HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 8.6s

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{fma}\left(1, 1 - \frac{2 - \frac{2 - \frac{1.3333333333333333}{v}}{v}}{v}, u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<=
      (* v (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))))
      -1.500000053056283e-6)
   (+
    1.0
    (*
     v
     (log
      (fma
       1.0
       (- 1.0 (/ (- 2.0 (/ (- 2.0 (/ 1.3333333333333333 v)) v)) v))
       u))))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u - ((-1.0f + u) * expf((-2.0f / v)))))) <= -1.500000053056283e-6f) {
		tmp = 1.0f + (v * logf(fmaf(1.0f, (1.0f - ((2.0f - ((2.0f - (1.3333333333333333f / v)) / v)) / v)), u)));
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u - Float32(Float32(Float32(-1.0) + u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.500000053056283e-6))
		tmp = Float32(Float32(1.0) + Float32(v * log(fma(Float32(1.0), Float32(Float32(1.0) - Float32(Float32(Float32(2.0) - Float32(Float32(Float32(2.0) - Float32(Float32(1.3333333333333333) / v)) / v)) / v)), u))));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1.50000005e-6

   1. Initial program 98.3%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. lower--.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 14.2

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

   5. Applied rewrites 14.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 24.8

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

   7. Applied rewrites 24.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 18.2%

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

   if -1.50000005e-6 < (*.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 99.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{fma}\left(1, 1 - \frac{2 - \frac{2 - \frac{1.3333333333333333}{v}}{v}}{v}, u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;1 + \frac{1 - {u}^{3}}{1 + \left(u \cdot u + u\right)} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (- u (* (+ -1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (+ 1.0 (* (/ (- 1.0 (pow u 3.0)) (+ 1.0 (+ (* u u) u))) -2.0))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u - ((-1.0f + u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = 1.0f + (((1.0f - powf(u, 3.0f)) / (1.0f + ((u * u) + u))) * -2.0f);
	} 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 ((v * log((u - (((-1.0e0) + u) * exp(((-2.0e0) / v)))))) <= (-1.0e0)) then
        tmp = 1.0e0 + (((1.0e0 - (u ** 3.0e0)) / (1.0e0 + ((u * u) + u))) * (-2.0e0))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u - Float32(Float32(Float32(-1.0) + u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - (u ^ Float32(3.0))) / Float32(Float32(1.0) + Float32(Float32(u * u) + u))) * Float32(-2.0)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u - ((single(-1.0) + u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = single(1.0) + (((single(1.0) - (u ^ single(3.0))) / (single(1.0) + ((u * u) + u))) * single(-2.0));
	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 94.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   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 91.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \leq -1:\\ \;\;\;\;1 + \frac{1 - {u}^{3}}{1 + \left(u \cdot u + u\right)} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u - ((single(-1.0) + u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = single(1.0) + ((single(1.0) / ((u + single(1.0)) / (single(1.0) - (u * u)))) * single(-2.0));
	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 94.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   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 91.9%

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

4. Final simplification 90.6%

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

Alternative 5: 90.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u - ((-1.0f + u) * expf((-2.0f / v)))))) <= -1.0f) {
		tmp = 1.0f + (((1.0f - (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 ((v * log((u - (((-1.0e0) + u) * exp(((-2.0e0) / v)))))) <= (-1.0e0)) then
        tmp = 1.0e0 + (((1.0e0 - (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(v * log(Float32(u - Float32(Float32(Float32(-1.0) + u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - Float32(u * u)) * Float32(-2.0)) / Float32(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 ((v * log((u - ((single(-1.0) + u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = single(1.0) + (((single(1.0) - (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 94.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   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 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 91.9%

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

4. Final simplification 90.6%

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

Alternative 6: 90.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u - ((single(-1.0) + u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = single(1.0) + (v * (((single(1.0) - u) / v) * single(-2.0)));
	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 94.0%

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

   3. Taylor expanded in v around inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-2 \cdot \frac{1 - u}{v}\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   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 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 91.9%

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

4. Final simplification 90.6%

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

Alternative 7: 90.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u - ((single(-1.0) + u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	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 94.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   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 91.9%

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

4. Final simplification 90.6%

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

Alternative 8: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + (-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 + (-u * exp(((-2.0e0) / v))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f32 N/A

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

   4. lower-neg.f32 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-exp.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 95.5

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

5. Applied rewrites 95.5%

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

Alternative 9: 42.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

5. Applied rewrites 37.8%

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

6. Taylor expanded in u around inf

   \[\leadsto 1 + v \cdot \log \color{blue}{\left(u \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)\right)} \]
7. 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. distribute-lft-in N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. associate-*r* N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. associate-*r/ N/A

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

   15. lower-exp.f32 N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 94.5

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

8. Applied rewrites 94.5%

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

Alternative 10: 87.0% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 87.4%

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

Alternative 11: 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 99.6%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.8%

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

Reproduce

herbie shell --seed 2024314 
(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))))))))