HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.7s

Alternatives: 18

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 91.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (+
    1.0
    (+
     (+
      -2.0
      (fma
       (- 1.0 u)
       (/ (fma (- 1.0 u) -4.0 4.0) (* v 2.0))
       (/
        (* (* u 0.16666666666666666) (fma u (fma u 16.0 -24.0) 8.0))
        (* v v))))
     (* 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 + ((-2.0f + fmaf((1.0f - u), (fmaf((1.0f - u), -4.0f, 4.0f) / (v * 2.0f)), (((u * 0.16666666666666666f) * fmaf(u, fmaf(u, 16.0f, -24.0f), 8.0f)) / (v * v)))) + (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) + Float32(Float32(Float32(-2.0) + fma(Float32(Float32(1.0) - u), Float32(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)) / Float32(v * Float32(2.0))), Float32(Float32(Float32(u * Float32(0.16666666666666666)) * fma(u, fma(u, Float32(16.0), Float32(-24.0)), Float32(8.0))) / Float32(v * v)))) + 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 93.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 81.4%

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

   6. Taylor expanded in u around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 81.4

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

   8. Simplified 81.4%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

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

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

   10. Applied egg-rr 81.9%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

4. Final simplification 92.5%

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

Alternative 3: 91.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (+
    1.0
    (fma
     0.16666666666666666
     (/ (* u (fma u (fma u 16.0 -24.0) 8.0)) (* v v))
     (fma u (+ 2.0 (fma -2.0 (/ u v) (/ 2.0 v))) -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(0.16666666666666666f, ((u * fmaf(u, fmaf(u, 16.0f, -24.0f), 8.0f)) / (v * v)), fmaf(u, (2.0f + fmaf(-2.0f, (u / v), (2.0f / v))), -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(0.16666666666666666), Float32(Float32(u * fma(u, fma(u, Float32(16.0), Float32(-24.0)), Float32(8.0))) / Float32(v * v)), fma(u, Float32(Float32(2.0) + fma(Float32(-2.0), Float32(u / v), Float32(Float32(2.0) / v))), 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 93.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 81.4%

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

   6. Taylor expanded in u around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 81.4

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

   8. Simplified 81.4%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. /-lowering-/.f32 81.8

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

   11. Simplified 81.8%

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

   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. Simplified 93.2%

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

Alternative 4: 91.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (/
    (fma
     v
     (fma v (fma u 2.0 -1.0) (* (- 1.0 u) (* (fma (- 1.0 u) -4.0 4.0) 0.5)))
     (* (* u 0.16666666666666666) (fma u (fma u 16.0 -24.0) 8.0)))
    (* 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(v, fmaf(v, fmaf(u, 2.0f, -1.0f), ((1.0f - u) * (fmaf((1.0f - u), -4.0f, 4.0f) * 0.5f))), ((u * 0.16666666666666666f) * fmaf(u, fmaf(u, 16.0f, -24.0f), 8.0f))) / (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(v, fma(v, fma(u, Float32(2.0), Float32(-1.0)), Float32(Float32(Float32(1.0) - u) * Float32(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)) * Float32(0.5)))), Float32(Float32(u * Float32(0.16666666666666666)) * fma(u, fma(u, Float32(16.0), Float32(-24.0)), Float32(8.0)))) / 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 93.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 81.4%

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

   6. Taylor expanded in u around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 81.4

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

   8. Simplified 81.4%

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

   9. Taylor expanded in v around 0

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

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

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

   11. Simplified 81.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

4. Final simplification 92.4%

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

Alternative 5: 91.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (+
    1.0
    (fma
     u
     (+
      2.0
      (+
       (/ 2.0 v)
       (fma u (- (/ -2.0 v) (/ 4.0 (* v v))) (/ 1.3333333333333333 (* v v)))))
     -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, (2.0f + ((2.0f / v) + fmaf(u, ((-2.0f / v) - (4.0f / (v * v))), (1.3333333333333333f / (v * v))))), -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(u, Float32(Float32(2.0) + Float32(Float32(Float32(2.0) / v) + fma(u, Float32(Float32(Float32(-2.0) / v) - Float32(Float32(4.0) / Float32(v * v))), Float32(Float32(1.3333333333333333) / Float32(v * v))))), 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 93.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 81.4%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

