HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.3s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 2: 91.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -1.0f) {
		tmp = fmaf(u, (2.0f - (fmaf(v, fmaf(2.0f, u, -2.0f), fmaf(4.0f, u, -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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(u, Float32(Float32(2.0) - Float32(fma(v, fma(Float32(2.0), u, Float32(-2.0)), fma(Float32(4.0), u, 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 94.0%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \left(v \cdot {\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2}\right) \cdot \left(\frac{u}{e^{2 \cdot \frac{-2}{v}}} \cdot -0.5\right)\right), -1\right)} \]

   6. Taylor expanded in v around -inf

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}}, -1\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(u, 2 + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)}, -1\right) \]

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   8. Simplified 65.9%

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

   9. Taylor expanded in v around 0

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

       1. lower-/.f32 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f32 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f32 N/A

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

       7. +-commutative N/A

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

       8. distribute-lft-in N/A

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

       9. associate-*r* N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 65.9

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

   11. Simplified 65.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

4. Final simplification 92.3%

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

Alternative 3: 91.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(u, Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) / v) - fma(Float32(2.0), u, 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 94.0%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \left(v \cdot {\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2}\right) \cdot \left(\frac{u}{e^{2 \cdot \frac{-2}{v}}} \cdot -0.5\right)\right), -1\right)} \]

   6. Taylor expanded in v around -inf

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}}, -1\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(u, 2 + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)}, -1\right) \]

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   8. Simplified 65.9%

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

   9. Taylor expanded in u around 0

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

   11. Simplified 65.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

4. Final simplification 92.2%

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

Alternative 4: 90.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-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 61.5

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

   5. Simplified 61.5%

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

   6. Taylor expanded in v around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-/.f32 63.6

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

   8. Simplified 63.6%

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

   9. Taylor expanded in v around -inf

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. lower-+.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f32 N/A

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

   11. Simplified 63.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

4. Final simplification 92.1%

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

Alternative 5: 90.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \left(v \cdot {\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2}\right) \cdot \left(\frac{u}{e^{2 \cdot \frac{-2}{v}}} \cdot -0.5\right)\right), -1\right)} \]

   6. Taylor expanded in v around -inf

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}}, -1\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(u, 2 + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)}, -1\right) \]

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   8. Simplified 65.9%

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

   9. Taylor expanded in u around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. distribute-lft-in N/A

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

       4. metadata-eval N/A

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

       5. neg-mul-1 N/A

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

       6. lower-+.f32 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

       11. lower-/.f32 63.6

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

   11. Simplified 63.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

4. Final simplification 92.1%

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

Alternative 6: 90.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(Float32(-2.0), Float32(Float32(1.0) - u), fma(Float32(2.0), Float32(u / 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 94.0%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f32 N/A

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

   5. Simplified 57.8%

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

   6. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 58.5

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

   8. Simplified 58.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

4. Final simplification 91.8%

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

Alternative 7: 90.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f32 N/A

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

   5. Simplified 57.8%

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

   6. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 58.5

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

   8. Simplified 58.5%

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

   9. Taylor expanded in v around 0

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

       1. +-commutative N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-neg-out N/A

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

       8. mul-1-neg N/A

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

       9. metadata-eval N/A

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

       10. associate-+l+ N/A

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

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

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       15. sub-neg N/A

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

   11. Simplified 58.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

4. Final simplification 91.8%

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

Alternative 8: 90.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-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 61.5

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

   5. Simplified 61.5%

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

   6. Taylor expanded in v around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. associate--l+ N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out N/A

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

      17. lower-fma.f32 N/A

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

      18. lower-+.f32 N/A

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

      19. lower-/.f32 63.3

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

   8. Simplified 63.3%

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

   9. Taylor expanded in v around 0

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

       1. lower-/.f32 N/A

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

   11. Simplified 63.3%

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

   12. Taylor expanded in v around inf

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

       1. sub-neg N/A

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

       2. associate-*r/ N/A

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

       3. associate-*l/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

       6. distribute-rgt-out N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f32 N/A

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

       11. sub-neg N/A

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

       12. remove-double-neg N/A

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

       13. lower-+.f32 N/A

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

       14. associate-*r/ N/A

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

       15. metadata-eval N/A

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

       16. lower-/.f32 58.4

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

   14. Simplified 58.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

4. Final simplification 91.8%

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

Alternative 9: 90.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.0))
		tmp = fma(Float32(2.0), Float32(u + Float32(u / 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 94.0%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f32 N/A

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

   5. Simplified 57.8%

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

   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. distribute-rgt-in N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-/.f32 58.4

