N/A

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

    2. lift-exp.f32 N/A

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

    3. lower-+.f32 N/A

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

    4. *-rgt-identity 96.4

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

12. Applied egg-rr 96.4%

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

13. Final simplification 96.4%

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

Alternative 14: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\log u, v, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(\frac{2}{v}\right), v \cdot u, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5) (fma (log u) v 1.0) (fma (expm1 (/ 2.0 v)) (* v u) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = fmaf(logf(u), v, 1.0f);
	} else {
		tmp = fmaf(expm1f((2.0f / v)), (v * u), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = fma(log(u), v, Float32(1.0));
	else
		tmp = fma(expm1(Float32(Float32(2.0) / v)), Float32(v * u), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   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 u around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-+r+ N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f32 N/A

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

   5. Simplified 100.0%

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

      1. lift-/.f32 N/A

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

      2. lift-exp.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-log.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 99.9

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Simplified 99.3%

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

   11. Taylor expanded in u around inf

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

       1. mul-1-neg N/A

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

       2. log-rec N/A

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

       3. remove-double-neg N/A

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

       4. lower-log.f32 99.0

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

   13. Simplified 99.0%

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

   if 0.5 < v

   1. Initial program 94.1%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 82.8

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

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{expm1}\left(\frac{2}{v}\right), v \cdot u, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   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 u around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-+r+ N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f32 N/A

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

   5. Simplified 100.0%

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

      1. lift-/.f32 N/A

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

      2. lift-exp.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-log.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 99.9

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Simplified 99.3%

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

   11. Taylor expanded in u around inf

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

       1. mul-1-neg N/A

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

       2. log-rec N/A

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

       3. remove-double-neg N/A

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

       4. lower-log.f32 99.0

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

   13. Simplified 99.0%

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

   if 0.5 < v

   1. Initial program 94.1%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 78.5%

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

   7. Applied egg-rr 78.7%

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

   8. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 78.7

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

   10. Simplified 78.7%

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

Alternative 16: 5.9% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 6.4%

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

Reproduce

herbie shell --seed 2024208 
(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))))))))