   8. Simplified 80.3%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 6: 91.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 81.4%

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

   6. Taylor expanded in u around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 81.4

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

   8. Simplified 81.4%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

   11. Simplified 80.3%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 7: 90.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (+
    1.0
    (fma
     0.16666666666666666
     (/ (* u 8.0) (* v v))
     (fma
      (* (- 1.0 u) (fma (- 1.0 u) -4.0 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(0.16666666666666666f, ((u * 8.0f) / (v * v)), fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 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(0.16666666666666666), Float32(Float32(u * Float32(8.0)) / Float32(v * v)), fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), 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 93.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 81.4%

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

   6. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 79.8

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

   8. Simplified 79.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 8: 90.7% accurate, 0.8× speedup

Math

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

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
      u
      -2.0
      (/ (fma u 1.3333333333333333 (/ (* u 0.6666666666666666) v)) (- 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(u, -2.0f, (fmaf(u, 1.3333333333333333f, ((u * 0.6666666666666666f) / v)) / -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(u, Float32(-2.0), Float32(fma(u, Float32(1.3333333333333333), Float32(Float32(u * Float32(0.6666666666666666)) / v)) / 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 93.6%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

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

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 77.5

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

   5. Simplified 77.5%

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

   6. Taylor expanded in v around -inf

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

   8. Simplified 78.4%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 9: 90.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.6%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

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

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 77.5

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

   5. Simplified 77.5%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

   8. Simplified 76.3%

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

   9. Taylor expanded in v around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f32 76.3

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

   11. Simplified 76.3%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 10: 90.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.6%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

   5. Simplified 72.1%

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

   6. Taylor expanded in u around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

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

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

      14. /-lowering-/.f32 N/A

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

   8. Simplified 71.9%

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

   9. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

   11. Simplified 72.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

4. Final simplification 91.9%

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

Alternative 11: 90.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (+ 1.0 (fma u (+ 2.0 (/ 2.0 v)) -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, (2.0f + (2.0f / v)), -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(u, Float32(Float32(2.0) + Float32(Float32(2.0) / v)), 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 93.6%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

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

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 77.5

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

   5. Simplified 77.5%

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

   6. Taylor expanded in v around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate--l+ N/A

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

      7. distribute-rgt-in N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f32 71.3

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

   8. Simplified 71.3%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 12: 90.4% 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 93.6%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

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

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 77.5

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

   5. Simplified 77.5%

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

   6. Taylor expanded in v around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f32 71.6

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

   8. Simplified 71.6%

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

   9. Taylor expanded in v around inf

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

       1. sub-neg N/A

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

       2. distribute-lft-out N/A

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

       3. metadata-eval N/A

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

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

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

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

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

       6. /-lowering-/.f32 71.2

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

   11. Simplified 71.2%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 13: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.6%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

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

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

      3. --lowering--.f32 60.7

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

   5. Simplified 60.7%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 14: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.6%

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

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

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

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

      4. +-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

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

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

      16. --lowering--.f32 93.3

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

   5. Simplified 93.3%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-+r+ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 60.7

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

   8. Simplified 60.7%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.2%

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

Alternative 15: 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. accelerator-lowering-fma.f32 N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

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

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

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

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

   16. --lowering--.f32 99.6

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

5. Simplified 99.6%

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

Alternative 16: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. /-lowering-/.f32 96.5

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

5. Simplified 96.5%

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

Alternative 17: 86.3% 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. Simplified 87.9%

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

Alternative 18: 6.0% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.9%

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

Reproduce

herbie shell --seed 2024204 
(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))))))))