HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 9.2s

Alternatives: 11

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 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 90.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float t_0 = u / (((1.0f / u) - 2.0f) - (2.0f / v));
	float tmp;
	if ((logf((u - ((-1.0f + u) * expf((-2.0f / v))))) * v) <= -1.0f) {
		tmp = (u * u) * (((t_0 * -2.0f) - v) / (t_0 * v));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = u / (((1.0e0 / u) - 2.0e0) - (2.0e0 / v))
    if ((log((u - (((-1.0e0) + u) * exp(((-2.0e0) / v))))) * v) <= (-1.0e0)) then
        tmp = (u * u) * (((t_0 * (-2.0e0)) - v) / (t_0 * v))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(u / Float32(Float32(Float32(Float32(1.0) / u) - Float32(2.0)) - Float32(Float32(2.0) / 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(u * u) * Float32(Float32(Float32(t_0 * Float32(-2.0)) - v) / Float32(t_0 * v)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = u / (((single(1.0) / u) - single(2.0)) - (single(2.0) / v));
	tmp = single(0.0);
	if ((log((u - ((single(-1.0) + u) * exp((single(-2.0) / v))))) * v) <= single(-1.0))
		tmp = (u * u) * (((t_0 * single(-2.0)) - v) / (t_0 * v));
	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 95.6%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-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 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in u around -inf

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

   10. Applied rewrites 55.6%

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

   12. Applied rewrites 55.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.2%

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

4. Final simplification 90.8%

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

Alternative 3: 90.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -1.0)
   (* (* (- (/ -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) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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)))))) < -1

   1. Initial program 95.6%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-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 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in u around -inf

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

   10. Applied rewrites 55.6%

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

   12. Applied rewrites 55.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.2%

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

4. Final simplification 90.8%

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

Alternative 4: 90.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -1.0)
   (* (- (/ -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) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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)))))) < -1

   1. Initial program 95.6%

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

   3. Taylor expanded in v around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-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 3.2

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

   5. Applied rewrites 3.2%

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

   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 55.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.2%

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

4. Final simplification 90.8%

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

Alternative 5: 90.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -1.0)
   (+ (/ 2.0 (/ -1.0 (- 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 / (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) / (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) / 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) / (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 95.6%

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

   3. Taylor expanded in v around inf

      \[\leadsto 1 + \color{blue}{-2 \cdot \left(1 - u\right)} \]
   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 49.1

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

   5. Applied rewrites 49.1%

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

   7. Applied rewrites 49.0%

      \[\leadsto 1 + \frac{1}{\frac{u + 1}{1 - u \cdot u}} \cdot -2 \]
   8. Step-by-step derivation

   9. Applied rewrites 49.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.2%

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

4. Final simplification 90.2%

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

Alternative 6: 90.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -1.0)
   (+ (* (* (/ (- 1.0 u) v) -2.0) v) 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 = ((((1.0f - u) / v) * -2.0f) * v) + 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 = ((((1.0e0 - u) / v) * (-2.0e0)) * v) + 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(Float32(Float32(Float32(1.0) - u) / v) * Float32(-2.0)) * v) + 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(1.0) - u) / v) * single(-2.0)) * v) + 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 95.6%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 49.1

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

   5. Applied rewrites 49.1%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.2%

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

4. Final simplification 90.2%

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

Alternative 7: 90.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -1.0)
   (+ (+ (* (/ -2.0 u) u) (* 2.0 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 / u) * u) + (2.0f * 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) / u) * u) + (2.0e0 * 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(Float32(Float32(-2.0) / u) * u) + Float32(Float32(2.0) * 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) / u) * u) + (single(2.0) * 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 95.6%

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

   3. Taylor expanded in v around inf

      \[\leadsto 1 + \color{blue}{-2 \cdot \left(1 - u\right)} \]
   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 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 49.0%

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

   10. Applied rewrites 49.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.2%

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

4. Final simplification 90.2%

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

Alternative 8: 90.3% 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 95.6%

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

   3. Taylor expanded in v around inf

      \[\leadsto 1 + \color{blue}{-2 \cdot \left(1 - u\right)} \]
   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 49.1

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

   5. Applied rewrites 49.1%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.2%

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

4. Final simplification 90.2%

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

Alternative 9: 46.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u}{\left(\frac{1}{u} - 2\right) - \frac{2}{v}}\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \frac{t\_0 \cdot -2 - v}{t\_0 \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (/ u (- (- (/ 1.0 u) 2.0) (/ 2.0 v)))))
   (if (<= v 0.10000000149011612)
     (+ (* (log (fma (exp (/ -2.0 v)) (- 1.0 u) u)) v) 1.0)
     (* (* u u) (/ (- (* t_0 -2.0) v) (* t_0 v))))))

C

float code(float u, float v) {
	float t_0 = u / (((1.0f / u) - 2.0f) - (2.0f / v));
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = (logf(fmaf(expf((-2.0f / v)), (1.0f - u), u)) * v) + 1.0f;
	} else {
		tmp = (u * u) * (((t_0 * -2.0f) - v) / (t_0 * v));
	}
	return tmp;
}

Julia

function code(u, v)
	t_0 = Float32(u / Float32(Float32(Float32(Float32(1.0) / u) - Float32(2.0)) - Float32(Float32(2.0) / v)))
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(log(fma(exp(Float32(Float32(-2.0) / v)), Float32(Float32(1.0) - u), u)) * v) + Float32(1.0));
	else
		tmp = Float32(Float32(u * u) * Float32(Float32(Float32(t_0 * Float32(-2.0)) - v) / Float32(t_0 * v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   4. Applied rewrites 99.8%

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

   if 0.100000001 < v

   1. Initial program 96.0%

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

   3. Taylor expanded in v around inf

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

      1. 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 5.6

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

   5. Applied rewrites 5.6%

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

   7. Applied rewrites 5.6%

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

   8. Taylor expanded in u around -inf

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

   10. Applied rewrites 53.5%

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

   12. Applied rewrites 53.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot u\right) \cdot \frac{\frac{u}{\left(\frac{1}{u} - 2\right) - \frac{2}{v}} \cdot -2 - v}{\frac{u}{\left(\frac{1}{u} - 2\right) - \frac{2}{v}} \cdot v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -1.0) -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 = -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 = -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(-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(-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 95.6%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 43.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 94.2%

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

4. Final simplification 89.7%

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

Alternative 11: 5.7% accurate, 231.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.8%

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

Reproduce

herbie shell --seed 2024254 
(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))))))))