HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.6s

Alternatives: 22

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

Alternative 2: 91.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 65.5%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.0%

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

Alternative 3: 91.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 65.4%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.0%

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

Alternative 4: 91.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 63.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.0%

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

Alternative 5: 91.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v)))))) <= (-1.0e0)) then
        tmp = u * (((2.0e0 / v) + (1.3333333333333333e0 / (v * v))) + (2.0e0 + ((-1.0e0) / u)))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = u * (((single(2.0) / v) + (single(1.3333333333333333) / (v * v))) + (single(2.0) + (single(-1.0) / u)));
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 61.4%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 61.6%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.0%

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

Alternative 6: 91.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = fma(u, Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + 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 94.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 61.4%

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

   10. Applied rewrites 61.4%

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

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 88.1%

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

Alternative 7: 90.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower--.f32 N/A

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

      13. lower-fma.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lower-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 56.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.0%

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

Alternative 8: 90.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 56.7%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.0%

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

Alternative 9: 90.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + ((1.0f - u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = fmaf(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(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= 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 94.2%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 61.4%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 56.7%

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

   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 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 90.0%

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

Alternative 10: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

Alternative 11: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.6

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

5. Applied rewrites 99.6%

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

Alternative 12: 95.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in v around -inf

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

10. Applied rewrites 94.8%

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

Alternative 13: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{fma}\left(\frac{1}{1 + \frac{2}{v \cdot v}}, 1 - u, u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, 8, \mathsf{fma}\left(1 - u, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot 16, \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -24\right)\right)\right), \frac{0.16666666666666666}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (+ 1.0 (* v (log (fma (/ 1.0 (+ 1.0 (/ 2.0 (* v v)))) (- 1.0 u) u))))
   (-
    (fma u 2.0 -1.0)
    (/
     (fma
      (fma
       (- 1.0 u)
       8.0
       (fma
        (- 1.0 u)
        (* (* (- 1.0 u) (- 1.0 u)) 16.0)
        (* (- 1.0 u) (* (- 1.0 u) -24.0))))
      (/ 0.16666666666666666 v)
      (* -0.5 (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f + (v * logf(fmaf((1.0f / (1.0f + (2.0f / (v * v)))), (1.0f - u), u)));
	} else {
		tmp = fmaf(u, 2.0f, -1.0f) - (fmaf(fmaf((1.0f - u), 8.0f, fmaf((1.0f - u), (((1.0f - u) * (1.0f - u)) * 16.0f), ((1.0f - u) * ((1.0f - u) * -24.0f)))), (0.16666666666666666f / v), (-0.5f * ((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(1.0) + Float32(v * log(fma(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(2.0) / Float32(v * v)))), Float32(Float32(1.0) - u), u))));
	else
		tmp = Float32(fma(u, Float32(2.0), Float32(-1.0)) - Float32(fma(fma(Float32(Float32(1.0) - u), Float32(8.0), fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)) * Float32(16.0)), Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(1.0) - u) * Float32(-24.0))))), Float32(Float32(0.16666666666666666) / v), Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

   3. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-+r+ N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-in N/A

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

      9. neg-mul-1 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f32 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 97.0%

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

   11. Taylor expanded in v around 0

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

   13. Applied rewrites 97.0%

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

   if 0.300000012 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 66.5%

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

Alternative 14: 93.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in v around inf

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

10. Applied rewrites 93.0%

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

11. Taylor expanded in v around 0

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

13. Applied rewrites 93.0%

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

Alternative 15: 91.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-+r+ N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-in N/A

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

   9. neg-mul-1 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in v around inf

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

10. Applied rewrites 89.4%

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

Alternative 16: 91.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2, -1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(1 - u, 8, \mathsf{fma}\left(1 - u, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot 16, \left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -24\right)\right)\right), \frac{0.16666666666666666}{v}, -0.5 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (-
    (fma u 2.0 -1.0)
    (/
     (fma
      (fma
       (- 1.0 u)
       8.0
       (fma
        (- 1.0 u)
        (* (* (- 1.0 u) (- 1.0 u)) 16.0)
        (* (- 1.0 u) (* (- 1.0 u) -24.0))))
      (/ 0.16666666666666666 v)
      (* -0.5 (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, 2.0f, -1.0f) - (fmaf(fmaf((1.0f - u), 8.0f, fmaf((1.0f - u), (((1.0f - u) * (1.0f - u)) * 16.0f), ((1.0f - u) * ((1.0f - u) * -24.0f)))), (0.16666666666666666f / v), (-0.5f * ((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(u, Float32(2.0), Float32(-1.0)) - Float32(fma(fma(Float32(Float32(1.0) - u), Float32(8.0), fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)) * Float32(16.0)), Float32(Float32(Float32(1.0) - u) * Float32(Float32(Float32(1.0) - u) * Float32(-24.0))))), Float32(Float32(0.16666666666666666) / v), Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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 90.2%