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

   8. Simplified 58.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

4. Final simplification 91.8%

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

Alternative 10: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{\frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right) - \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(u, 9.333333333333334, \mathsf{fma}\left(u, 32, u \cdot -32\right)\right), -0.6666666666666666\right)}{v}}{v} - \mathsf{fma}\left(2, u, -2\right)}{v}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (fma v (log (* (expm1 (/ -2.0 v)) (- u))) 1.0)
   (fma
    u
    (+
     2.0
     (/
      (-
       (/
        (-
         (fma u -4.0 1.3333333333333333)
         (/
          (fma
           0.5
           (fma u 9.333333333333334 (fma u 32.0 (* u -32.0)))
           -0.6666666666666666)
          v))
        v)
       (fma 2.0 u -2.0))
      v))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = fmaf(v, logf((expm1f((-2.0f / v)) * -u)), 1.0f);
	} else {
		tmp = fmaf(u, (2.0f + ((((fmaf(u, -4.0f, 1.3333333333333333f) - (fmaf(0.5f, fmaf(u, 9.333333333333334f, fmaf(u, 32.0f, (u * -32.0f))), -0.6666666666666666f) / v)) / v) - fmaf(2.0f, u, -2.0f)) / v)), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = fma(v, log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u))), Float32(1.0));
	else
		tmp = fma(u, Float32(Float32(2.0) + Float32(Float32(Float32(Float32(fma(u, Float32(-4.0), Float32(1.3333333333333333)) - Float32(fma(Float32(0.5), fma(u, Float32(9.333333333333334), fma(u, Float32(32.0), Float32(u * Float32(-32.0)))), Float32(-0.6666666666666666)) / v)) / v) - fma(Float32(2.0), u, Float32(-2.0))) / v)), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)} \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in u around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. neg-mul-1 N/A

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

      11. lower-*.f32 N/A

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

   8. Simplified 99.5%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 81.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \left(v \cdot {\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2}\right) \cdot \left(\frac{u}{e^{2 \cdot \frac{-2}{v}}} \cdot -0.5\right)\right), -1\right)} \]

   6. Taylor expanded in v around -inf

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \left(-1 \cdot \frac{\frac{1}{2} \cdot \left(\frac{28}{3} \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - \frac{2}{3}}{v} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}}, -1\right) \]

   8. Simplified 76.5%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 - \frac{\mathsf{fma}\left(2, u, -2\right) - \frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right) - \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(u, 9.333333333333334, \mathsf{fma}\left(u, 32, u \cdot -32\right)\right), -0.6666666666666666\right)}{v}}{v}}{v}}, -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{\frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right) - \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(u, 9.333333333333334, \mathsf{fma}\left(u, 32, u \cdot -32\right)\right), -0.6666666666666666\right)}{v}}{v} - \mathsf{fma}\left(2, u, -2\right)}{v}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{\frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right) - \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(u, 9.333333333333334, \mathsf{fma}\left(u, 32, u \cdot -32\right)\right), -0.6666666666666666\right)}{v}}{v} - \mathsf{fma}\left(2, u, -2\right)}{v}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (fma
    u
    (+
     2.0
     (/
      (-
       (/
        (-
         (fma u -4.0 1.3333333333333333)
         (/
          (fma
           0.5
           (fma u 9.333333333333334 (fma u 32.0 (* u -32.0)))
           -0.6666666666666666)
          v))
        v)
       (fma 2.0 u -2.0))
      v))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, (2.0f + ((((fmaf(u, -4.0f, 1.3333333333333333f) - (fmaf(0.5f, fmaf(u, 9.333333333333334f, fmaf(u, 32.0f, (u * -32.0f))), -0.6666666666666666f) / v)) / v) - fmaf(2.0f, u, -2.0f)) / v)), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = fma(u, Float32(Float32(2.0) + Float32(Float32(Float32(Float32(fma(u, Float32(-4.0), Float32(1.3333333333333333)) - Float32(fma(Float32(0.5), fma(u, Float32(9.333333333333334), fma(u, Float32(32.0), Float32(u * Float32(-32.0)))), Float32(-0.6666666666666666)) / v)) / v) - fma(Float32(2.0), u, Float32(-2.0))) / v)), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.1%

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

   if 0.0500000007 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \left(v \cdot {\left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)}^{2}\right) \cdot \left(\frac{u}{e^{2 \cdot \frac{-2}{v}}} \cdot -0.5\right)\right), -1\right)} \]

   6. Taylor expanded in v around -inf

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \left(-1 \cdot \frac{\frac{1}{2} \cdot \left(\frac{28}{3} \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - \frac{2}{3}}{v} + \frac{1}{2} \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}}, -1\right) \]

   8. Simplified 70.5%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{2 - \frac{\mathsf{fma}\left(2, u, -2\right) - \frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right) - \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(u, 9.333333333333334, \mathsf{fma}\left(u, 32, u \cdot -32\right)\right), -0.6666666666666666\right)}{v}}{v}}{v}}, -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{\frac{\mathsf{fma}\left(u, -4, 1.3333333333333333\right) - \frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(u, 9.333333333333334, \mathsf{fma}\left(u, 32, u \cdot -32\right)\right), -0.6666666666666666\right)}{v}}{v} - \mathsf{fma}\left(2, u, -2\right)}{v}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.9% 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 v around 0

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

5. Simplified 88.1%

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

Alternative 13: 5.8% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.7%

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

Reproduce

herbie shell --seed 2024215 
(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))))))))