HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 9.7s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 2: 49.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-2}{v}}\\ \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot t\_0\right) \cdot v \leq -0.75:\\ \;\;\;\;\left(u \cdot u\right) \cdot \left(\frac{-2}{v} - \frac{\frac{1}{u} - \left(\frac{2}{v} + 2\right)}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(t\_0, 1 - u, u\right)\right) \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (exp (/ -2.0 v))))
   (if (<= (* (log (- u (* (+ -1.0 u) t_0))) v) -0.75)
     (* (* u u) (- (/ -2.0 v) (/ (- (/ 1.0 u) (+ (/ 2.0 v) 2.0)) u)))
     (+ (* (log (fma t_0 (- 1.0 u) u)) v) 1.0))))

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v));
	float tmp;
	if ((logf((u - ((-1.0f + u) * t_0))) * v) <= -0.75f) {
		tmp = (u * u) * ((-2.0f / v) - (((1.0f / u) - ((2.0f / v) + 2.0f)) / u));
	} else {
		tmp = (logf(fmaf(t_0, (1.0f - u), u)) * v) + 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	t_0 = exp(Float32(Float32(-2.0) / v))
	tmp = Float32(0.0)
	if (Float32(log(Float32(u - Float32(Float32(Float32(-1.0) + u) * t_0))) * v) <= Float32(-0.75))
		tmp = Float32(Float32(u * u) * Float32(Float32(Float32(-2.0) / v) - Float32(Float32(Float32(Float32(1.0) / u) - Float32(Float32(Float32(2.0) / v) + Float32(2.0))) / u)));
	else
		tmp = Float32(Float32(log(fma(t_0, Float32(Float32(1.0) - u), u)) * v) + 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)))))) < -0.75

   1. Initial program 92.9%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower--.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 55.8%

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

   9. Taylor expanded in u around -inf

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

   11. Applied rewrites 55.8%

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

   if -0.75 < (*.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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. div-inv N/A

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

      4. exp-prod N/A

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

      5. inv-pow N/A

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

      6. sqr-pow N/A

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

      7. pow-unpow N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-exp.f32 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f32 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f32 N/A

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

      18. metadata-eval 99.9

          \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\left({\left(e^{-2}\right)}^{\left({v}^{-0.5}\right)}\right)}^{\left({v}^{\color{blue}{-0.5}}\right)}\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left({\left(e^{-2}\right)}^{\left({v}^{-0.5}\right)}\right)}^{\left({v}^{-0.5}\right)}}\right) \]
   5. Step-by-step derivation

      1. lift-pow.f32 N/A

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

      2. sqr-pow N/A

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

      3. pow-sqr N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-pow.f32 N/A

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

      8. log-pow N/A

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

      9. lift-exp.f32 N/A

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

      10. rem-log-exp N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f32 99.9

          \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\left({v}^{-0.5} \cdot -2\right) \cdot \left(2 \cdot \color{blue}{\left({v}^{-0.5} \cdot 0.5\right)}\right)}\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\left({v}^{-0.5} \cdot -2\right) \cdot \left(2 \cdot \left({v}^{-0.5} \cdot 0.5\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. rem-log-exp N/A

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

      6. log-pow N/A

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

      7. lift-*.f32 N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. pow-sqr N/A

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

      12. sqr-pow N/A

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

      13. log-pow N/A

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

      14. rem-log-exp N/A

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

      15. lift-*.f32 N/A

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

      16. lower-*.f32 99.9

          \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{{v}^{-0.5} \cdot \left({v}^{-0.5} \cdot -2\right)}}\right) \]

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f32 99.9

          \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{{v}^{-0.5} \cdot \color{blue}{\left(-2 \cdot {v}^{-0.5}\right)}}\right) \]

   8. Applied rewrites 99.9%

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{{v}^{-0.5} \cdot \left(-2 \cdot {v}^{-0.5}\right)}}\right) \]
   9. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 99.9

         \[\leadsto \color{blue}{v \cdot \log \left(u + \left(1 - u\right) \cdot e^{{v}^{-0.5} \cdot \left(-2 \cdot {v}^{-0.5}\right)}\right) + 1} \]

   10. Applied rewrites 98.4%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -0.75:\\ \;\;\;\;\left(u \cdot u\right) \cdot \left(\frac{-2}{v} - \frac{\frac{1}{u} - \left(\frac{2}{v} + 2\right)}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right) \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -0.75:\\ \;\;\;\;\left(u \cdot u\right) \cdot \left(\frac{-2}{v} - \frac{\frac{1}{u} - \left(\frac{2}{v} + 2\right)}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -0.75)
   (* (* u u) (- (/ -2.0 v) (/ (- (/ 1.0 u) (+ (/ 2.0 v) 2.0)) u)))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((logf((u - ((-1.0f + u) * expf((-2.0f / v))))) * v) <= -0.75f) {
		tmp = (u * u) * ((-2.0f / v) - (((1.0f / u) - ((2.0f / v) + 2.0f)) / u));
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((log((u - (((-1.0e0) + u) * exp(((-2.0e0) / v))))) * v) <= (-0.75e0)) then
        tmp = (u * u) * (((-2.0e0) / v) - (((1.0e0 / u) - ((2.0e0 / v) + 2.0e0)) / u))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(u - Float32(Float32(Float32(-1.0) + u) * exp(Float32(Float32(-2.0) / v))))) * v) <= Float32(-0.75))
		tmp = Float32(Float32(u * u) * Float32(Float32(Float32(-2.0) / v) - Float32(Float32(Float32(Float32(1.0) / u) - Float32(Float32(Float32(2.0) / v) + Float32(2.0))) / u)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((log((u - ((single(-1.0) + u) * exp((single(-2.0) / v))))) * v) <= single(-0.75))
		tmp = (u * u) * ((single(-2.0) / v) - (((single(1.0) / u) - ((single(2.0) / v) + single(2.0))) / u));
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.75

