HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 13.3s

Alternatives: 17

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: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 87.3%

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

8. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

10. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. 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 99.5

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

5. Applied rewrites 99.5%

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

Alternative 4: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 87.3%

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

8. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in u around 0

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

13. Applied rewrites 96.5%

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

Alternative 5: 91.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around -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. Applied rewrites 81.6%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 81.6%

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

4. Final simplification 91.9%

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

Alternative 6: 91.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(v, 2, \mathsf{fma}\left(u, -2.6666666666666665, 4\right)\right)}{v}, -2 + \frac{-1.3333333333333333}{v}\right)}{-v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/
      (*
       u
       (fma
        u
        (/ (fma v 2.0 (fma u -2.6666666666666665 4.0)) v)
        (+ -2.0 (/ -1.3333333333333333 v))))
      (- v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), ((u * fmaf(u, (fmaf(v, 2.0f, fmaf(u, -2.6666666666666665f, 4.0f)) / v), (-2.0f + (-1.3333333333333333f / v)))) / -v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(u * fma(u, Float32(fma(v, Float32(2.0), fma(u, Float32(-2.6666666666666665), Float32(4.0))) / v), Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)))) / Float32(-v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around -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. Applied rewrites 81.6%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 81.6%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, -\frac{u \cdot \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(v, 2, \mathsf{fma}\left(u, -2.6666666666666665, 4\right)\right)}{v}, -2 + \frac{-1.3333333333333333}{v}\right)}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \mathsf{fma}\left(u, \frac{\mathsf{fma}\left(v, 2, \mathsf{fma}\left(u, -2.6666666666666665, 4\right)\right)}{v}, -2 + \frac{-1.3333333333333333}{v}\right)}{-v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \mathsf{fma}\left(u, 2, \frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.6666666666666665, 4\right), -1.3333333333333333\right)}{v} - 2\right)}{-v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/
      (*
       u
       (fma
        u
        2.0
        (-
         (/ (fma u (fma u -2.6666666666666665 4.0) -1.3333333333333333) v)
         2.0)))
      (- v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), ((u * fmaf(u, 2.0f, ((fmaf(u, fmaf(u, -2.6666666666666665f, 4.0f), -1.3333333333333333f) / v) - 2.0f))) / -v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(u * fma(u, Float32(2.0), Float32(Float32(fma(u, fma(u, Float32(-2.6666666666666665), Float32(4.0)), Float32(-1.3333333333333333)) / v) - Float32(2.0)))) / Float32(-v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around -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. Applied rewrites 81.6%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 81.6%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, -\frac{u \cdot \mathsf{fma}\left(u, 2, \frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.6666666666666665, 4\right), -1.3333333333333333\right)}{v} - 2\right)}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \mathsf{fma}\left(u, 2, \frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.6666666666666665, 4\right), -1.3333333333333333\right)}{v} - 2\right)}{-v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around -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. Applied rewrites 81.6%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 75.1%

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

4. Final simplification 91.5%

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

Alternative 9: 90.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 73.3%

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

   13. Applied rewrites 73.3%

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

Alternative 10: 90.7% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 73.3%

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

   13. Applied rewrites 73.3%

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

Alternative 11: 90.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 73.3%

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

   12. Taylor expanded in v around 0

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

   14. Applied rewrites 73.3%

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

Alternative 12: 90.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 73.3%

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

   12. Taylor expanded in v around -inf

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

   14. Applied rewrites 73.3%

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

Alternative 13: 90.5% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-/.f32 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 68.6%

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

Alternative 14: 89.9% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 61.5

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

   5. Applied rewrites 61.5%

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

Alternative 15: 89.9% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   4. Applied rewrites 92.1%

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

   6. Applied rewrites 91.5%

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

   7. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 61.5

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

   9. Applied rewrites 61.5%

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

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 61.5%

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

Alternative 16: 86.6% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 87.3%

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

Alternative 17: 6.1% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.9%

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

Reproduce

herbie shell --seed 2024219 
(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))))))))