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

   if 0.300000012 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 66.5%

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

Alternative 17: 91.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{0.16666666666666666}{v} \cdot \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (fma
      (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0))
      -0.5
      (*
       (/ 0.16666666666666666 v)
       (fma
        (* (- 1.0 u) (- 1.0 u))
        (fma (- 1.0 u) 16.0 -24.0)
        (fma 8.0 (- u) 8.0))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - (fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), -0.5f, ((0.16666666666666666f / v) * fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), 16.0f, -24.0f), fmaf(8.0f, -u, 8.0f)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(-0.5), Float32(Float32(Float32(0.16666666666666666) / v) * fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)), fma(Float32(8.0), Float32(-u), Float32(8.0))))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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 90.2%

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

   if 0.300000012 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 66.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), -0.5, \frac{0.16666666666666666}{v} \cdot \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 91.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(u, 2, -2\right) - \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, -16, 24\right), \mathsf{fma}\left(u, 8, -8\right)\right)}{v}, \left(1 - u\right) \cdot \mathsf{fma}\left(-0.5, \left(1 - u\right) \cdot -4, -2\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+
    1.0
    (-
     (fma u 2.0 -2.0)
     (/
      (fma
       -0.16666666666666666
       (/
        (fma
         (* (- 1.0 u) (- 1.0 u))
         (fma (- 1.0 u) -16.0 24.0)
         (fma u 8.0 -8.0))
        v)
       (* (- 1.0 u) (fma -0.5 (* (- 1.0 u) -4.0) -2.0)))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (fmaf(u, 2.0f, -2.0f) - (fmaf(-0.16666666666666666f, (fmaf(((1.0f - u) * (1.0f - u)), fmaf((1.0f - u), -16.0f, 24.0f), fmaf(u, 8.0f, -8.0f)) / v), ((1.0f - u) * fmaf(-0.5f, ((1.0f - u) * -4.0f), -2.0f))) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(fma(u, Float32(2.0), Float32(-2.0)) - Float32(fma(Float32(-0.16666666666666666), Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), fma(Float32(Float32(1.0) - u), Float32(-16.0), Float32(24.0)), fma(u, Float32(8.0), Float32(-8.0))) / v), Float32(Float32(Float32(1.0) - u) * fma(Float32(-0.5), Float32(Float32(Float32(1.0) - u) * Float32(-4.0)), Float32(-2.0)))) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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 90.2%

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

   if 0.300000012 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 66.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(u, 2, -2\right) - \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, -16, 24\right), \mathsf{fma}\left(u, 8, -8\right)\right)}{v}, \left(1 - u\right) \cdot \mathsf{fma}\left(-0.5, \left(1 - u\right) \cdot -4, -2\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 91.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(0.16666666666666666, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 16, -24\right), 8\right)}{v \cdot v}, \mathsf{fma}\left(-2, -u, -2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+
    1.0
    (fma
     0.5
     (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
     (fma
      0.16666666666666666
      (/ (* u (fma u (fma u 16.0 -24.0) 8.0)) (* v v))
      (fma -2.0 (- u) -2.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), fmaf(0.16666666666666666f, ((u * fmaf(u, fmaf(u, 16.0f, -24.0f), 8.0f)) / (v * v)), fmaf(-2.0f, -u, -2.0f)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), fma(Float32(0.16666666666666666), Float32(Float32(u * fma(u, fma(u, Float32(16.0), Float32(-24.0)), Float32(8.0))) / Float32(v * v)), fma(Float32(-2.0), Float32(-u), Float32(-2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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 90.2%

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

   if 0.300000012 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 66.0%

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

Alternative 20: 90.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(u \cdot \mathsf{fma}\left(u, -4, 4\right), \frac{0.5}{v}, \mathsf{fma}\left(2, u, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   1.0
   (+ 1.0 (fma (* u (fma u -4.0 4.0)) (/ 0.5 v) (fma 2.0 u -2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf((u * fmaf(u, -4.0f, 4.0f)), (0.5f / v), fmaf(2.0f, u, -2.0f));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(u * fma(u, Float32(-4.0), Float32(4.0))), Float32(Float32(0.5) / v), fma(Float32(2.0), u, Float32(-2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   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 90.2%

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

   if 0.300000012 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f32 N/A

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

      12. lower--.f32 N/A

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

      13. lower-fma.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lower-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 57.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 57.1

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

   10. Applied rewrites 57.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(u \cdot \mathsf{fma}\left(u, -4, 4\right), \frac{0.5}{v}, \mathsf{fma}\left(2, u, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 87.3% 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. Applied rewrites 84.2%

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

Alternative 22: 5.7% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.7%

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

Reproduce

herbie shell --seed 2024226 
(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))))))))