   1. Initial program 92.9%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower--.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 55.8%

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

   9. Taylor expanded in u around -inf

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

   11. Applied rewrites 55.8%

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

   if -0.75 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -0.75:\\ \;\;\;\;\left(u \cdot u\right) \cdot \left(\frac{-2}{v} - \frac{\frac{1}{u} - \left(\frac{2}{v} + 2\right)}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -0.75:\\ \;\;\;\;\left(\frac{-2}{v} - \frac{\left(\frac{1}{u} - 2\right) - \frac{2}{v}}{u}\right) \cdot \left(u \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -0.75)
   (* (- (/ -2.0 v) (/ (- (- (/ 1.0 u) 2.0) (/ 2.0 v)) u)) (* u u))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((logf((u - ((-1.0f + u) * expf((-2.0f / v))))) * v) <= -0.75f) {
		tmp = ((-2.0f / v) - ((((1.0f / u) - 2.0f) - (2.0f / v)) / u)) * (u * u);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((log((u - (((-1.0e0) + u) * exp(((-2.0e0) / v))))) * v) <= (-0.75e0)) then
        tmp = (((-2.0e0) / v) - ((((1.0e0 / u) - 2.0e0) - (2.0e0 / v)) / u)) * (u * u)
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(u - Float32(Float32(Float32(-1.0) + u) * exp(Float32(Float32(-2.0) / v))))) * v) <= Float32(-0.75))
		tmp = Float32(Float32(Float32(Float32(-2.0) / v) - Float32(Float32(Float32(Float32(Float32(1.0) / u) - Float32(2.0)) - Float32(Float32(2.0) / v)) / u)) * Float32(u * u));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((log((u - ((single(-1.0) + u) * exp((single(-2.0) / v))))) * v) <= single(-0.75))
		tmp = ((single(-2.0) / v) - ((((single(1.0) / u) - single(2.0)) - (single(2.0) / v)) / u)) * (u * u);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.75

   1. Initial program 92.9%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower--.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 55.8%

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

   if -0.75 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -0.75:\\ \;\;\;\;\left(\frac{-2}{v} - \frac{\left(\frac{1}{u} - 2\right) - \frac{2}{v}}{u}\right) \cdot \left(u \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -0.75:\\ \;\;\;\;\left(\left(\frac{-2}{v} - \frac{\left(\frac{1}{u} - 2\right) - \frac{2}{v}}{u}\right) \cdot u\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -0.75)
   (* (* (- (/ -2.0 v) (/ (- (- (/ 1.0 u) 2.0) (/ 2.0 v)) u)) u) u)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((logf((u - ((-1.0f + u) * expf((-2.0f / v))))) * v) <= -0.75f) {
		tmp = (((-2.0f / v) - ((((1.0f / u) - 2.0f) - (2.0f / v)) / u)) * u) * u;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((log((u - (((-1.0e0) + u) * exp(((-2.0e0) / v))))) * v) <= (-0.75e0)) then
        tmp = ((((-2.0e0) / v) - ((((1.0e0 / u) - 2.0e0) - (2.0e0 / v)) / u)) * u) * u
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(u - Float32(Float32(Float32(-1.0) + u) * exp(Float32(Float32(-2.0) / v))))) * v) <= Float32(-0.75))
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) / v) - Float32(Float32(Float32(Float32(Float32(1.0) / u) - Float32(2.0)) - Float32(Float32(2.0) / v)) / u)) * u) * u);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((log((u - ((single(-1.0) + u) * exp((single(-2.0) / v))))) * v) <= single(-0.75))
		tmp = (((single(-2.0) / v) - ((((single(1.0) / u) - single(2.0)) - (single(2.0) / v)) / u)) * u) * u;
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.75

   1. Initial program 92.9%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower--.f32 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 4.3

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

   5. Applied rewrites 4.3%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 55.8%

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

   10. Applied rewrites 55.8%

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

   if -0.75 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -0.75:\\ \;\;\;\;\left(\left(\frac{-2}{v} - \frac{\left(\frac{1}{u} - 2\right) - \frac{2}{v}}{u}\right) \cdot u\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.0%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 49.4

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

   5. Applied rewrites 49.4%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 93.2%

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

4. Final simplification 90.1%

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

Alternative 7: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u around inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto 1 + v \cdot \log \left(u \cdot \left(\color{blue}{\left(0 - e^{\frac{-2}{v}}\right)} + 1\right)\right) \]

   4. associate-+l- N/A

      \[\leadsto 1 + v \cdot \log \left(u \cdot \color{blue}{\left(0 - \left(e^{\frac{-2}{v}} - 1\right)\right)}\right) \]

   5. neg-sub0 N/A

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

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

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

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

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

   8. neg-mul-1 N/A

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

   9. lower-*.f32 N/A

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

   10. neg-mul-1 N/A

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

   11. lower-neg.f32 N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

   16. lower-expm1.f32 N/A

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

   17. associate-*r/ N/A

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

   18. metadata-eval N/A

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

   19. distribute-neg-frac N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f32 44.2

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

5. Applied rewrites 41.7%

   \[\leadsto 1 + v \cdot \log \color{blue}{\left(\left(-u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 94.0%

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

8. Final simplification 94.0%

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

Alternative 8: 87.2% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 86.8%

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

Alternative 9: 5.8% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.0%

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

Reproduce

herbie shell --seed 2024295 
(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))))))))